| problem_id,problem_statement,final_answer,difficulty_tier |
| 8afd11cf5203,"4. Let set $A=\left\{(x, y) \mid y=x^{2}\right\}, B=$ $\left\{(x, y) \mid x^{2}+(y-m)^{2}=1\right\}$. If $A \cap B \neq \varnothing$, then the range of values for $m$ is $\qquad$.",See reasoning trace,medium |
| 13fc6d39724f,"[Ratio of areas of triangles with a common angle] Auxiliary similar triangles $\quad]$ |
| |
| In triangle $A B C$, angle $C$ is $30^{\circ}$, and angle $A$ is acute. A line perpendicular to side $B C$ is drawn, cutting off triangle $C N M$ from triangle $A B C$ (point $N$ lies between vertices $B$ and $C$). The areas of triangles $C N M$ and $A B C$ are in the ratio $3: 16$. Segment $M N$ is half the height $B H$ of triangle $A B C$. Find the ratio $A H: H C$.","In triangle $A B C$, a line is drawn from point $E$ on side $B C$, parallel to the altitude $B D$, and intersects side $A C$ at point $F$",medium |
| 1f6244bae244,"25. Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it. |
| How long, in seconds, would it take Thomas to walk down the broken escalator? |
| A 30 |
| B 40 |
| C 45 |
| D 50 |
| E 60",See reasoning trace,medium |
| 6517c26296eb,"2. Find the equation of the ellipse centered at the origin, with foci on the y-axis, eccentricity equal to 0.8, and focal distance of 4.",1$.,easy |
| 3a577c10e5c2,7.051. $\frac{\lg 8-\lg (x-5)}{\lg \sqrt{x+7}-\lg 2}=-1$.,29,medium |
| 3f2ebe3cc970,"3. Given the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ with its right focus at $F$, the upper vertex at $A$, and $P$ as a moving point on the ellipse in the first quadrant. Then the maximum value of the area of $\triangle A P F$ is $\qquad$ .",See reasoning trace,medium |
| ef2236264b85,"Which pair of numbers does NOT have a product equal to $36$? |
| $\text{(A)}\ \{-4,-9\}\qquad\text{(B)}\ \{-3,-12\}\qquad\text{(C)}\ \left\{\frac{1}{2},-72\right\}\qquad\text{(D)}\ \{ 1,36\}\qquad\text{(E)}\ \left\{\frac{3}{2},24\right\}$",C,easy |
| 50c5cfb81fdb,"14. B. Color the five sides and five diagonals of the convex pentagon $A B C D E$, such that any two segments sharing a common vertex are of different colors. Find the minimum number of colors needed.",See reasoning trace,medium |
| b254a2b32834,"3. In a regular triangular prism $A B C-A_{1} B_{1} C_{1}$, $D, E$ are points on the side edges $B B_{1}, C C_{1}$ respectively, $E C=B C=2 B D$, then the size of the dihedral angle formed by the section $A D E$ and the base $A B C$ is $\qquad$ .","BC = FB$, thus $\angle CAF = 90^{\circ} \Rightarrow \angle CAE$ is the plane angle of the dihedral a",easy |
| 465ee077db1d,"7. $[7]$ A student at Harvard named Kevin |
| Was counting his stones by 11 |
| He messed up $n$ times |
| And instead counted $9 \mathrm{~s}$ |
| And wound up at 2007. |
| How many values of $n$ could make this limerick true?",21,easy |
| f2882c092373,"4. At a table, there are four people: Luca, Maria, Nicola, and Paola. Each of the four always lies or never lies. Moreover, they do not like to talk about themselves, but rather about their friends; so when asked who among them always lies, their answers are: |
| |
| Luca: ""every girl is always honest"" |
| |
| Maria: ""every boy is always a liar"" |
| |
| Nicola: ""there is a girl who always lies, the other is always honest"" |
| |
| Paola: ""one of the boys is always honest, the other never is"". |
| |
| Could you tell how many at the table are always honest? |
| (A) None |
| (B) 1 |
| (C) 2 |
| (D) 3 |
| (E) all.",$\mathbf{( C )}$,medium |
| 20c4f84b1618,"Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.",31,medium |
| a6949db67c30,"10. Given the function $f(x)=4^{x}-3 \cdot 2^{x}+3$ has a range of $[1,7]$, the domain of $x$ is $\qquad$ .",See reasoning trace,easy |
| 3ea00889c0c0,"Find all functions $f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}$ such that $$f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right) |
| = xyf(\frac{1}{x})f(\frac{1}{y}),$$ for all $x, y \in \Bbb{Q}_{>0,}$ where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b.$",f(x) = \text{numerator of,medium |
| 49ccb90d48dd,3. Find the minimum value of the function $f(x)=\sqrt{\frac{1}{4} x^{4}-2 x^{2}-4 x+13}+\frac{1}{2} x^{2}+\frac{1}{2}$.,"2 x$, and point $M(3,2)$ is inside this parabola, while point $N\left(\frac{1}{2}, 0\right)$ is exac",medium |
| c95548e4d9c5,"G2 Let $A D, B F$ and $C E$ be the altitudes of $\triangle A B C$. A line passing through $D$ and parallel to $A B$ intersects the line $E F$ at the point $G$. If $H$ is the orthocenter of $\triangle A B C$, find the angle $\widehat{C G H}$.",See reasoning trace,medium |
| d92e5a5f6202,"19・16 Let the sum of the interior angles of polygon $P$ be $S$. It is known that each interior angle is $7 \frac{1}{2}$ times its corresponding exterior angle, then |
| (A) $S=2660^{\circ}$, and $P$ could be a regular polygon. |
| (B) $S=2660^{\circ}$, and $P$ is not a regular polygon. |
| (C) $S=2700^{\circ}$, and $P$ is a regular polygon. |
| (D) $S=2700^{\circ}$, and $P$ is not a regular polygon. |
| (E) $S=2700^{\circ}$, and $P$ could be a regular polygon or not a regular polygon. |
| (11th American High School Mathematics Examination, 1960)",$(E)$,easy |
| f40dd1feae96,"5. On the cells of a chessboard of size $8 \times 8$, 5 identical pieces are randomly placed. Find the probability that four of them will be located either on the same row, or on the same column, or on one of the two main diagonals.",$P(A)=\frac{18 \cdot\left(C_{8}^{4} \cdot C_{56}^{1}+C_{8}^{5}\right)}{C_{64}^{5}}=\frac{18 \cdot 56 \cdot 71}{31 \cdot 61 \cdot 63 \cdot 64}=\frac{71}{4 \cdot 31 \cdot 61}=\frac{71}{7564} \approx 0,easy |
| 62369f6dfa5e,"Example 7. In a sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. How many terms can such a sequence have at most? |
| |
| 保留源文本的换行和格式，翻译结果如下： |
| |
| Example 7. In a sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. How many terms can such a sequence have at most?",16$.,medium |
| 8503df10bff4,"$$ |
| \begin{array}{l} |
| \text { 10. Given } f(x)=\left\{\begin{array}{ll} |
| -2 x, & x<0 ; \\ |
| x^{2}-1, & x \geqslant 0, |
| \end{array}\right. \text { the equation } \\ |
| f(x)+2 \sqrt{1-x^{2}}+\left|f(x)-2 \sqrt{1-x^{2}}\right|-2 a x-4=0 |
| \end{array} |
| $$ |
| |
| has three real roots $x_{1}<x_{2}<x_{3}$. If $x_{3}-x_{2}=2\left(x_{2}-x_{1}\right)$, then the real number $a=$ $\qquad$",\frac{12 a}{a^{2}+4} \Rightarrow a=\frac{\sqrt{17}-3}{2}$.,medium |
| 624f3d1a7d54,"4. In the acute triangle $\triangle A B C$, $2 \angle B=\angle C, A B: A C$ the range of values ( ). |
| A. $(0,2)$ |
| B. $(0, \sqrt{2})$ |
| C. $(\sqrt{2}, 2)$ |
| D. $(\sqrt{2}, \sqrt{3})$","2 R \sin \angle C, A C=2 R \sin \angle B$, then $A B: A C=\frac{\sin \angle C}{\sin \angle B}=2 \cos",easy |
| 8b8a7096cd19,"Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$",177,medium |
| 301cc2d6346d,"2. Among all quadruples of natural numbers $(k, l, m, n), k>l>m>n$, find the one such that the sum $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}+\frac{1}{n}$ is less than one and as close to it as possible.","$(2,3,7,43)$",medium |
| ffb5f8a9983c,"For any 4 distinct points $P_{1}, P_{2}, P_{3}, P_{4}$ in the plane, find the minimum value of the ratio $\frac{\sum_{1 \leq i<j \leq 4} P_{i} P_{j}}{\min _{1 \leq i<j \leq 4} P_{i} P_{j}}$.",See reasoning trace,medium |
| c9980b49c691,"4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$.",See reasoning trace,easy |
| 42c7d9ccb1cb,10.47 Given complex numbers $z_{1}=1+a i$ and $z_{2}=2^{\frac{3}{4}}\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right)$. Find all real values of $a$ for which $z_{1}^{3}=z_{2}^{2}$.,1$.,medium |
| 2d22ccac9f76,"There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white. |
| These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color? |
| |
| $ \textbf{(A)}\ 1006 \qquad \textbf{(B)}\ 1204 \qquad \textbf{(C)}\ 1340 \qquad \textbf{(D)}\ 2011 \qquad \textbf{(E)}\ \text{None}$",1006,medium |
| d05b2e463bae,"17. Solve the congruence equations using methods that both utilize and do not utilize primitive roots: |
| (i) $x^{4} \equiv 41(\bmod 37) ;$ |
| (ii) $x^{4} \equiv 37(\bmod 41)$.",See reasoning trace,easy |
| de789876f787,"4. Antonio, Beppe, Carlo, and Duccio randomly distribute the 40 cards of a deck, 10 to each. Antonio has the ace, two, and three of coins. Beppe has the ace of swords and the ace of clubs. Carlo has the ace of cups. Who is more likely to have the seven of coins? |
| (A) Antonio |
| (B) Beppe |
| (C) Carlo |
| (D) Duccio |
| (E) two or more players have the same probability of having it.",(D),medium |
| 8909b4a2da34,"5. Represent as an algebraic expression (simplify) the sum $8+88+888+8888+\ldots+8 \ldots 8$, if the last term in its notation contains p times the digit eight. |
| |
| #",- 0 points,medium |
| 35436ff5dc42,"$[$ [Arithmetic. Mental calculation, etc.] $]$ |
| |
| Authors: Gaityerri G.A., Grierenko D.: |
| |
| 2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 and 9999 years)?",109 years,medium |
| 7dee83c6ab97,"How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$ |
| $$ |
| \mathrm a. ~ 180\qquad \mathrm b.~184\qquad \mathrm c. ~186 \qquad \mathrm d. ~189 \qquad \mathrm e. ~191 |
| $$",180,medium |
| b849e799cb63,"If $p$ and $q$ are positive integers, $\max (p, q)$ is the maximum of $p$ and $q$ and $\min (p, q)$ is the minimum of $p$ and $q$. For example, $\max (30,40)=40$ and $\min (30,40)=30$. Also, $\max (30,30)=30$ and $\min (30,30)=30$. |
| |
| Determine the number of ordered pairs $(x, y)$ that satisfy the equation |
| |
| $$ |
| \max (60, \min (x, y))=\min (\max (60, x), y) |
| $$ |
| |
| where $x$ and $y$ are positive integers with $x \leq 100$ and $y \leq 100$. |
| |
| ## PART B |
| |
| For each question in Part B, your solution must be well organized and contain words of explanation or justification when appropriate. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.",See reasoning trace,medium |
| 6fd59e2ad0b0,"4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \perp A M$, then $\angle B A C$ equals ( ). |
| (A) $30^{\circ}$ |
| (B) $45^{\circ}$ |
| (C) $60^{\circ}$ |
| (D) $90^{\circ}$",See reasoning trace,easy |
| 7e70a969531a,"5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$.",$A C=13$,medium |
| 3bb26201def1,"19. (GBR 6) The $n$ points $P_{1}, P_{2}, \ldots, P_{n}$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $d_{n}$ between any two of these points has its largest possible value $D_{n}$. Calculate $D_{n}$ for $n=2$ to 7 and justify your answer.",7$ we have $D_{7} \leq D_{6}=1$. This value is attained if six of the seven points form a regular he,medium |
| dd4431aa4ce8,"9. The sequence $\left\{a_{n}\right\}$ satisfies $(n-1) a_{n+1}=(n+1) a_{n}-2(n-1), n=$ $1,2, \cdots$ and $a_{100}=10098$. Find the general term formula of the sequence $\left\{a_{n}\right\}$.","\frac{a_{n}}{n(n-1)}-\frac{2}{(n+1) n}$, i.e., $\frac{a_{n+1}}{(n+1) n}-\frac{a_{n}}{n(n-1)}=\frac{2",medium |
| ce9bdb58abda,"[Example 5.2.2] Solve the equation: |
| $$ |
| x^{4}-x^{2}+8 x-16=0 . |
| $$",See reasoning trace,medium |
| e496c81837e2,"If $f(x)$ is a linear function with $f(k)=4, f(f(k))=7$, and $f(f(f(k)))=19$, what is the value of $k$ ?",$\frac{13}{4}$,easy |
| 29cb1c59aaa1,"Evdokimov M.A. |
| |
| Dominoes $1 \times 2$ are placed without overlapping on a chessboard $8 \times 8$. In this case, dominoes can extend beyond the board's edges, but the center of each domino must lie strictly inside the board (not on the edge). Place |
| |
| a) at least 40 dominoes; |
| |
| b) at least 41 dominoes; |
| |
| c) more than 41 dominoes.",See reasoning trace,medium |
| 50cb1ae2687d,"The diameter of the base circle of a straight cone and its slant height are both $20 \mathrm{~cm}$. What is the maximum length of a strip that can be attached to the cone's lateral surface without bending, breaking (cutting), or overlapping, if the strip's width is $2 \mathrm{~cm}$?",523&width=485&top_left_y=519&top_left_x=798),medium |
| eadb8d70e5dd,"【Question 5】 |
| Two line segments are parallel, forming a pair of parallel line segments. As shown in the figure, among the 12 edges of a rectangular prism, there are $\qquad$ pairs of parallel line segments.","\frac{4 \times 3}{2 \times 1}=6$ pairs of parallel line segments; similarly, the 4 widths of the rec",easy |
| f203aa36851f,690. Find the divisibility rule for 2 in a number system with any odd base.,A number is divisible by 2 if and only if the sum of its digits is divisible by 2,medium |
| af70ec604eb0,"7. Given positive integers $a, b$ satisfy |
| $$ |
| \sqrt{\frac{a b}{2 b^{2}-a}}=\frac{a+2 b}{4 b} \text {. } |
| $$ |
| |
| Then $|10(a-5)(b-15)|+2=$",2012$.,medium |
| 645fe323bd97,"Task B-1.5. Marko and his ""band"" set off on a tour. On the first day, they headed east, on the second day, they continued north, on the third day, they continued west, on the fourth day, they headed south, on the fifth day, they headed east, and so on. If on the n-th day of the tour they walked $\frac{n^{2}}{2}$ kilometers, how many km were they from the starting point at the end of the fortieth day?",See reasoning trace,medium |
| 57fc549e100d,"17. For which positive integers $n$ is it true that |
| $$1^{2}+2^{2}+3^{2}+\cdots+(n-1)^{2} \equiv 0(\bmod n) ?$$",See reasoning trace,easy |
| 3914524a86b5,"24. Given $a \in \mathbf{Z}_{+}$. The area of the quadrilateral enclosed by the curves |
| $$ |
| (x+a y)^{2}=4 a^{2} \text { and }(a x-y)^{2}=a^{2} |
| $$ |
| |
| is ( ). |
| (A) $\frac{8 a^{2}}{(a+1)^{2}}$ |
| (B) $\frac{4 a}{a+1}$ |
| (C) $\frac{8 a}{a+1}$ |
| (D) $\frac{8 a^{2}}{a^{2}+1}$ |
| (E) $\frac{8 a}{a^{2}+1}$",See reasoning trace,medium |
| 2900c0494d8e,"\section*{ |
| |
| A spherical cap is cut from a sphere of radius \(r\), consisting of a cone of height \(h\) and the associated spherical segment. |
| |
| a) What is the length \(h\) of the height of the cone if the area of the cut spherical cap is equal to one third of the surface area of the sphere? |
| |
| b) What is the length \(h\) of the height of the cone if the volume of the spherical sector is equal to one third of the volume of the sphere?",V_{\text {sector }} \rightarrow \frac{2}{3} \pi \cdot r^{2} \cdot(r-h)=\frac{4}{9} \pi \cdot r^{3}\),medium |
| c8e760cc87b9,"6. (10 points) Arrange 8 numbers in a row from left to right. Starting from the third number, each number is exactly the sum of the two numbers before it. If the 5th number and the 8th number are 53 and 225, respectively, then the 1st number is $\qquad$ .",: 7,easy |
| ad37c2cacd6e,Tadeo draws the rectangle with the largest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$ and the rectangle with the smallest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$. What is the difference between the perimeters of the rectangles Tadeo drew?,3840,medium |
| 7f4261efe36b,"G7.1 Figure 1 shows a cone and a hemisphere. |
| G7.2 What is the volume of the hemisphere shown in figure 1 ? Give your answer in terms of $\pi$. |
| G7.3 In figure 2, a right circular cone stands inside a right cylinder of same base radius $r$ and height $h$. Express the volume of the space between them in terms of $r$ and $h$. |
| |
| G7.4 Find the ratio of the volume of the cylinder to that of the cone.",See reasoning trace,easy |
| e01a49a1c52c,"$11 、$ Satisfy all 7 conditions below, the five different natural numbers $\mathrm{A} 、 \mathrm{~B} 、 \mathrm{C} 、 \mathrm{D} 、 \mathrm{E}$ are $\mathrm{A}=(\quad), \mathrm{B}=(\quad)$, $\mathrm{C}=(\quad), \mathrm{D}=(\quad), \mathrm{E}=(\quad)$ respectively. What are they? |
| （1）These numbers are all less than 10; |
| (2) A is greater than 5; |
| (3) $\mathrm{A}$ is a multiple of $\mathrm{B}$; |
| (4) $\mathrm{C}$ plus $\mathrm{A}$ equals $\mathrm{D}$; |
| (5) The sum of $B 、 C 、 E$ equals $A$; |
| (6) The sum of B and $\mathrm{C}$ is less than $\mathrm{E}$; |
| (7) The sum of $\mathrm{C}$ and $\mathrm{E}$ is less than $\mathrm{B}+5$.","7$, at this point (3) is not satisfied, so $\mathrm{E}$ is not 4. If $\mathrm{E}=6$, from (5) we get",medium |
| fab9469fa6e8,"Alphonso and Karen started out with the same number of apples. Karen gave twelve of her apples to Alphonso. Next, Karen gave half of her remaining apples to Alphonso. If Alphonso now has four times as many apples as Karen, how many apples does Karen now have? |
| (A) 12 |
| (B) 24 |
| (C) 36 |
| (D) 48 |
| (E) 72",(B),medium |
| ae9512a2bf97,"4. Solve the system of equations $\left\{\begin{array}{l}2 \cos ^{2} x+2 \sqrt{2} \cos x \cos ^{2} 4 x+\cos ^{2} 4 x=0, \\ \sin x=\cos y .\end{array}\right.$. |
| |
| #","$\left(\frac{3 \pi}{4}+2 \pi k, \pm \frac{\pi}{4}+2 \pi n\right),\left(-\frac{3 \pi}{4}+2 \pi k, \pm \frac{3 \pi}{4}+2 \pi n\right), \quad k, n \in Z$",medium |
| 2f1c26470094,"Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root?",0,medium |
| 8a8887b414c1,"10. Given a sequence $\left\{a_{n}\right\}$ whose terms are all non-negative real numbers, and satisfies: for any integer $n \geqslant 2$, we have $a_{n+1}=$ $a_{n}-a_{n-1}+n$. If $a_{2} a_{2022}=1$, find the maximum possible value of $a_{1}$.",\frac{4051}{2025}$.,medium |
| 4fc808da26ab,Example 1-19 Select 3 numbers from 1 to 300 so that their sum is exactly divisible by 3. How many schemes are there?,See reasoning trace,medium |
| bec6d841e62e,"Let the curve $ C: y \equal{} |x^2 \plus{} 2x \minus{} 3|$ and the line $ l$ passing through the point $ ( \minus{} 3,\ 0)$ with a slope $ m$ in $ x\minus{}y$ plane. Suppose that $ C$ intersects to $ l$ at distinct two points other than the point $ ( \minus{} 3,\ 0)$, find the value of $ m$ for which the area of the figure bounded by $ C$ and $ l$ is minimized.",12 - 8\sqrt{2,medium |
| dc42fa4ab2f0,"2. (15 points) A wooden cube with edge $\ell=30$ cm floats in a lake. The density of wood $\quad \rho=750 \mathrm{kg} / \mathrm{m}^{3}, \quad$ the density of water $\rho_{0}=1000 \mathrm{kg} / \mathrm{m}^{3}$. What is the minimum work required to completely pull the cube out of the water?",See reasoning trace,medium |
| 34e4cb26cbfd,"Example 4 Given a unit cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$, $E$ is a moving point on $B C$, and $F$ is the midpoint of $A B$. Determine the position of point $E$ such that $C_{1} F \perp A_{1} E$.","\frac{1}{2} \lambda$, i.e., $a=\frac{1}{2}$. This shows that $E$ is the midpoint of $B C$. At this p",medium |
| 6fc196d65b51,"4. [6] Let $A B C D$ be a square of side length 2. Let points $X, Y$, and $Z$ be constructed inside $A B C D$ such that $A B X, B C Y$, and $C D Z$ are equilateral triangles. Let point $W$ be outside $A B C D$ such that triangle $D A W$ is equilateral. Let the area of quadrilateral $W X Y Z$ be $a+\sqrt{b}$, where $a$ and $b$ are integers. Find $a+b$.",\sqrt{12}-2$.,easy |
| c3601e35ca49,"4. (7 points) On the board, 45 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 45 minutes?",990,medium |
| c18da74e439d,"For what $n$ can the following system of inequalities be solved? |
| |
| $$ |
| 1<x<2 ; \quad 2<x^{2}<3 ; \quad \ldots, \quad n<x^{n}<n+1 |
| $$",See reasoning trace,medium |
| b4313b6d3611,"6. A blacksmith is building a horizontal iron fence consisting of many vertical bars, parallel to each other, each of which is positioned $18 \mathrm{~cm}$ apart from the two adjacent ones. The blacksmith connects the ends of each pair of adjacent bars with a curved bar in the shape of an arc of a circle, placed in the plane of the bars, the highest point of which is $3 \sqrt{3} \mathrm{~cm}$ from the line (dashed in the figure) that passes through the upper ends of all the bars, and is perpendicular to the bars themselves. How long is each of the bars used to build the arcs? |
| |
|  |
| (A) $8 \pi(\sqrt{3}-1) \mathrm{cm}$ |
| (B) $6 \pi \sqrt{3} \mathrm{~cm}$ |
| (C) $12 \pi(\sqrt{3}-1) \mathrm{cm}$ |
| (D) $4 \pi \sqrt{3} \mathrm{~cm}$ |
| (E) $8 \pi \sqrt{3} \mathrm{~cm}$.",(D),medium |
| e13b85e86fd0,"7. Tangents are drawn from point $A$ to a circle with a radius of 10 cm, touching the circle at points $B$ and $C$ such that triangle $A B C$ is equilateral. Find its area.",$S_{\triangle A B C}=75 \sqrt{3}$,easy |
| c992b5097156,"11. Prime numbers $p$, $q$, $r$ satisfy $p+q=r$, and $(r-p) \cdot$ $(q-p)-27 p$ is a perfect square. Then all the triples $(p, q, r)=$ $\qquad$","(2,29,31)$.",easy |
| 4ab9de2c74d1,118. Find the remainders of $a^{4}$ divided by 5 for all possible integer values of $a$.,See reasoning trace,easy |
| 712487cf2704,"For $k=1,2,\dots$, let $f_k$ be the number of times |
| \[\sin\left(\frac{k\pi x}{2}\right)\] |
| attains its maximum value on the interval $x\in[0,1]$. Compute |
| \[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]",\frac{1,medium |
| 6d2f94ad4bca,"Five integers form an arithmetic sequence. Whether we take the sum of the cubes of the first four terms, or the sum of the cubes of the last four terms, in both cases we get 16 times the square of the sum of the considered terms. Determine the numbers.","32 / 3, d= \pm 16$ that arose during the solution do not only consist of non-integers, but also do n",medium |
| f2cac2e3d3b1,"$A B C$ triangle, perpendiculars are raised at $A$ and $B$ on side $AB$, which intersect $BC$ and $AC$ at $B_{1}$ and $A_{1}$, respectively. What is the area of the triangle if $AB=c$, $AB_{1}=m$, and $BA_{1}=n$?",See reasoning trace,easy |
| dc266fd3cabf,"## 202. Math Puzzle $3 / 82$ |
| |
| On one day, a cooperative delivered 1680 kilograms of milk to the dairy. The cream obtained from this amounted to $1 / 8$ of this quantity. The butter produced amounted to $1 / 3$ of the cream quantity. How much cream and butter were produced? |
| |
| How many kilograms of butter did the dairy workers produce the next day when they obtained 240 kilograms of cream? How many kilograms of milk did the cooperative farmers deliver $\mathrm{ab}$ ?","1920$, i.e., $1920 \mathrm{~kg}$ of milk were delivered.",easy |
| 1dc4d9c43361,"11. The blackboard is written with 1989 consecutive natural numbers $1, 2, 3, \ldots, 1989$. First, perform the following transformation: erase any two numbers on the blackboard, and add the remainder obtained by dividing the sum of the erased two numbers by 19. After several transformations, there are two numbers left on the blackboard, one of which is 89, and the other is a single-digit number. This single-digit number is ( ).","104160 \ldots \ldots .15$; $89 \div 19=4 \ldots \ldots 13$; thus, the remainder when the remaining n",easy |
| 5eae532847e6,"5. The diagonals of quadrilateral $A B C D$ intersect at point $O$. It is known that $A B=B C=$ $=C D, A O=8$ and $\angle B O C=120^{\circ}$. What is $D O ?$",$D O=8$,medium |
| 3d33d7112c47,10. $\sin 20^{\circ} \cdot \sin 40^{\circ} \cdot \sin 80^{\circ}=$,4\left(\cos 20^{\circ}-\cos 60^{\circ}\right) \sin 80^{\circ} \\ =4 \sin 80^{\circ} \cdot \cos 20^{\,easy |
| 77e434ad60f1,"21.3.9 $\star \star$ Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of $1,2, \cdots, n$, satisfying $a_{i} \neq i(i=1$, $2, \cdots, n)$, there are $D_{n}$ such groups of $a_{1}, a_{2}, \cdots, a_{n}$, find $D_{n}$.",See reasoning trace,medium |
| ccc233d35cc5,"The equation $\sqrt {x + 10} - \frac {6}{\sqrt {x + 10}} = 5$ has: |
| $\textbf{(A)}\ \text{an extraneous root between } - 5\text{ and } - 1 \\ \textbf{(B)}\ \text{an extraneous root between }-10\text{ and }-6\\ \textbf{(C)}\ \text{a true root between }20\text{ and }25\qquad \textbf{(D)}\ \text{two true roots}\\ \textbf{(E)}\ \text{two extraneous roots}$",\textbf{(B),easy |
| d544b26d0e2b,"4. In the Cartesian coordinate system $x O y$, let $F_{1}, F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, respectively. $P$ is a point on the right branch of the hyperbola, $M$ is the midpoint of $P F_{2}$, and $O M \perp P F_{2}, 3 P F_{1}=4 P F_{2}$. Then the eccentricity of the hyperbola is $\qquad$.",See reasoning trace,easy |
| 40663be9b159,"Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is $6$ more than the number of balls from the white one or the number of balls from the white one is $16$ more than the number of balls from the red one. With that given information find all possible values of $n$",n = 11m,medium |
| 9750c173300a,"906. Compute surface integrals of the first kind (over the area of the surface): |
| 1) $I=\iint_{\sigma}(6 x+4 y+3 z) d s$, where $\sigma$ is the part of the plane $x+2 y+$ $+3 z=6$ located in the first octant. |
| 2) $K=\iint_{W}\left(y+z+V \sqrt{a^{2}-x^{2}}\right) d s$, where $W-$ is the surface of the cylinder $x^{2}+y^{2}=a^{2}$, bounded by the planes $z=0$ and $z=h$.",See reasoning trace,medium |
| 28cf684eb6b6,"# Task 5. Maximum 15 points |
| |
| In the treasury of the Magic Kingdom, they would like to replace all old banknotes with new ones. There are a total of 3,628,800 old banknotes in the treasury. Unfortunately, the machine that prints new banknotes requires major repairs and each day it can produce fewer banknotes: on the first day, it can only produce half of the banknotes that need to be replaced; on the second day, only a third of the remaining old banknotes in the treasury; on the third day, only a quarter, and so on. Each run of the machine in any state costs the treasury 90,000 monetary units (m.u.), and major repairs will cost 800,000 m.u. After major repairs, the machine can produce no more than one million banknotes per day. The kingdom has allocated no more than 1 million m.u. for the renewal of banknotes in the treasury. |
| |
| (a) After how many days will 80% of the old banknotes be replaced? |
| |
| (b) Will the kingdom be able to replace all the old banknotes in the treasury?",See reasoning trace,medium |
| 4fa28b5860cd,"Tairova |
| |
| A father and son are skating in a circle. From time to time, the father overtakes the son. After the son changed the direction of his movement to the opposite, they started meeting 5 times more frequently. How many times faster does the father skate compared to the son?",1,easy |
| 8d2b47a36805,【Question 1】Calculate: $2 \times(999999+5 \times 379 \times 4789)=$,.,easy |
| 258e9ab8e6b8,"5. (8 points) On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$ on the circle, points $B$ and $C$ are chosen such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is $12$, and $\cos \alpha = \frac{3}{4}$?",18,medium |
| 759ebe3916d4,"## |
| |
| Calculate the limit of the numerical sequence: |
| |
| $\lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{3}-7}+\sqrt[3]{n^{2}+4}}{\sqrt[4]{n^{5}+5}+\sqrt{n}}$",See reasoning trace,medium |
| cbf4a51e849d,"On the sides $A B$ and $B C$ of an equilateral triangle $A B C$, two points $D$ and $E$ are fixed, respectively, such that $\overline{A D}=\overline{B E}$. |
| |
|  |
| |
| If the segments $A E$ and $C D$ intersect at point $P$, determine $\measuredangle A P C$. |
| |
| #","60^{\circ}-\measuredangle D A P=60^{\circ}-\measuredangle P C A$. That is, $\angle P A C+\measuredan",medium |
| 4f610f4e94da,"2. Given that $a$ is an integer, $14 a^{2}-12 a-27 \mid$ is a prime number. Then the sum of all possible values of $a$ is ( ). |
| (A) 3 |
| (B) 4 |
| (C) 5 |
| (D) 6",See reasoning trace,easy |
| 4ffb971ff251,"14. The graph of $\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right)^{2}$ meets the $x$-axis in $p$ different places and meets the $y$-axis in $q$ different places. |
| What is the value of $100 p+100 q$ ?",400$.,medium |
| 6643c4824de8,"Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of |
| \[AB^2 + 2AC^2 - 3AD^2.\]",6,medium |
| 42de35e92fb3,"Example 1 Let $a, b, c, d \geqslant 0$, satisfying $\sum a=32$. Find the maximum and minimum values of the function |
| $$ |
| f(a, b, c, d)=\sum \sqrt{5 a+9} |
| $$",See reasoning trace,medium |
| 998f22fdad35,"7. How many four-digit numbers $\overline{a b c d}$ are there such that the three-digit number $\overline{a b c}$ is divisible by 4 and the three-digit number $\overline{b c d}$ is divisible by 3? |
| |
| The use of a pocket calculator or any reference materials is not allowed. |
| |
| ## Ministry of Science and Education of the Republic of Croatia |
| |
| Agency for Education and Upbringing |
| |
| Croatian Mathematical Society |
| |
| ## SCHOOL/CITY COMPETITION IN MATHEMATICS January 26, 2023. 7th grade - elementary school |
| |
| In addition to the final result, the process is also graded. To receive full credit, all solutions must be found and it must be confirmed that there are no others, the process must be written down, and the conclusions must be justified. |
| |
| ##",See reasoning trace,medium |
| 9a47b6bd42fa,"Kathy owns more cats than Alice and more dogs than Bruce. Alice owns more dogs than Kathy and fewer cats than Bruce. Which of the statements must be true? |
| |
| (A) Bruce owns the fewest cats. |
| |
| (B) Bruce owns the most cats. |
| |
| (C) Kathy owns the most cats. |
| |
| (D) Alice owns the most dogs. |
| |
| (E) Kathy owns the fewest dogs.",(A) is not true,medium |
| 469ea3b276e0,"$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$?",2019,medium |
| a45882656da2,"2. Given the function $y=\frac{a-x}{x-a-1}$, the graph of its inverse function is symmetric about the point $(-1,4)$. Then the value of the real number $a$ is $\qquad$ .","4$, which means $a=3$.",easy |
| ff23113ac6e9,223. Calculate the area of the figure bounded by the lines $y=$ $=2 x-x^{2}$ and $y=0$ (Fig. 143).,See reasoning trace,medium |
| 29a325f269f9,5. How many diagonals in a regular 32-sided polygon are not parallel to any of its sides,240,easy |
| a27aff699713,"A merchant sold goods for 39 K. How much did the goods cost, if the profit margin was the same percentage as the cost of the goods? |
| |
| |
| |
| |
| |
| Translate the above text back into Hungarian, please keep the original text's line breaks and format, and output the translation result directly. |
| preneurial context. |
| A kereskedő 39 K-ért adott el árut. Mennyiért vette az árút, ha ugyanannyi százalékot nyert, mint a mennyibe az áru került?",See reasoning trace,easy |
| 42f64be14af4,"In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2.$ Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}.$ What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$ |
| |
| $\textbf{(A) } \frac{4}{9} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{5}{9} \qquad \textbf{(D) } \frac{3}{5} \qquad \textbf{(E) } \frac{2}{3}$",\textbf{(A),easy |
| 923bf1f5dc7f,"Fibonacci numbers Euclidean algorithm |
| |
| For each natural $n$, provide an example of a rectangle that would be cut into exactly $n$ squares, among which there should be no more than two identical ones.","For example, a rectangle $F_{n} \times F_{n+1}$",easy |
| ea26da0a1ecc,"Let $\{a_n\}$ be a sequence of integers satisfying the following conditions. |
| [list] |
| [*] $a_1=2021^{2021}$ |
| [*] $0 \le a_k < k$ for all integers $k \ge 2$ |
| [*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. |
| [/list] |
| Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.",0,medium |
| 92b2ad9fb932,"## |
| |
| Calculate the limit of the function: |
| |
| $\lim _{x \rightarrow 0} \frac{\arcsin 2 x}{\sin 3(x+\pi)}$",See reasoning trace,medium |
| 2fd62733fe8c,"## |
| |
| Calculate the limit of the function: |
| |
| $\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{\sqrt[4]{x}-1}$",See reasoning trace,medium |
| 2ea3b6f30064,"9. Let set $A=\left\{x \mid x^{2}+x-6=0\right\}, B=\{x \mid m x+1=0\}$, then a sufficient but not necessary condition for $B \varsubsetneqq A$ is $\qquad$ .",-\frac{1}{2}$ (or $m=-\frac{1}{3}$ ).,easy |
| 859168f94d67,"6. The volume of the solid of revolution obtained by rotating the figure bounded by the curves $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ around the $y$-axis is $V_{1}$: The volume of the solid of revolution obtained by rotating the figure composed of points $(x, y)$ that satisfy $x^{2} y \leqslant 16 \cdot x^{2}+(y-2)^{2} \geqslant 4 \cdot x^{2}+(y+2)^{2} \geqslant 4$ around the $y$-axis is $V_{2}$, then |
| A. $V_{1}=\frac{1}{2} V_{2}$ |
| B. $V_{1}=\frac{2}{3} V_{2}$ |
| C. $V_{1}=V_{2}$ |
| D. $V_{1}=2 V_{2}$",V_{2}$.,medium |
| 3de16d3514fb,"## Task 4 - 210824 |
| |
| Regarding the membership status of a company sports club (BSG), which consists of exactly five sections, the following statements have been made: |
| |
| - Exactly 22 members of the BSG belong to the Chess section. |
| - Exactly one third of all BSG members belong to the Football section. |
| - Exactly one fifth of all BSG members belong to the Athletics section. |
| - Exactly three sevenths of all BSG members belong to the Table Tennis section. |
| - Exactly two ninths of all BSG members belong to the Gymnastics section. |
| - Exactly 8 members of the BSG belong to exactly three different sections. |
| - Exactly 72 members of the BSG belong to at least two different sections. |
| - No member of the BSG belongs to more than three sections, but each member belongs to at least one section. |
| |
| Investigate whether there is a composition of membership numbers both for the entire BSG and for the five individual sections such that all these statements are true! Investigate whether these membership numbers are uniquely determined by the statements! If this is the case, provide the membership numbers!",See reasoning trace,medium |
| ce85f7329f17,"Which one does not divide the numbers of $500$-subset of a set with $1000$ elements? |
| |
| $ \textbf{(A)}\ 3 |
| \qquad\textbf{(B)}\ 5 |
| \qquad\textbf{(C)}\ 11 |
| \qquad\textbf{(D)}\ 13 |
| \qquad\textbf{(E)}\ 17 |
| $",11,medium |
| a9d07ec1f725,"11. (This question is worth 20 points) Determine all complex numbers $\alpha$ such that for any complex numbers $z_{1}, z_{2}\left(\left|z_{1}\right|,\left|z_{2}\right|<1, z_{1} \neq z_{2}\right)$, we have |
| $$ |
| \left(z_{1}+\alpha\right)^{2}+\alpha \overline{z_{1}} \neq\left(z_{2}+\alpha\right)^{2}+\alpha \overline{z_{2}} . |
| $$",See reasoning trace,medium |
| f767d69b2731,"In a tournament, any two players play against each other. Each player gets one point for a win, 1/2 for a draw, and 0 points for a loss. Let $S$ be the set of the 10 lowest scores. We know that each player obtained half of their score playing against players from $S$. |
| |
| a) What is the sum of the scores of the players in $S$? |
| |
| b) Determine how many participants are in the tournament. |
| |
| Note: Each player plays only once against each opponent.",See reasoning trace,medium |
| 16e38f829178,"4. Let the function $y=f(x)$ satisfy $f(3+x)=f(3-x)$ for all real numbers $x$, and the equation $f(x)=0$ has exactly 6 distinct real roots. Then the sum of these 6 real roots is |
| A. 18 |
| B. 12 |
| C. 9 |
| D. 0",$A$,easy |
| ad6f1c981501,,See reasoning trace,medium |
| 86d4c84da03a,Example 8 Find the value of $\cos ^{5} \frac{\pi}{9}+\cos ^{5} \frac{5 \pi}{9}+\cos ^{5} \frac{7 \pi}{9}$.,See reasoning trace,medium |
| 9e3ba67501bb,"1. If $x, y$ are natural numbers for which $x-y-\frac{1}{x}+\frac{1}{y}=0$, calculate the value of the expression $\left(\frac{x}{y}\right)^{2018}+\left(\frac{y}{x}\right)^{2018}$.","0$, i.e., to $(x-y) \frac{x y+1}{x y}=0$. The numbers are natural, so $x y+1 \neq 0$. Therefore, $x=",medium |
| c819b6976f89,"4. Let $f(x)=a x+b$ where $a$ and $b$ are integers. If $f(f(0))=0$ and $f(f(f(4)))=9$, find the value of $f(f(f(f(1))))+f(f(f(f(2))))+\cdots+f(f(f(f(2014))))$. |
| (1 mark) |
| 設 $f(x)=a x+b$, 其中 $a 、 b$ 為整數。若 $f(f(0))=0$ 而 $f(f(f(4)))=9$, 求 $f(f(f(f(1))))+f(f(f(f(2))))+\cdots+f(f(f(f(2014)))$ 的值。",See reasoning trace,easy |
| 713fb4a0165f,"Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.","(8, 3), (2, 9), (-7, 9), (-7, 3), (2, -6), (8, -6)",medium |
| 044cd71f2480,"Let $n$ be a positive integer with the following property: $2^n-1$ divides a number of the form $m^2+81$, where $m$ is a positive integer. Find all possible $n$.",n = 2^k,medium |
| b7fe52d72909,"1. Let $-1<a<0, \theta=\arcsin a$, then the solution set of the inequality $\sin x<a$ is |
| (A) $\{x \mid 2 n \pi+\theta<x<(2 n+1) \pi-\theta, n \in Z\}$; |
| (B) $\{x \mid 2 n \pi-\theta<x<(2 n+1) \pi+\theta, n \in Z\}$; |
| (C) $\{x \mid(2 n-1) \pi+\theta<x<2 n \pi-\theta, n \in Z\}$; |
| (D) $\{x \mid(2 n-1) \pi-\theta<x<2 n \pi+\theta, n \in Z\}$.","a$ are $x=-\pi-\arcsin a$ or $x=\arcsin a$. Therefore, the solution set of the inequality $\sin x<a$",easy |
| f321ce3a554a,"5. Find all triples of real numbers $a, b, c$, for which |
| |
| $$ |
| 27^{a^{2}+b+c+1}+27^{b^{2}+c+a+1}+27^{c^{2}+a+b+1}=3 |
| $$",b=c=-1$.,easy |
| 2c92279f5dfb,"Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins? |
| $\textbf{(A)}$ Barbara will win with $2013$ coins and Jenna will win with $2014$ coins. |
| $\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins. |
| $\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins. |
| $\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins. |
| $\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.",\textbf{(B),medium |
| f5e9e80942ff,"10. In $\triangle A B C$, $A B=\sqrt{2}, A C=\sqrt{3}$, $\angle B A C=30^{\circ}$, and $P$ is any point in the plane of $\triangle A B C$. Then the minimum value of $\mu=\overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A}$ is $\qquad$",See reasoning trace,medium |
| 84fc47c17f25,"10.9 Given $x$ and $y$ as real numbers, solve the equation |
| |
| $$ |
| (-5+2 i) x-(3-4 i) y=2-i |
| $$","$x=-\frac{5}{14}, y=-\frac{1}{14}$",easy |
| e6449a45eb46,"6.23 The sum of the first three terms of an increasing arithmetic progression is 21. If 1 is subtracted from the first two terms of this progression, and 2 is added to the third term, the resulting three numbers will form a geometric progression. Find the sum of the first eight terms of the geometric progression.",765,medium |
| 1b0e33800109,"【Example 5】 There are $n$ young people and $r$ elderly people, $n>2r$, to be arranged in a row. It is required that each elderly person has a young person on both sides to support them (but each pair of young people only supports one elderly person). How many different arrangements are there?",See reasoning trace,medium |
| b0d37b7d0071,"\section*{ |
| |
| Imagine three planes (not necessarily distinct from each other) passing through the center of a sphere. |
| |
| Into how many regions can the surface of the sphere be divided by such planes? Consider different cases to determine all possible numbers of regions!",See reasoning trace,medium |
| bf6cd523dac1,"17 Given that $a+\frac{1}{a+1}=b+\frac{1}{b-1}-2$ and $a-b+2 \neq 0$, find the value of $a b-a+b$.","a+1, y=b-1(x-y \neq 0)$, then $x-1+\frac{1}{x}=y+1+\frac{1}{y}-2 \Rightarrow x+\frac{1}{x}=y+\frac{1",easy |
| 9c281d81793a,"Four boys, $A, B, C$, and $D$, made 3 statements each about the same number $x$. We know that each of them has at least one correct statement, but also that at least one of their statements is false. Can $x$ be determined? The statements are: |
| |
| A: (1) The reciprocal of $x$ is not less than 1. |
| |
| (2) The decimal representation of $x$ does not contain the digit 6. |
| |
| (3) The cube of $x$ is less than 221. |
| |
| $B$: (4) $x$ is an even number. |
| |
| (5) $x$ is a prime number. |
| |
| (6) $x$ is an integer multiple of 5. |
| |
| $C$: (7) $x$ cannot be expressed as the quotient of two integers, |
| |
| (8) $x$ is less than 6. |
| |
| (9) $x$ is the square of a natural number. |
| |
| $D$: (10) $x$ is greater than 20. |
| |
| (11) $x$ is positive and its base-10 logarithm is at least 2. |
| |
| (12) $x$ is not less than 10.",See reasoning trace,medium |
| df5985ac8106,"3. The increasing sequence $T=2356781011$ consists of all positive integers which are not perfect squares. What is the 2012th term of $T$ ? |
| (A) 2055 |
| (B) 2056 |
| (C) 2057 |
| (D) 2058 |
| (E) 2059","1936,45^{2}=2025$ and $46^{2}=2116$. So $2,3, \ldots, 2012$ has at most 2012 - 44 terms. For the 201",easy |
| 1fc0978c31e8,"Example 3 In $\triangle A B C$, $\angle C=90^{\circ}, B C=2$, $P$ is a point inside $\triangle A B C$ such that the minimum value of $P A+P B+P C$ is $2 \sqrt{7}$. Find the degree measure of $\angle A B C$. |
| |
| Analysis: The key to this",See reasoning trace,medium |
| 16abfbfdd6f7,"## 1. Aljmez |
| |
| On the planet Aljmez, a day is called NAD, an hour is called TAS, a minute is called NIM, and a second is called KES. One NAD lasts ten TAS, one TAS lasts ten NIM, and one NIM lasts eight KES. How many Earth seconds does one Aljmezian KES last, if one NAD lasts as long as one day? |
| |
| Result: $\quad \mathbf{1 0 8}$",86400: 800=864: 8=108$ seconds,easy |
| 8b99a04e4055,"Solve the following system of equations without using logarithms: |
| |
| $$ |
| a^{7 x} \cdot a^{15 y}=\sqrt{a^{19}}, \quad \sqrt[3]{a^{25 y}}: \sqrt{a^{13 x}}=\sqrt[12]{a} |
| $$",See reasoning trace,easy |
| 99136848ec62,"6. [9] Petya had several hundred-ruble bills, and no other money. Petya started buying books (each book costs an integer number of rubles) and receiving change in small denominations (1-ruble coins). When buying an expensive book (not cheaper than 100 rubles), Petya paid only with hundred-ruble bills (the minimum necessary number of them), and when buying a cheap book (cheaper than 100 rubles), he paid with small denominations if he had enough, and if not, with a hundred-ruble bill. By the time the hundred-ruble bills ran out, Petya had spent exactly half of his money on books. Could Petya have spent at least 5000 rubles on books? |
| |
| (Tatyana Kazitsyna)","98$ rubles. For each other purchase, this difference is negative and even (since the sum of the pric",medium |
| 605305b96602,"6. Find the sum |
| |
| $$ |
| \begin{aligned} |
| & \frac{1}{(\sqrt[4]{1}+\sqrt[4]{2})(\sqrt{1}+\sqrt{2})}+\frac{1}{(\sqrt[4]{2}+\sqrt[4]{3})(\sqrt{2}+\sqrt{3})}+ \\ |
| & +\ldots+\frac{1}{(\sqrt[4]{9999}+\sqrt[4]{10000})(\sqrt{9999}+\sqrt{10000})} |
| \end{aligned} |
| $$",9,easy |
| 04f00d5f0b8f,"1. Given $a+b+c=2, a^{2}+b^{2}+c^{2}=2$. Try to compare the size of the following three expressions: |
| $$ |
| \begin{array}{l} |
| a(1-a)^{2}-b(1-b)^{2}-c(1-c)^{2} \\ |
| \text { (fill in “>”, “= ” or “<”). } |
| \end{array} |
| $$","b(1-b)^{2}, a b c=a(1-a)^{2}$.",easy |
| e2393b655683,"The centers of the faces of a certain cube are the vertices of a regular octahedron. The feet of the altitudes of this octahedron are the vertices of another cube, and so on to infinity. What is the sum of the volumes of all the cubes, if the edge of the first cube is $a$?",See reasoning trace,medium |
| 4b9fa9f5d3db,,$52,medium |
| 022c4583e9da,"Find the least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ and $b$.",70,medium |
| f1327978d9da,"Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$.",1932,medium |
| ce9d91640d71,"Example 9 Solve the equation |
| \[ |
| \begin{array}{l} |
| \frac{36}{\sqrt{x-2}}+\frac{4}{\sqrt{y-1}}+4 \sqrt{x-2}+\sqrt{y-1}-28 \\ |
| =0 . |
| \end{array} |
| \] |
| (1988, Xuzhou City Junior High School Mathematics Competition)","11, \\ y=5 .\end{array}\right.$",medium |
| 3720971c84d3,"The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",139,medium |
| 6007676d6bb5,"a) Find $\angle A C M$. |
| b) Prove that $C M=\frac{A B \cdot B C}{2 A C}$.",See reasoning trace,medium |
| ef3fad866df6,"7.1. Two spheres are inscribed in a dihedral angle, touching each other. The radius of one sphere is twice that of the other, and the line connecting the centers of the spheres forms an angle of $45^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. Write the cosine of this angle in your answer, rounding it to two decimal places if necessary.",0,medium |
| 75b2f376986a,"## Task Condition |
| |
| Find the $n$-th order derivative. |
| |
| $y=2^{3 x+5}$",See reasoning trace,medium |
| 19872e5ac860,"Solve the following equation: |
| |
| $$ |
| \sin x + \sin 2x + \sin 3x = 1 + \cos x + \cos 2x |
| $$",See reasoning trace,medium |
| 9f62c4bb34db,"Example 3: 10 people go to a bookstore to buy books. It is known that each person bought three types of books, and any two people have at least one book in common. Question: What is the maximum number of people who bought the most purchased book, at a minimum? (No. 8",See reasoning trace,medium |
| 62a72c5bb987,"1. Multiplying the first and fourth, and the second and third factors, we get the equation: $\left(x^{2}+5 x+4\right)\left(x^{2}+5 x+6\right)=360$. |
| |
| By making the substitution $y=x^{2}+5 x+4$, we get $y^{2}+2 y-360=0$, from which $y_{1}=-20$, $y_{2}=18$. Therefore, we have the equations: |
| |
| $x^{2}+5 x+24=0, x^{2}+5 x-14=0$ |
| |
| The first equation has no solutions, from the second we get: $x_{1}=-7, x_{2}=2$.","$x_{1}=-7, x_{2}=2$",easy |
| 192540bb5294,"6. In a rectangle $A B C D$ (including the boundary) with an area of 1, there are 5 points, among which no three points are collinear. Find the minimum number of triangles, with these 5 points as vertices, whose area is not greater than $\frac{1}{4}$.",See reasoning trace,medium |
| 7feb98396707,(B. Frenkin),Cloudy,medium |
| 623c8498e8fb,Example 4 Find the maximum of $y=\sin ^{2} x+2 \sin x \cos x+3 \cos ^{2} x$.,"2+2 \sin 2 x+\cos 2 x$. Since $\sin ^{2} 2 x+\cos ^{2} 2 x=1$, by the corollary, we get $2 \geqslant",easy |
| 054d9f6b99ea,"Task B-1.2. Write as a power with base 5: |
| |
| $$ |
| 3\left(5^{n}-5\right)\left(5^{n}+5\right)+2\left(25+5^{2 n}\right)+25^{n+1}: 5^{2 n} |
| $$",See reasoning trace,easy |
| 9bf88d96d16f,"Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? |
| |
| $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$",13,medium |
| 5a127f55bae3,"2. Person A and Person B are playing a card game. There are 40 cards in total, each card has a number from 1 to 10, and each number has four different suits. At the beginning, each person has 20 cards. Each person removes pairs of cards that differ by 5. In the end, Person B has two cards left, with the numbers 4 and $a$, and Person A also has two cards left, with the numbers 7 and $b$. What is the value of $|a-b|$? |
| (A) 3 |
| (B) 4 |
| (C) 6 |
| (D) 7",7$.,easy |
| cc94a5cc441c,"5-6. On a rectangular table of size $x$ cm $\times 80$ cm, identical sheets of paper of size 5 cm $\times 8$ cm are placed. The first sheet touches the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length $x$ in centimeters? |
| |
| ",77,medium |
| 4822dc393774,"## |
| |
| Calculate the limit of the function: |
| |
| $$ |
| \lim _{x \rightarrow 2} \frac{\ln (x-\sqrt[3]{2 x-3})}{\sin \left(\frac{\pi x}{2}\right)-\sin ((x-1) \pi)} |
| $$",See reasoning trace,medium |
| 861e67b64415,"## Task A-3.6. |
| |
| Calculate the product |
| |
| $$ |
| \left(1-\frac{\cos 61^{\circ}}{\cos 1^{\circ}}\right)\left(1-\frac{\cos 62^{\circ}}{\cos 2^{\circ}}\right) \ldots\left(1-\frac{\cos 119^{\circ}}{\cos 59^{\circ}}\right) |
| $$",See reasoning trace,medium |
| 2701be9a59af,"4. Let triangle $ABC$ be an isosceles triangle with base $\overline{AB}$ of length $10 \, \text{cm}$ and legs of length $13 \, \text{cm}$. Let $D$ be a point on side $\overline{BC}$ such that $|BD|:|DC|=1:2$ and let $E$ be a point on side $\overline{CA}$ such that $|CE|:|EA|=1:2$. Calculate $|DE|$.",\sqrt{5^{2}+4^{2}}=\sqrt{41} \mathrm{~cm}$.,medium |
| 64ae882fda5b,"A triangle has side lengths $a, a$ and $b$. It has perimeter $P$ and area $A$. Given that $b$ and $P$ are integers, and that $P$ is numerically equal to $A^{2}$, find all possible pairs $(a, b)$.",See reasoning trace,medium |
| ca8da6fce24b,"Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?",42,medium |
| a30e396450c2,Example 1 Find all positive integers $n$ such that $2^{n}-1$ is divisible by 7.,"(\underbrace{100 \cdots 0}_{n \uparrow})_{2}$, then $2^{n}-1=(\underbrace{11 \cdots 1}_{n \uparrow})",easy |
| e5dc7ffed8df,"5. (10 points) Rearrange the 37 different natural numbers $1,2,3, \cdots, 37$ into a sequence, denoted as $a_{1}, a_{2}, \cdots, a_{37}$, where $a_{1}=37, a_{2}=1$, and ensure that $a_{1}+a_{2}+\cdots+a_{k}$ is divisible by $a_{k+1}$ $(k=1,2, \cdots, 36)$. Find $a_{3}=? a_{37}=?$","37+1=38$, so 38 is a multiple of $a_{3}$, hence $a_{3}=2$.",medium |
| bc8592f13e4e,"2. In triangle $\triangle A B C$, the sides $A B=5$ and $A C=6$ are known. What should the side $B C$ be so that the angle $\angle A C B$ is as large as possible? Provide the length of side $B C$, rounded to the nearest integer.",See reasoning trace,medium |
| 74910de79928,2. Find all natural numbers $n$ for which there exists a prime number $p$ such that the number $p^{2}+7^{n}$ is a perfect square.,"3, n=1$ ..... 1 point",medium |
| 1d2a4543b267,"1. If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$. |
| (1 mark) |
| If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$. |
| (1 mark)",by substituting suitable values of $x$ and $y$,easy |
| dfe04a2130d0,"6. Among the 8 vertices, 12 midpoints of edges, 6 centers of faces, and the center of the cube, a total of 27 points, the number of groups of three collinear points is ( ). |
| (A) 57 |
| (B) 49 |
| (C) 43 |
| (D) 37",28$ (groups); the number of collinear triplets with both endpoints being the centers of faces is $\f,easy |
| 6a45fa290ba6,"14. (2 marks) In $\triangle A B C, \angle A C B=3 \angle B A C, B C=5, A B=11$. Find $A C$. |
| (2 分) 在 $\triangle A B C$ 中 $\angle A C B=3 \angle B A C, B C=5, A B=11$ 。求 $A C$ 。",\sin A \cos B+\cos A \sin B$ is used repeatedly here.),medium |
| 387b4439a114,"1. The number of integer solutions to the equation $\left(x^{2}-x-1\right)^{x+2}=1$ is ( ). |
| (A) 5 |
| (B) 4 |
| (C) 3 |
| (D) 2",See reasoning trace,easy |
| 2295bfe6c714,"For example, as shown in Figure $24-2$, a tunnel is designed for four lanes in both directions, with a total lane width of 22 meters, and the vehicle height limit is 4.5 meters. The total length of the tunnel is 25 kilometers, and the arch of the tunnel is approximately a half-ellipse shape. |
| (1) If the maximum arch height $h$ is 6 meters, what is the arch width $l$ of the tunnel design?",See reasoning trace,medium |
| e4ca65a8487f,"10. In a drawer, there are 6 red socks and 2 blue socks placed in a messy manner. These 8 socks are identical except for their colors. Now the room is pitch black, and the minimum number of socks that need to be taken out to ensure getting a pair of red socks is $\qquad$.",See reasoning trace,easy |
| d48d30dfed7c,"[Example 5.2.4] Let $x, y, z, w$ be real numbers, and satisfy: |
| (1) $x+y+z+w=0$; |
| (2) $x^{7}+y^{7}+z^{7}+w^{7}=0$. |
| |
| Find the value of $w(w+x)(w+y)(w+z)$.",0$.,medium |
| e8da9c5f70fc,"Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length $1$, what is the total area of all the individual circles? |
| |
| [i]2015 CCA Math Bonanza Lightning Round #4.4[/i]",\frac{\pi,medium |
| 626c2084c8e7,"93. A small train puzzle. The express from Basltown to Ironchester travels at a speed of 60 km/h, and the express from Ironchester to Basltown, which departs simultaneously with it, travels at a speed of 40 km/h. |
| |
| How far apart will they be one hour before meeting? |
| |
| I couldn't find these cities on any map or in any reference, so I don't know the exact distance between them. Let's assume it does not exceed 250 km.",See reasoning trace,easy |
| 5385359e0344,"Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)",f(x) = kx + n,medium |
| 57b83b1fb55b,"Three students named João, Maria, and José took a test with 100 questions, and each of them answered exactly 60 of them correctly. A question is classified as difficult if only one student answered it correctly, and it is classified as easy if all three answered it correctly. We know that each of the 100 questions was solved by at least one student. Are there more difficult or easy questions? Additionally, determine the difference between the number of difficult and easy questions.",See reasoning trace,medium |
| 9eb50786e65b,"## Task Condition |
| |
| Find the derivative of the specified order. |
| |
| $$ |
| y=x \ln (1-3 x), y^{(I V)}=? |
| $$",See reasoning trace,medium |
| d29b764bae5d,"## |
| |
| Find the indefinite integral: |
| |
| $$ |
| \int \frac{\sqrt{1+\sqrt[3]{x}}}{x \sqrt{x}} d x |
| $$",See reasoning trace,medium |
| c313f3b5328e,"[ Examples and counterexamples. Constructions $\quad]$ [ Linear inequalities and systems of inequalities $]$ |
| |
| Can 20 numbers be written in a row so that the sum of any three consecutive numbers is positive, while the sum of all 20 numbers is negative?",We can,easy |
| d2d810e4217f,"17・120 $P$ is any point on side $BC$ of equilateral $\triangle ABC$, $PX \perp AB, PY \perp AC$, connect $XY$, and let the perimeter of $\triangle XAY$ be $L$, and the perimeter of quadrilateral $BCYX$ be $S$, then the relationship between $L$ and $S$ is |
| (A) $S>L$. |
| (B) $S=L$. |
| (C) $S<L$. |
| (D) Uncertain. |
| (Chinese Zhejiang Province Junior High School Mathematics Competition, 1992)",$(B)$,easy |
| bcd4a210991f,"The function $f$ maps the set of positive integers into itself, and satisfies the equation |
| |
| $$ |
| f(f(n))+f(n)=2 n+6 |
| $$ |
| |
| What could this function be?",x+2$ is also the only solution.,medium |
| b778bd871c53,"Find all periodic sequences $x_1,x_2,\dots$ of strictly positive real numbers such that $\forall n \geq 1$ we have $$x_{n+2}=\frac{1}{2} \left( \frac{1}{x_{n+1}}+x_n \right)$$","a, \frac{1",medium |
| 807b07aa2a1a,"7 There are 6 seats arranged in a row, and three people are to be seated. The number of different seating arrangements where exactly two empty seats are adjacent is ( ). |
| (A) 48 ways |
| (B) 60 ways |
| (C) 72 ways |
| (D) 96 ways",$\mathrm{C}$,medium |
| f85fa55cbcb5,Example 7 How many positive integer factors does 20! have?,See reasoning trace,medium |
| 4b061c59d2ea,"A10 How many of the following statements are correct? |
| $20 \%$ of $40=8$ |
| $2^{3}=8$ |
| $3^{2}-1^{2}=8$ |
| $7-3 \cdot 2=8$ |
| $2 \cdot(6-4)^{2}=8$ |
| (A) 1 |
| (B) 2 |
| (C) 3 |
| (D) 4 |
| (E) 5 |
| |
|  |
| |
| km traveled",See reasoning trace,easy |
| f2913f24120f,"## Task 6 - 110836 |
| |
| For a rectangle $A B C D$ with side lengths $\overline{A B}=a$ and $\overline{B C}=b, a>b$, a parallelogram $E F G H$ is inscribed such that the sides $D A$ and $B C$ of the rectangle are divided by the vertices of the parallelogram in the ratio $2: 3$ or $3: 2$, and the sides $A B$ and $C D$ in the ratio $3: 4$ or $4: 3$, with $E$ on $A B, F$ on $B C, G$ on $C D, H$ on $D A$. |
| |
| Determine whether this is possible in one or more ways! In each of the possible cases, determine the ratio of the areas of the rectangle and the parallelogram to each other!",See reasoning trace,medium |
| 1069ffd58c0a,"Consider a square $ABCD$ with center $O$. Let $E, F, G$, and $H$ be points on the interiors of sides $AB, BC, CD$, and $DA$, respectively, such that $AE = BF = CG = DH$. It is known that $OA$ intersects $HE$ at point $X$, $OB$ intersects $EF$ at point $Y$, $OC$ intersects $FG$ at point $Z$, and $OD$ intersects $GH$ at point $W$. Let $x$ and $y$ be the lengths of $AE$ and $AH$, respectively. |
| |
| a) Given that Area $(EFGH) = 1 \text{ cm}^2$, calculate the value of $x^2 + y^2$. |
| |
| b) Verify that $HX = \frac{y}{x+y}$. Then, conclude that $X, Y, Z$, and $W$ are vertices of a square. |
| |
| c) Calculate |
| |
| Area $(ABCD) \cdot$ Area $(XYZW)$.",See reasoning trace,medium |
| 37144e7ef919,"Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.",\frac{a^2 + b^2 + c^2,medium |
| 5f3e0b066656,"(7) Let $z \in \mathbf{C}$, and satisfy $|z-\mathrm{i}| \leqslant 1, A=\operatorname{Re}(z)\left(|z-\mathrm{i}|^{2}-1\right)$. Find $\max A$ and $\min A$.","When $z=-\frac{\sqrt{3}}{3}+\mathrm{i}$, $\max A=\frac{2 \sqrt{3}}{9}$; when $z=\frac{\sqrt{3}}{3}+\mathrm{i}$, $\min A=-\frac{2 \sqrt{3}}{9}$ )",easy |
| f540aaf98f03,"2、The average height of boys in Class 5(1) is $149 \mathrm{~cm}$, and the average height of girls is $144 \mathrm{~cm}$. The average height of the whole class is $147 \mathrm{~cm}$. Then, how many times is the number of boys in Class 5(1) compared to the number of girls?","3 y$, which means the number of boys is 1.5 times the number of girls.",easy |
| 6932ccc6ae13,4. A household raises chickens and rabbits. The chickens and rabbits have a total of 50 heads and 140 legs. How many chickens and how many rabbits are there in this household?,See reasoning trace,easy |
| fd9c473cfcbf,"2. Given real numbers $x, y$ satisfy $x^{2}-x y+2 y^{2}=1$. Then the sum of the maximum and minimum values of $x^{2}+$ $2 y^{2}$ is equal to ( ). |
| (A) $\frac{8}{7}$ |
| (B) $\frac{16}{7}$ |
| (C) $\frac{8-2 \sqrt{2}}{7}$ |
| (D) $\frac{8+2 \sqrt{2}}{7}$","\sqrt{2} y$ or $x=-\sqrt{2} y$, the two equalities in equation (1) are respectively satisfied. There",medium |
| 12289f20f9b8,"4.41 Given that $n$ is a positive integer, determine the number of solutions in ordered pairs of positive integers $(x, y)$ for the equation $\frac{x y}{x+y}=n$.",See reasoning trace,medium |
| 681d044104b0,"## Task 3 - 050723 |
| |
| Compare the sum of all three-digit natural numbers divisible by 4 with the sum of all three-digit even natural numbers not divisible by 4! |
| |
| a) Which of the two sums is greater? |
| |
| b) What is the difference between the two sums in absolute value?",See reasoning trace,easy |
| 9484c9aa561d,"$15 \cdot 18$ in simplest form has a denominator of 30, find the sum of all such positive rational numbers less than 10. |
| (10th American Invitational Mathematics Examination, 1992)",See reasoning trace,medium |
| 6dea3cfa450b,"[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part [Pythagorean Theorem (direct and inverse). |
| |
| A circle is tangent to side $B C$ of triangle $A B C$ at its midpoint $M$, passes through point $A$, and intersects segments $A B$ and $A C$ at points $D$ and $E$ respectively. Find the angle $A$, if it is known that $B C=12, A D=3.5$ and $E C=\frac{9}{\sqrt{5}}$. |
| |
| #",\( 90^{\circ} \),medium |
| b7145b571287,"We want to arrange on a shelf $k$ mathematics books (distinct), $m$ physics books, and $n$ chemistry books. In how many ways can this arrangement be done: |
| |
| 1. if the books must be grouped by subjects |
| 2. if only the mathematics books must be grouped.",See reasoning trace,medium |
| 34dae8f716b8,"Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$? |
| $\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$",\textbf{(A),medium |
| 1308b7d68e15,"16. Find the value of |
| $$ |
| \frac{1}{1+11^{-2011}}+\frac{1}{1+11^{-2009}}+\frac{1}{1+11^{-2007}}+\cdots+\frac{1}{1+11^{2009}}+\frac{1}{1+11^{2011}} . |
| $$",See reasoning trace,easy |
| c37c7dab8e5c,"6. Let $f(x)$ be defined on $\mathbf{N}_{+}$, with its range $B \subseteq$ $\mathbf{N}_{+}$, and for any $n \in \mathbf{N}_{+}$, we have |
| $$ |
| f(n+1)>f(n) \text {, and } f(f(n))=3 n \text {. } |
| $$ |
| |
| Then $f(10)+f(11)=$ $\qquad$",39$.,easy |
| c57c30b5a420,"4. Find the greatest and least values of the expression $x-2 y$, if $(x ; y)$ are related by the relation $\sqrt{x-2}+\sqrt{y-3}=3$. For which $(x ; y)$ are they achieved?","$(x-2 y)_{\max }=5$ when $x=11, y=3,(x-2 y)_{\min }=-22$ when $x=2, y=12$",easy |
| 3f691b726bd5,"A2. The car's odometer reads 13,833. Marko has thought about the smallest number of kilometers he must drive so that the odometer will again show three identical digits. Between which numbers does this number lie? |
| (A) between 1 and 30 |
| (B) between 31 and 70 |
| (C) between 71 and 120 |
| (D) between 121 and 500 |
| (E) between 501 and 1000",See reasoning trace,easy |
| e4c9dfc3e0a9,"14. Let $k$ be a positive integer, such that for any positive numbers $a, b, c$ satisfying the inequality $k(a b+b c+c a)>5\left(a^{2}+b^{2}\right.$ $+c^{2}$), there must exist a triangle with side lengths $a, b, c$. Find the minimum value of $k$.","6$, the inequality becomes $6(a b+a c+b c)>5\left(a^{2}+b^{2}+c^{2}\right)$. Without loss of general",medium |
| 5116da91b308,"In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in case of a tie, the order is arbitrary). It is then noticed that each participant has won half of their points against the bottom ten in the ranking. How many people participated in the tournament",See reasoning trace,medium |
| c8a47ef0cc2a,"3. Given that the two angle bisectors $B D$ and $C E$ of $\triangle A B C$ intersect at point $I, I D=I E, \angle A B C=70^{\circ}$. Then the degree measure of $\angle A$ is $\qquad$ .",See reasoning trace,medium |
| 9b2405803e17,"14. Two segments, each $1 \mathrm{~cm}$ long, are marked on opposite sides of a square of side $8 \mathrm{~cm}$. The ends of the segments are joined as shown in the diagram. What is the total shaded area? |
| A $2 \mathrm{~cm}^{2}$ |
| B $4 \mathrm{~cm}^{2}$ |
| C $6.4 \mathrm{~cm}^{2}$ |
| D $8 \mathrm{~cm}^{2}$ |
| E $10 \mathrm{~cm}^{2}$",$ $\frac{1}{2} \times(h+8-h)=4$.,easy |
| c2a7937fbd6d,"Example 6 Let real numbers $a$, $b$, $c$ satisfy |
| $$ |
| \left\{\begin{array}{l} |
| a^{2}-b c-8 a+7=0, \\ |
| b^{2}+c^{2}+b c-6 a+6=0 . |
| \end{array}\right. |
| $$ |
| |
| Find the range of values for $a$. |
| (1995, Jilin Province Junior High School Mathematics Competition)",See reasoning trace,medium |
| b93f11eb65f9,"Example 3. Find the logarithmic residue of the function |
| |
| $$ |
| f(z)=\frac{1+z^{2}}{1-\cos 2 \pi z} |
| $$ |
| |
| with respect to the circle $|z|=\pi$.",\sum_{k=0}^{n} a_{k} z^{k}$.,hard |
| 3311f85d5629,"G1.2 On the coordinate plane, there are $T$ points $(x, y)$, where $x, y$ are integers, satisfying $x^{2}+y^{2}<10$, find the value of $T$. (Reference: 2002 FI4.3)",See reasoning trace,medium |
| eac87f81ddd4,"How many integer values of $x$ satisfy $|x|<3\pi$? |
| $\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~18 \qquad\textbf{(D)} ~19 \qquad\textbf{(E)} ~20$",\textbf{(D),easy |
| 946c47217df4,"14. The perimeter of a rectangle is 20 decimeters. If it is cut along the line connecting the midpoints of the longer sides, it is divided into two identical smaller rectangles, and the sum of their perimeters is 6 decimeters more than the original perimeter. The area of the original rectangle is $\qquad$ square decimeters.",21,easy |
| 04441452911f,"As shown in the figure, a rectangular table has 8 columns. Numbers $1, 2, \cdots$ are filled into the table in a certain order (filled from left to right, and when a row is full, move to the next row, still filling from left to right). A student first colors the cell with the number 1 black, then skips 1 cell, and colors the cell with the number 3 black; then skips 2 cells, and colors the cell with the number 6 black; then skips 3 cells, and colors the cell with the number 10 black. This continues until every column contains at least one black cell (no more coloring after that). Therefore, the number in the last black cell he colored is $\qquad$",(1+15) \times 15 \div 2=120$.,medium |
| a538383aa991,"rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?",2n - 1,medium |
| 10579a6f2253,"A4. For quadrilateral $A B C D$, it is given: $|A B|=3,|B C|=4,|C D|=5$, $|D A|=6$ and $\angle A B C=90^{\circ}$. ( $|A B|$ stands for the length of line segment $A B$, etc.) What is the area of quadrilateral $A B C D$? |
| A) 16 |
| B) 18 |
| C) $18 \frac{1}{2}$ |
| D) 20 |
| E) $6+5 \sqrt{11}$ |
| |
| ",318&width=312&top_left_y=1592&top_left_x=1573),easy |
| 6ee0b6d0cd7a,"2. [5] Given right triangle $A B C$, with $A B=4, B C=3$, and $C A=5$. Circle $\omega$ passes through $A$ and is tangent to $B C$ at $C$. What is the radius of $\omega$ ?","\angle B A C$, and so triangles $A B C$ and $C M O$ are similar. Then, $C O / C M=$ $A C / A B$, fro",easy |
| 187550fa953d,"8. The circle $\rho=D \cos \theta+E \sin \theta$ is tangent to the line of the polar axis if and only if ( ). |
| (A) $D \cdot E=0$ |
| (B) $D \cdot E \neq 0$ |
| (C) $D=0, E \neq 0$ |
| (D) $D \neq 0, E=0$","D \cos \theta+E \sin \theta$, the rectangular coordinate equation is $\left(x-\frac{D}{2}\right)^{2}",easy |
| ca97475b4329,"Example 14 Let the three sides of $\triangle ABC$ be $a, b, c$ with corresponding altitudes $h_{a}$, $h_{b}$, $h_{c}$, and the radius of the incircle of $\triangle ABC$ be $r=2$. If $h_{a}+h_{b}+h_{c}=18$, find the area of $\triangle ABC$.",\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times(4 \sqrt{3})^{2}=12 \sqrt{3}$.,medium |
| 83b617794530,"a) Verify that if we choose 3 or more integers from the set $\{6 k+1,6 k+2,6 k+3,6 k+4,6 k+5,6 k+6\}$, at least two will differ by 1, 4, or 5. |
| |
| b) What is the largest number of positive integers less than or equal to 2022 that we can choose so that there are no two numbers whose difference is 1, 4, or 5?","674$. Since the difference between any two multiples of 3 is a multiple of 3, it follows that none o",medium |
| 358f85de9176,"Calculate $\binom{n}{0},\binom{n}{1}$ and $\binom{n}{2}$.",See reasoning trace,medium |
| 72f2a9e6021b,"N2. Find the maximum number of natural numbers $x_{1}, x_{2}, \ldots, x_{m}$ satisfying the conditions: |
| |
| a) No $x_{i}-x_{j}, 1 \leq i<j \leq m$ is divisible by 11 ; and |
| |
| b) The sum $x_{2} x_{3} \ldots x_{m}+x_{1} x_{3} \ldots x_{m}+\cdots+x_{1} x_{2} \ldots x_{m-1}$ is divisible by 11 .",See reasoning trace,medium |
| c1c834f42477,"Example 3 Find all values of $a$ such that the roots $x_{1}, x_{2}, x_{3}$ of the polynomial $x^{3}-6 x^{2}+a x+a$ satisfy |
| $$ |
| \left(x_{1}-3\right)^{2}+ |
| \left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0 \text {. } |
| $$",-9$.,medium |
| 1dea014f0c03,What is the smallest natural number $n$ such that the base 10 representation of $n!$ ends with ten zeros,45$.,medium |
| cf1bc3fd52db,"(22) For the sequence $\left\{a_{n}\right\}$, we define $\left\{\Delta a_{n}\right\}$ as the first-order difference sequence of $\left\{a_{n}\right\}$, where $\Delta a_{n} = a_{n+1} - a_{n} \left(n \in \mathbf{N}^{*}\right)$. For a positive integer $k$, we define $\left\{\Delta^{k} a_{n}\right\}$ as the $k$-th order difference sequence of $\left\{a_{n}\right\}$, where $\Delta^{k} a_{n} = \Delta^{k-1} a_{n+1} - \Delta^{k-1} a_{n} = \Delta\left(\Delta^{k-1} a_{n}\right)$. |
| (1) If the sequence $\left\{a_{n}\right\}$ has the first term $a_{1} = 1$, and satisfies |
| $$ |
| \Delta^{2} a_{n} - \Delta a_{n+1} + a_{n} = -2^{n}, |
| $$ |
| |
| find the general term formula of the sequence $\left\{a_{n}\right\}$; |
| (2) For the sequence $\left\{a_{n}\right\}$ in (1), does there exist an arithmetic sequence $\left\{b_{n}\right\}$ such that |
| $$ |
| b_{1} \mathrm{C}_{n}^{1} + b_{2} \mathrm{C}_{n}^{2} + \cdots + b_{n} \mathrm{C}_{n}^{n} = a_{n} |
| $$ |
| |
| holds for all positive integers $n \in \mathbf{N}^{*}$? If it exists, find the general term formula of the sequence $\left\{b_{n}\right\}$; if not, explain the reason; |
| (3) Let $c_{n} = (2 n - 1) b_{n}$, and set |
| $$ |
| T_{n} = \frac{c_{1}}{a_{1}} + \frac{c_{2}}{a_{2}} + \cdots + \frac{c_{n}}{a_{n}}, |
| $$ |
| |
| if $T_{n} < M$ always holds, find the smallest positive integer $M$.",See reasoning trace,medium |
| 02fb9c094d00,,5,medium |
| 61afb0cbec44,"2 Given that $\sqrt{2 x+y}+\sqrt{x^{2}-9}=0$, find the value(s) of $y-x$. |
| (A) -9 |
| (B) -6 |
| (C) -9 or 9 |
| (D) -3 or 3 |
| (E) None of the above","0, \sqrt{2 x+y}=0$ and $\sqrt{x^{2}-9}=0$. So we have $x=3$ or -3 and $y=-2 x=-6$ or $6 \Rightarrow ",easy |
| baa62e7383f5,"## 7. Gold Coins |
| |
| Ante, Branimir, Celestin, and Dubravko are sitting around a round table (in that order). Together, they have 1600 gold coins. First, Ante divides half of his gold coins into two equal parts and gives one part to his left neighbor and the other part to his right neighbor, while keeping the other half for himself. Then, Branimir does the same, followed by Celestin, and finally Dubravko. In the end, all four of them have an equal number of gold coins. How many gold coins did Branimir have at the beginning? |
| |
| Result: $\quad 575$",See reasoning trace,medium |
| 8c869005425e,6.,1$. The function we seek is $f(x)=x+1$.,medium |
| 43c95281260a,"II. (25 points) Let $D$ be the midpoint of the base $BC$ of isosceles $\triangle ABC$, and $E$, $F$ be points on $AC$ and its extension, respectively. Given that $\angle EDF=90^{\circ}$, $DE=a$, $DF=b$, $AD=c$. Try to express the length of $BC$ in terms of $a$, $b$, and $c$. |
| |
| --- |
| |
| The translation preserves the original text's formatting and structure.",2 B D=\frac{2 a b c}{\sqrt{a^{2} c^{2}+b^{2} c^{2}-a^{2} b^{2}}}$.,medium |
| 6d833e5e16b3,"13. (10 points) In the equation below, $A, B, C, D, E, F, G, H, I$ each represent different digits from $1 \sim 9$. |
| $$ |
| \overline{\mathrm{ABCD}}+\overline{\mathrm{EF}} \times \overline{\mathrm{GH}}-I=X |
| $$ |
| |
| Then the minimum value of $X$ is . $\qquad$","2369$,",medium |
| be16c8b61057,"9. As shown in the figure, in the right triangle $A B C$, $\angle A C B=\frac{\pi}{2}, A C=B C=2$, point $P$ is a point on the hypotenuse $A B$, and $B P=2 P A$, then $\overrightarrow{C P} \cdot \overrightarrow{C A}+\overrightarrow{C P} \cdot \overrightarrow{C B}=$ $\qquad$ .",4,medium |
| 1a1cf51a5ff1,"| $A$ | $B$ | $C$ | $D$ | $E$ | |
| | :--- | :--- | :--- | :--- | :--- | |
| | $E$ | $D$ | $C$ | $B$ | $A$ | |
| | $F$ | $F$ | $F$ | $F$ | $F$ |$+$ |
| |
| A fragment of a conversation on the beach: |
| |
| Feri: Don't rush me now, I need to write digits here instead of the letters : What's the fuss about this? I can write down 77 solutions for you. - F: I doubt that. - Gy: I'll buy you as many ice creams as the number of solutions you miss, just come and swim now. - F: I'll go, but is the bet on? - How did the bet turn out?",See reasoning trace,medium |
| d421c58625a1,". For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ (e.g. $d(12)=6$ ) and let $s(k)$ denote the digit sum of $k$ (e.g. $s(12)=3$ ). A positive integer $n$ is said to be amusing if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than 1 ?",9,medium |
| 8b0c28e60cae,"3. A ball sliding on a smooth horizontal surface catches up with a block moving along the same surface. The ball's velocity is perpendicular to the edge of the block, against which it collides. The mass of the ball is much less than the mass of the block. After an elastic collision, the ball slides along the surface in the opposite direction with a speed that is half of its initial speed. |
| |
| Find the ratio of the ball's and the block's velocities before the collision.",349&width=645&top_left_y=1810&top_left_x=1274),easy |
| 54e2bf190ad1,"1. From the integers 1 to 100, select two numbers without repetition to form an ordered pair $(x, y)$, such that the product $x y$ is not divisible by 3. How many such pairs can be formed?",See reasoning trace,easy |
| 65e21c7775e1,"Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. |
| |
| [i]V. Senderov[/i]",k = 1,medium |
| eb1572564d47,"1. Given the sequence $\left\{a_{n}\right\}$ with the general term formula $a_{n}=\log _{3}\left(1+\frac{2}{n^{2}+3 n}\right)$, then $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)=$",See reasoning trace,medium |
| 4c393bc4d8b3,"## Task B-1.2. |
| |
| In a sequence of six natural numbers, the third and each subsequent number is equal to the sum of the two preceding ones. Determine all such sequences of numbers, if the fifth number in the sequence is equal to 25.",See reasoning trace,medium |
| 783cbe131b8d,"## Subject IV. (20 points) |
| |
| Emil is waiting in line at a ticket booth, along with other people, standing in a row. Andrei, who is right in front of Emil, says: ""Behind me, there are 5 times as many people as in front of me."" Mihai, who is right behind Emil, says: ""Behind me, there are 3 times as many people as in front of me."" How many people are waiting at the ticket booth? |
| |
| Prof. Sorin Borodi, ""Alexandru Papiu Ilarian"" Theoretical High School, Dej |
| |
| ## Grading Scale for 5th Grade (OLM 2014 - local stage) |
| |
| ## Official $10 \mathrm{p}$",See reasoning trace,medium |
| 2bb849252d7e,"8. If for any $x \in(-\infty,-1)$, we have |
| $$ |
| \left(m-m^{2}\right) 4^{x}+2^{x}+1>0 \text {, } |
| $$ |
| |
| then the range of real number $m$ is $\qquad$",See reasoning trace,medium |
| e07a7a769433,"In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.","2, 3, 4",medium |
| 4dd9e0ab6c87,"B1. Witch Čiračara specialized in mathematical spells. The ingredients for the magic spell are: 3, 133, 38, 42, 2, 56, 9, 120, and 6. The magic number is calculated as follows: |
| |
| Divide the largest even number by the smallest odd number to get the devilish number. Then multiply the smallest even number by the largest odd number to get the wizardly number. Finally, multiply by ten the difference you get when you subtract the wizardly number from twice the devilish number. The resulting product is the magic number. |
| |
| What is the magic number of Witch Čiračara? Write down your answer.",-1860$.,easy |
| df1d01bc0de8,"4.017. Find four numbers that form a geometric progression, where the third term is 9 more than the first, and the second term is 18 more than the fourth.","$3,-6,12,-24$",medium |
| 5e434c918afb,"11. (This question is worth 20 points) Find all pairs of positive real numbers $(a, b)$ such that the function $f(x)=a x^{2}+b$ satisfies: for any real numbers $x, y$, |
| $$ |
| f(x y)+f(x+y) \geq f(x) f(y) . |
| $$",See reasoning trace,medium |
| 28acfa720b65,"B3. Let $p(x)=3 x^{3}-2 x^{2}-3$ and $q(x)=x+1$. |
| |
| a) Calculate $3 p(-2)+2 q(3)$. |
| |
| b) Write the leading term of the polynomial $2(p(x))^{2}$. |
| |
| c) Calculate $p(x) \cdot(q(x))^{2}$. |
| |
| d) Divide $p(x)$ by $q(x)$.",\left(3 x^{2}-5 x+5\right)(x+1)-8 \ldots \ldots \ldots .2$ points,medium |
| a5a562335c34,30. Find the remainder when the 2018-digit number $\underbrace{\overline{55 \cdots}}_{2018 \text { 555 }}$ is divided by 13.,3,easy |
| 10794d8b8825,"$3 \cdot 12$ If $(x, y)$ is a solution to the system of equations |
| $$ |
| \left\{\begin{array}{l} |
| x y=6 \\ |
| x^{2} y+x y^{2}+x+y=63 |
| \end{array}\right. |
| $$ |
| |
| then $x^{2}+y^{2}$ equals |
| (A) 13 . |
| (B) $\frac{1173}{32}$. |
| (C) 55 . |
| (D) 69 . |
| (E) 81 . |
| (38th American High School Mathematics Examination, 1987)",$(D)$,easy |
| 7709c8c5dc10,"51 Given a regular quadrilateral prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, with the base edge length being $a$, and the side edge length being $b, P$ is a moving point on the diagonal $A C_{1}$, let the angle between $P C$ and the plane $A B C D$ be $\alpha$, and the angle between $P D$ and the plane $C D D_{1} C_{1}$ be $\beta$. Find the value of $\tan \alpha \cdot \tan \beta$.",See reasoning trace,medium |
| 6e236bf811fa,"Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.",298,medium |
| 36d2e2323e7e,"Task A-1.5. (4 points) |
| |
| How many elements at least need to be removed from the set $\{2,4,6,8,10,12,14,16\}$ so that the product of the remaining elements is a square of a natural number?",See reasoning trace,medium |
| 5ec164f12d2e,"18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his original riding speed, and arrives at $B$ 17 minutes later than the expected time. What is Li Shuang's walking speed in meters per minute?",Li Shuang's speed of pushing the cart while walking is 72 meters/minute,easy |
| 6b07c8af02ef,"3-4. How many pairs of integers $x, y$, lying between 1 and 1000, are there such that $x^{2}+y^{2}$ is divisible by 7.",$142^{2}=20164$,easy |
| 0f2666327c0c,"11. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with its left focus at $F$. A tangent line is drawn through a point $A$ on the ellipse $C$, intersecting the $y$-axis at point $Q$, and $O$ is the origin. If $\angle Q F O=45^{\circ}, \angle Q F A=30^{\circ}$, then the eccentricity of the ellipse is $\qquad$.",See reasoning trace,medium |
| 5940056d833d,"12. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2, a_{n+1}=3 a_{n}-2 a_{n-1}, a_{2002}=$","3 a_{n}-2 a_{n-1}$, we can respectively get $a_{n+1}-2 a_{n}=a_{n}-2 a_{n-1}, a_{n}-2 a_{n-1}=0$, so",easy |
| a2ecaaa1b212,"266. Inside the square $A B C D$, an isosceles triangle $A B L$ is constructed with base $A B$ and base angles of $15^{\circ}$. Under what angle is the side $C D$ seen from the vertex $L$?",See reasoning trace,medium |
| 9b2a5bba89ab,"(12) Given the function $f(x)=a^{x}-x(a>1)$. |
| (1) If $f(3)<0$, try to find the range of values for $a$; |
| (2) Write a set of numbers $a, x_{0}\left(x_{0} \neq 3\right.$, keep 4 significant digits), such that $f\left(x_{0}\right)<0$ holds.",See reasoning trace,easy |
| 68c65ed8eb78,"10. Draw a diagram! Two ferries leave simultaneously from opposite banks of a river and cross it perpendicular to the banks. The speeds of the ferries are constant. The ferries meet each other 720 meters from the nearest bank. Upon reaching the bank, they immediately head back. On the return trip, they meet 400 meters from the other bank. What is the width of the river?",See reasoning trace,medium |
| c67f7ff9e8bd,Emil Kolev,$ $\frac{2^{2005}}{2^{11}}=2^{1994}$ and this is the required number.,medium |
| 097173cc91a8,"83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation)",1536,easy |
| 2441b3f8bafc,"8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \%$ of satisfied customers leave positive reviews. A certain online store has received 60 angry and 20 positive reviews. Using this statistics, estimate the probability that the next customer will be satisfied with the service in this online store[^0] |
| |
| ",approximately 0,easy |
| a135eff435f9,"7. As shown in the figure, the beads on the bracelet are numbered from 1 to 22 in a counterclockwise direction starting from the pendant bead. Xiao Ming is playing a bead counting game, with the rule being: starting from bead 1, count natural numbers in a clockwise direction, but skip any number that contains the digit 7 or is a multiple of 7, and directly count the next number. For example: after counting to 6, the next number is 8; after counting to 13, the next number is 15, and so on. So, when counting to 100, which bead number $\qquad$ will it land on?","14$ multiples of 7 within 100; when counting to 100, 86 numbers should be counted $100-14=86,86 \div",easy |
| 1855f3701c1f,"Let $k$ be the smallest positive real number such that for all positive real numbers $x$, we have |
| |
| $$ |
| \sqrt[3]{x} \leq k(x+1) |
| $$ |
| |
| What is the value of $k^{3}$?","\frac{1}{2}$. In conclusion, $k^{3}=\frac{4}{27}$.",medium |
| 8bd4057fae02,"1. Andrei, Boris, and Valentin participated in a 1 km race. (We assume that each of them ran at a constant speed). Andrei was 100 m ahead of Boris at the finish line. And Boris was 50 m ahead of Valentin at the finish line. What was the distance between Andrei and Valentin at the moment Andrei finished?",$145 \mathrm{m}$,easy |
| 3afb7f44aa20,"Three nonnegative real numbers satisfy $a,b,c$ satisfy $a^2\le b^2+c^2, b^2\le c^2+a^2$ and $c^2\le a^2+b^2$. Prove the inequality |
| \[(a+b+c)(a^2+b^2+c^2)(a^3+b^3+c^3)\ge 4(a^6+b^6+c^6).\] |
| When does equality hold?",(a+b+c)(a^2+b^2+c^2)(a^3+b^3+c^3) \geq 4(a^6+b^6+c^6),medium |
| 300c236ac01a,"C2. Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move. |
| |
| Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board.",$K=20^{2} / 4=100$,medium |
| 65aad02368b1,"3. Determine all triples $(a, b, c)$ of positive integers for which |
| |
| $$ |
| 2^{a+2 b+1}+4^{a}+16^{b}=4^{c} . |
| $$",". For the second approach, award 3 points for expressing the left side of the equation as a square, 2 points for the reasoning leading to the equality $a=2 b$, and 1 point for calculating $c$ and the correct answer",medium |
| bf37c0913366,"For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist? |
| |
| [i](A. Golovanov)[/i]",2,medium |
| c5be209e9643,"Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value? |
| $\textbf{(A) } 5 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$",\textbf{(C),medium |
| d21a29a1642a,"5. In the sequence $\left\{a_{n}\right\}$, $a_{1}=2, a_{n}+a_{n+1}=1\left(n \in \mathbf{N}_{+}\right)$, let $S_{n}$ be the sum of the first $n$ terms of the sequence $a_{n}$, then the value of $S_{2017}-$ $2 S_{2018}+S_{2019}$ is $\qquad$",See reasoning trace,easy |
| d0e4edd5e19b,"6. If $(2 x+4)^{2 n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}\left(n \in \mathbf{N}^{*}\right)$, then the remainder when $a_{2}+a_{4}+\cdots+a_{2 n}$ is divided by 3 is $\qquad$",1,easy |
| 15cc7e51edf8,"2. Inside the area of $\measuredangle A O B$, a ray $O C$ is drawn such that $\measuredangle A O C$ is $40^{\circ}$ less than $\measuredangle C O B$ and is equal to one third of $\measuredangle A O B$. Determine $\measuredangle A O B$. |
| |
| 翻译完成，保留了原文的换行和格式。",x$ and $\measuredangle C O B=y$. Then $\measuredangle A O B=x+y$ and it holds that $x=y-40^{\circ}$ ,easy |
| 60063e96ad81,2. Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ have radii 3 and 7 respectively. The circles intersect at distinct points $A$ and $B$. A point $P$ outside $\mathcal{C}_{2}$ lies on the line determined by $A$ and $B$ at a distance of 5 from the center of $\mathcal{C}_{1}$. Point $Q$ is chosen on $\mathcal{C}_{2}$ so that $P Q$ is tangent to $\mathcal{C}_{2}$ at $Q$. Find the length of the segment $P Q$.,See reasoning trace,easy |
| 6cec42106298,"What is the maximum number of subsets of $S = {1, 2, . . . , 2n}$ such that no one is contained in another and no two cover whole $S$? |
| |
| [i]Proposed by Fedor Petrov[/i]",\binom{2n,medium |
| ca4055c60e98,"## Task 3 - 030723 |
| |
| A wooden cube with an edge length of $30 \mathrm{~cm}$ is to be cut into cubes with an edge length of $10 \mathrm{~cm}$. |
| |
| a) How many cuts must be made? (Sawing in a package is not allowed.) |
| |
| b) How many cubes will you get?",See reasoning trace,easy |
| eb263cee51ab,"9. Let $a, b$ be real numbers, for any real number $x$ satisfying $0 \leqslant x \leqslant 1$ we have $|a x+b| \leqslant 1$. Then the maximum value of $|20 a+14 b|+|20 a-14 b|$ is $\qquad$.","2, b=-1$.",medium |
| 8fb0173cd008,"4. Suppose $x, y$ are real numbers such that $\frac{1}{x}-\frac{1}{2 y}=\frac{1}{2 x+y}$. Find the value of $\frac{y^{2}}{x^{2}}+\frac{x^{2}}{y^{2}}$. |
| (A) $\frac{2}{3}$ |
| (B) $\frac{9}{2}$ |
| (C) $\frac{9}{4}$ |
| (D) $\frac{4}{9}$ |
| (E) $\frac{2}{9}$",See reasoning trace,easy |
| 5dc10c767f80,14th Australian 1993,"b = c. Obviously (a,a,a) is a possible solution. So assume the numbers are not all equal. Then a+b i",medium |
| 88dde2f89386,"Example 11 Given $m^{2}=m+1, n^{2}=n+1$, and $m \neq n$. Then $m^{5}+n^{5}=$ $\qquad$ . |
| (From the riverbed Jiangsu Province Junior High School Mathematics Competition)",See reasoning trace,easy |
| 82c0203e83ba,"Solve the triangle whose area $t=357.18 \mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.",See reasoning trace,medium |
| c82478429759,"8. (10 points) When withdrawing money from an $A T M$ machine, one needs to input the bank card password to proceed to the next step. The password is a 6-digit number ranging from 000000 to 999999. A person forgot the password but remembers that it contains the digits 1, 3, 5, 7, 9 and no other digits. If there is no limit to the number of incorrect password attempts, the person can input $\qquad$ different passwords at most to proceed to the next step.",: 1800,easy |
| 34d578644d0c,"11. Bing Dwen Dwen practiced skiing for a week, during which the average length of skiing per day for the last four days was 4 kilometers more than the average length of skiing per day for the first three days, and the average length of skiing per day for the last three days was 3 kilometers more than the average length of skiing per day for the first four days. The total length of skiing for the last three days was $\qquad$ kilometers more than the total length of skiing for the first three days.",See reasoning trace,easy |
| af8ccfcd7fee,"6. A die is rolled four times in succession, and from the second roll onwards, the number of points that appear each time is not less than the number of points that appeared in the previous roll. The probability of this happening is $\qquad$ .",\frac{7}{72}$.,medium |
| 2945bf091668,"## |
| |
| Calculate the limit of the function: |
| |
| $\lim _{x \rightarrow 0} \frac{1+\cos (x-\pi)}{\left(e^{3 x}-1\right)^{2}}$",See reasoning trace,medium |
| f1b458db385c,"## Task 5 - 030715 |
| |
| How many zeros does the product of all natural numbers from 1 to 40 end with? (Justification!)",See reasoning trace,easy |
| ae0295284436,"## |
| |
| Calculate the limit of the numerical sequence: |
| |
| $$ |
| \lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+2 n+3}{2 n^{2}+2 n+1}\right)^{3 n^{2}-7} |
| $$",See reasoning trace,medium |
| e143fef19079,"Dudeney, Amusements in Mathematics",144/(x+2). Hence x 2 + 2x - 288 = 0. This factorises as (x - 16)(x + 18) = 0. Obviously x is positiv,easy |
| ca8a8bf21e66,"2.12. (GDR, 77). How many pairs of values $p, q \in \mathbf{N}$, not exceeding 100, exist for which the equation |
| |
| $$ |
| x^{5}+p x+q=0 |
| $$ |
| |
| has solutions in rational numbers?","-x^5 - px$, and since $p$ and $q$ are positive integers, $p$ must be less than or equal to 99. If $x",medium |
| 4c0b37070c8a,"1. If the graph of the function $y=f(x)$ passes through the point $(2,4)$, then the inverse function of $y=f(2-2x)$ must pass through the point","4$, the function $y=f(2-2x)$ passes through the point $(0,4)$, so its inverse function passes throug",easy |
| ae666d2939d2,"Subject 2. Let $x \neq 1, \quad y \neq-2, \quad z \neq-3$ be rational numbers such that $\frac{2015}{x+1}+\frac{2015}{y+2}+\frac{2015}{z+3}=2014$. Calculate $\frac{x-1}{x+1}+\frac{y}{y+2}+\frac{z+1}{z+3}$.",3-2$ | $\left.\frac{1}{x+1}+\frac{1}{y+2}+\frac{1}{z+3}\right)=\frac{2017}{2015} \ldots$ | p |,medium |
| f7682918520d,"1. Given $f(x) \in\left[\frac{3}{8}, \frac{4}{9}\right]$, then the range of $y=f(x)+\sqrt{1-2 f(x)}$ is $\qquad$","t \in\left[\frac{1}{3}, \frac{1}{2}\right]$, then $f(x)=\frac{1}{2}\left(1-t^{2}\right), \therefore ",medium |
| 6fcd9bd69e49,"## Task Condition |
| |
| Find the derivative. |
| |
| $y=\sqrt{(3-x)(2+x)}+5 \arcsin \sqrt{\frac{x+2}{5}}$",See reasoning trace,medium |
| dbb83b8c1eb0,"Example 22 Skew lines $a, b, a \perp b, c$ forms a $30^{\circ}$ angle with $a$, find the range of the angle $\theta$ formed between $c$ and $b$.","90^{\circ}$, hence the range of the angle $\theta$ formed by $c$ and $b$ is $\left[60^{\circ}, 90^{\",medium |
| 57022e7d683e,"5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $30^{\circ}$.",$14 \sqrt[4]{3}$,medium |
| a4ad79f5ebf9,"[ Case Analysis ] [ Proof by Contradiction ] |
| |
| In the cells of a $3 \times 3$ table, numbers are arranged such that the sum of the numbers in each column and each row is zero. What is the smallest number of non-zero numbers that can be in this table, given that this number is odd? |
| |
| #",7 numbers,medium |
| a334ce9d5625,"Senderov V.A. |
| |
| 1999 numbers stand in a row. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of its two neighbors. |
| |
| Find the last number. |
| |
| #",1,easy |
| 2dff135f75ad,"## Task Condition |
| |
| Find the derivative. |
| |
| $$ |
| y=x(\arcsin x)^{2}+2 \sqrt{1-x^{2}} \arcsin x-2 x |
| $$",See reasoning trace,easy |
| 71e72d31df19,". Some $1 \times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n \times n$ so that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is 2008 , find the least possible value of $n$. |
| |
| Answer: 77",185&width=263&top_left_y=393&top_left_x=905),medium |
| febb550c5c48,"Four, (50 points) Let $T$ be the set of all positive divisors of $2020^{100}$, and set $S$ satisfies: |
| (1) $S$ is a subset of $T$; |
| (2) No element in $S$ is a multiple of another element in $S$. |
| Find the maximum number of elements in $S$.",10201$.,medium |
| 83bff1d6dd2f,"Example 2. Compute the integral |
| |
| $$ |
| \int_{C}\left(z^{2}+z \bar{z}\right) d z |
| $$ |
| |
| where $C-$ is the arc of the circle $\{z \mid=1(0 \leqslant \arg z \leqslant \pi)$.",See reasoning trace,medium |
| 6152342151c7,"3. Let's call a number $x$ semi-integer if the number $2x$ is an integer. The semi-integer part of a number $x$ is defined as the greatest semi-integer number not exceeding $x$, and we will denote it as $] x[$. Solve the equation $x^{2} + 2 \cdot ] x[ = 6$. (20 points)","$\sqrt{3},-\sqrt{14}$",medium |
| 195bf373ac14,"A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white? |
| $\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3}$",\textbf{(D),easy |
| 8d04bd2aad56,"Matilda counted the birds that visited her bird feeder yesterday. She summarized the data in the bar graph shown. The percentage of birds that were goldfinches is |
| (A) $15 \%$ |
| (B) $20 \%$ |
| (C) $30 \%$ |
| (D) $45 \%$ |
| (E) $60 \%$ |
|
|
| ",(C),easy |
| b9d5c26e3669,"Example 11 Let real numbers $a, b$ satisfy $0<a<1,0<b<1$, and $ab=\frac{1}{36}$, find the minimum value of $u=\frac{1}{1-a}+\frac{1}{1-b}$. (Example 4 from [2])","\frac{1}{3}, u=\frac{1}{1-a}+$ $\frac{1}{1-b}=\frac{1^{2}}{1-a}+\frac{1^{2}}{1-b} \geqslant \frac{(1",easy |
| 383665714058,"Find all natural numbers $n> 1$ for which the following applies: |
| The sum of the number $n$ and its second largest divisor is $2013$. |
|
|
| (R. Henner, Vienna)",n = 1342,medium |
| 73e7cac0ce67,"The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers? |
| $\text{(A)}\ 162 \qquad \text{(B)}\ 172 \qquad \text{(C)}\ 173 \qquad \text{(D)}\ 174 \qquad \text{(E)}\ 222$",\text{(A),easy |
| b5d80e2d3d89,"In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG + JH + CD$? |
|
|
| $\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10}$",\mathbf{(D),medium |
| e4e1bc110689,"7. Given the sets |
| $$ |
| \begin{array}{l} |
| A=\left\{(x, y) \mid x=m, y=-3 m+2, m \in \mathbf{Z}_{+}\right\}, \\ |
| B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\} . |
| \end{array} |
| $$ |
|
|
| Then the number of integers $a$ such that $A \cap B \neq \varnothing$ is $\qquad$.",See reasoning trace,medium |
| 8cc9f7fcc369,3. The solution set of the inequality $\sin x \cdot|\sin x|>\cos x \cdot|\cos x|$ is . $\qquad$,See reasoning trace,easy |
| a1bb851bd7c8,"Zaslavsky A.A. |
|
|
| Two ants each crawled along their own closed path on a $7 \times 7$ board. Each ant only crawled along the sides of the cells and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ants crawled along?",16 sides,easy |
| 4fee866f30ca,"1. (Option 1) The decimal representation of the natural number $n$ contains sixty-three digits. Among these digits, there are twos, threes, and fours. No other digits are present. The number of twos is 22 more than the number of fours. Find the remainder when $n$ is divided by 9.",. 5,easy |
| 3413a52f4583,Let $1 = d_1 < d_2 < ...< d_k = n$ be all natural divisors of the natural number $n$. Find all possible values of the number $k$ if $n=d_2d_3 + d_2d_5+d_3d_5$.,"k \in \{8, 9\",medium |
| 502cc25a30ce," |
|
|
| Determine all natural numbers $n$ for which the expression |
|
|
| $$ |
| T(n)=n^{2}+(n+1)^{2}+(n+2)^{2}+(n+3)^{2} |
| $$ |
|
|
| is divisible by 10 without a remainder.","1$ must hold if $T(n) \equiv 0(\bmod 5)$ is to be true. Therefore, the natural numbers $n$ are of th",medium |
| 92257a43b4fc,"13.355. A car, having traveled a distance from $A$ to $B$, equal to 300 km, turned back and after 1 hour 12 minutes from leaving $B$, increased its speed by 16 km/h. As a result, it spent 48 minutes less on the return trip than on the trip from $A$ to $B$. Find the original speed of the car.",60 km/h,easy |
| ead22e2d1398,7.103. $\frac{\lg (2 x-19)-\lg (3 x-20)}{\lg x}=-1$.,10,easy |
| efa8b16ee6cd,"2. Let $O=(0,0), Q=(13,4), A=(a, a), B=(b, 0)$, where $a$ and $b$ are positive real numbers with $b \geq a$. The point $Q$ is on the line segment $A B$. |
| (a) Determine the values of $a$ and $b$ for which $Q$ is the midpoint of $A B$. |
| (b) Determine all values of $a$ and $b$ for which $Q$ is on the line segment $A B$ and the triangle $O A B$ is isosceles and right-angled. |
| (c) There are infinitely many line segments $A B$ that contain the point $Q$. For how many of these line segments are $a$ and $b$ both integers?",See reasoning trace,medium |
| 7f81fc7fbbdf,Example 6 Find the value of $\left(\frac{7}{3}\right)^{999} \sqrt{\frac{3^{1998}+15^{1998}}{7^{1998}+35^{1998}}}$.,\left(\frac{7}{3}\right)^{999} \sqrt{\frac{3^{1998}\left(1+5^{1998}\right)}{7^{1998}\left(1+5^{1998},easy |
| 761a41342df9,"3. The tetrahedron S-ABC is empty, with three pairs of edges being equal, sequentially $\sqrt{34}, \sqrt{41}, 5$. Then the volume of the tetrahedron is ( ). |
| (A) 20 |
| (B) $10 \sqrt{7}$ |
| (C) $20 \sqrt{3}$ |
| (D) 30",See reasoning trace,easy |
| 7f5169ec7731,"Find an integer $x$ such that $x \equiv 2(\bmod 3), x \equiv 3(\bmod 4)$ and $x \equiv 1(\bmod 5)$.","11$. In particular, it follows that $x = 11$ satisfies simultaneously the three equations $x \equiv ",medium |
| 12ed6085c36b,"1. (2 points) Among six different quadratic trinomials that differ by the permutation of coefficients, what is the maximum number that can have two distinct roots",23>0$.,easy |
| c53c8f1be659,"28. As shown in the figure, the edge length of the large cube is 2 cm, and the edge lengths of the two smaller cubes are both 1 cm. Therefore, the total surface area (including the base) of the combined solid figure is $\qquad$ square cm.",32,easy |
| a1a75f24960f,"In the cube $A B C D$ $A_{1} B_{1} C_{1} D_{1}$ with edge length $a\left(a \in \mathbf{R}_{+}\right)$ as shown in Figure 4, $E$ is the center of the square $A B B_{1} A_{1}$, and $F, G$ are points moving along the edges $B_{1} C_{1}$ and $D D_{1}$, respectively. Question: What is the maximum value of the area of the projection of the spatial quadrilateral DEFG onto the six faces of the cube?",See reasoning trace,medium |
| 19658b12ece5,"Three, (50 points) Find the maximum value of the positive integer $r$ such that: for any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ elements in common. |
|
|
| Find the maximum value of the positive integer $r$ such that: for any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ elements in common.",See reasoning trace,medium |
| 13be54c7cbb9,Example 1 Solve the congruence equation $x^{8} \equiv 41(\bmod 23)$.,See reasoning trace,medium |
| 7c6b50cc4a3b,"1. Given non-empty set $A=\{x \mid m+1 \leqslant x \leqslant 2 m-1\}, B=\left\{x \mid x^{2}-2 x-15 \leqslant 0\right\}$, and $A \subseteq B$, then the range of real number $m$ is $\qquad$ .","(x+3)(x-5) \leqslant 0 \Rightarrow B=[-3,5]$, thus $\left\{\begin{array}{l}m+1 \leqslant 2 m-1, \\ m",easy |
| 47e11c687845,"Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$? |
|
|
| ${ \textbf{(A)}\ \dfrac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \dfrac52\qquad\textbf{(C)}\ \dfrac{3+\sqrt{5}}{2}\qquad\textbf{(D)}}\ \dfrac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3 $",\frac{3 + \sqrt{5,medium |
| 348c48b82e27,37th Putnam 1976,"p(x) - p(x-1). If p(x) is of order n with leading coefficient a x n , then Δp(x) is of order n-1 wit",medium |
| e0c0466931ac,"3. Two people simultaneously step onto an escalator from opposite ends, which is moving downward at a speed of $u=1.5 \mathrm{~m} / \mathrm{s}$. The person moving downward has a speed of $v=3 \mathrm{~m} / \mathrm{s}$ relative to the escalator, while the person moving upward has a speed of $2 v / 3$ relative to the escalator. At what distance from the bottom of the escalator will they meet? The length of the escalator is $l=100 \mathrm{~m}$.",See reasoning trace,easy |
| 99dd15cfe6ea,"Example 3 (Question from the 2nd ""Hope Cup"" Invitational Competition) If sets $M$ and $N$ each contain $m$ and $n$ elements, respectively, then the number of possible mappings from $M$ to $N$ is (). |
| A. $m+n$ |
| B. $m \cdot n$ |
| C. $m^{n}$ |
| D. $n^{m}$","\left\{a_{1}, a_{2}, \cdots, a_{m}\right\}, N=\left\{b_{1}, b_{2}, \cdots, b_{n}\right\}$, then $f\l",easy |
| 78f3675ecabc,Example 2 Let the perfect square $y^{2}$ be the sum of the squares of 11 consecutive integers. Then the minimum value of $|y|$ is $\qquad$ .,),medium |
| bfa8ca3e2c36,,x ; \quad x=9 \mathrm{~km}$.,medium |
| 2f95fff05b05,"12. Let the equation $x y=6(x+y)$ have all positive integer solutions $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, then $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=$ |
| $\qquad$ .","(7,42),(8,24),(9,18),(10,15),(12,12),(15,10),(18,9),(24,8),(42,7)$. So $\sum_{k=1}^{n}\left(x_{k}+y_",medium |
| 26ab0006ed03," |
|
|
| Call a set of 3 distinct elements which are in arithmetic progression a trio. What is the largest number of trios that can be subsets of a set of $n$ distinct real numbers? |
|
|
| |
| fdbb8f40ce3f,"4. Choose any two numbers from $2,4,6,7,8,11,12,13$ to form a fraction. Then, there are $\qquad$ irreducible fractions among these fractions.",See reasoning trace,easy |
| b452a3406107,"B. If $k$ numbers are chosen from 2, $, 8, \cdots, 101$ these 34 numbers, where the sum of at least two of them is 43, then the minimum value of $k$ is: $\qquad$",See reasoning trace,medium |
| 6e3854bfb012,"In a large hospital with several operating rooms, ten people are each waiting for a 45 minute operation. The first operation starts at 8:00 a.m., the second at 8:15 a.m., and each of the other operations starts at 15 minute intervals thereafter. When does the last operation end? |
| (A) 10:15 a.m. |
| (B) 10:30 a.m. |
| (C) 10:45 a.m. |
| (D) 11:00 a.m. |
| (E) 11:15 a.m.",See reasoning trace,easy |
| 9f570fa2b41b,"12. The Musketeers' Journey. The distance between Athos and Aramis, riding on the road, is 20 leagues. In one hour, Athos travels 4 leagues, and Aramis - 5 leagues. What distance will be between them after an hour?",s are: 29 leagues; 19 leagues; 21 leagues,easy |
| dd132a6069f3,"4. Find all values of the parameter $a$ for which the equation |
| |
| $$ |
| \left(\sqrt{6 x-x^{2}-4}+a-2\right)((a-2) x-3 a+4)=0 |
| $$ |
| |
| has exactly two distinct real roots. |
| |
| (P. Alishev)",See reasoning trace,medium |
| 5d285d119516,"A triangle has sides $a, b, c$, and its area is $\frac{(a+b+c)(a+b-c)}{4}$. What is the measure of the largest angle of the triangle?",See reasoning trace,medium |
| b3dd9f45e8ef,"Question 15: Let the set $M=\{1,2, \ldots, 100\}$ be a 100-element set. If for any n-element subset $A$ of $M$, there are always 4 elements in $A$ that are pairwise coprime, find the minimum value of $\mathrm{n}$.","13$ numbers must come from these four subsets. By the pigeonhole principle, at least one of these su",medium |
| 6e7ce8a32a83,"6. (10 points) There are 12 students playing a card game, with 4 participants each time, and any 2 students can participate together at most 1 time. How many times can they play $\qquad$ at most.",: 9,medium |
| 5e0c962504a0,"Raskina I.V. |
| |
| At the exchange office, two types of operations are carried out: |
| |
| 1) give 2 euros - get 3 dollars and a candy as a gift; |
| 2) give 5 dollars - get 3 euros and a candy as a gift. |
| |
| When the wealthy Pinocchio came to the exchange office, he only had dollars. When he left, he had fewer dollars, no euros appeared, but he received 50 candies. How much did such a ""gift"" cost Pinocchio in dollars?",10 dollars,easy |
| b9eec7786bd4,"3.1.1. (12 points) Calculate $\sin (\alpha+\beta)$, if $\sin \alpha+\cos \beta=\frac{1}{4}$ and $\cos \alpha+\sin \beta=-\frac{8}{5}$. Answer: $\frac{249}{800} \approx 0.31$.","\frac{1049}{400}$, i.e., $\sin (\alpha+\beta)=\frac{249}{800} \approx 0.31$.",easy |
| 29a63a1dc2a4,"Let's determine the first term and the common difference of an arithmetic progression, if the sum of the first $n$ terms of this progression is $\frac{n^{2}}{2}$ for all values of $n$.",See reasoning trace,medium |
| 27e135acdfc8,8.200. $\frac{3(\cos 2 x+\operatorname{ctg} 2 x)}{\operatorname{ctg} 2 x-\cos 2 x}-2(\sin 2 x+1)=0$.,"$x=(-1)^{k+1} \frac{\pi}{12}+\frac{\pi k}{2}, k \in Z$",medium |
| 24d7d5446bde,The straight line $y=ax+16$ intersects the graph of $y=x^3$ at $2$ distinct points. What is the value of $a$?,12,medium |
| 9d90aadcdb7b,"12 bags contain 8 white balls and 2 red balls. Each time, one ball is randomly taken out, and then 1 white ball is put back. The probability that exactly all red balls are taken out by the 4th draw is $\qquad$ .",See reasoning trace,easy |
| caba5fbdd648,"18. Let the variable $x$ satisfy the inequality: $x^{2}+b x \leqslant-x(b<-1)$, and the minimum value of $f(x)=x^{2}+b x$ is $-\frac{1}{2}$, find the value of the real number $b$.",-\frac{3}{2}$.,medium |
| 1a4a1094733e,"1. In the village of Big Vasyuki, there is a riverside promenade 50 meters long, running along the river. A boat 10 meters long passes by it in 5 seconds when traveling downstream and in 4 seconds when traveling upstream. How many seconds will it take for a paper boat to float from one end of the promenade to the other?",$33 \frac{1}{3}$ sec,easy |
| a3f053ac9f58,"10. A rectangular floor measuring 17 feet by 10 feet is covered with 170 square tiles, each $1 \times 1$ square foot. A bug crawls from one corner to the opposite corner. Starting from the first tile it leaves, how many tiles does it cross in total? |
| (A) 17 |
| (B) 25 |
| (C) 26 |
| (D) 27 |
| (E) 28",See reasoning trace,easy |
| 679077ee8f0e,"Example 4 In Figure 1, there are 8 vertices, each with a real number. The real number at each vertex is exactly the average of the numbers at the 3 adjacent vertices (two vertices connected by a line segment are called adjacent vertices). Find |
| $$ |
| a+b+c+d-(e+f+g+h) |
| $$",See reasoning trace,medium |
| 89765e2d6d1c,"10. A convex octagon has 8 interior angles. |
| |
| How many of them, at most, can be right angles? |
| (A) 4 |
| (B) 2 |
| (C) 3 |
| (D) 5 |
| (E) 1",is $(C)$,medium |
| e03fb5eeef15,"8. Given $x, y \in [0,+\infty)$. Then the minimum value of $x^{3}+y^{3}-5 x y$ is $\qquad$ .","y=\frac{5}{3}$, the minimum value of the original expression is $-\frac{125}{27}$.",medium |
| ac001a228a8c,"8・165 Given a sequence of natural numbers $\left\{x_{n}\right\}$ that satisfies |
| $$ |
| x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, n=1,2,3, \cdots |
| $$ |
| |
| If one of the terms in the sequence is 1000, what is the smallest possible value of $a+b$?",See reasoning trace,medium |
| ec0d934164df,"4. Given that point $P(x, y)$ lies on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$, find the maximum value of $2 x-y$. |
| |
| untranslated text remains the same as the",See reasoning trace,easy |
| 8586b8164bdc,How many positive integers of at most $n$ digits are there whose square ends in the same $n$ digits (possibly with some leading zeros)?,See reasoning trace,medium |
| 76a0194fc920,"23. It is possible to choose, in two different ways, six different integers from 1 to 9 inclusive such that their product is a square. Let the two squares so obtained be $p^{2}$ and $q^{2}$, where $p$ and $q$ are both positive. What is the value of $p+q$ ? |
| A 72 |
| B 84 |
| C 96 |
| D 108 |
| E 120",36+72=108$.,medium |
| 67e3509070f4,"3. On the side $AB$ of an acute-angled triangle $ABC$, a point $M$ is marked. A point $D$ is chosen inside the triangle. Circles $\omega_{A}$ and $\omega_{B}$ are circumscribed around triangles $AMD$ and $BMD$ respectively. The side $AC$ intersects the circle $\omega_{A}$ again at point $P$, and the side $BC$ intersects the circle $\omega_{B}$ again at point $Q$. The ray $PD$ intersects the circle $\omega_{B}$ again at point $R$, and the ray $QD$ intersects the circle $\omega_{A}$ again at point $S$. Find the ratio of the areas of triangles $ACR$ and $BCS$.",1,medium |
| 810f27ecbda9,"6. Xiao Hu placed chess pieces on the grid points of a $19 \times 19$ Go board, first forming a solid rectangular array. Then, by adding 45 more chess pieces, he formed a larger solid rectangular array with one side unchanged. How many chess pieces did Xiao Hu use at most? |
| (A) 285 |
| (B) 171 |
| (C) 95 |
| (D) 57",285$.,easy |
| ae8c04d86582,"$19 \cdot 28$ convex polygon, the interior angles less than $120^{\circ}$ cannot be more than |
| (A) 2. |
| (B) 3. |
| (C) 4. |
| (D) 5. |
| (China Zhejiang Ningbo Junior High School Mathematics Competition, 1987)",$(D)$,easy |
| 138c6aafff9b,"$17^{2}-15^{2}$ equals |
| (A) $8^{2}$ |
| (B) $2^{2}$ |
| (C) $4^{2}$ |
| (D) $7^{2}$ |
| (E) $6^{2}$",289-225=64=8^{2}$.,easy |
| 5393d8a6d6c3,". Find all polynomials $P(x)$ with real coefficients that satisfy the equality $P(a-$ $b)+P(b-c)+P(c-a)=2 P(a+b+c)$ for all real numbers $a, b, c$ such that $a b+b c+c a=0$.","p-2 q, v=q$). A known argument then shows that $Q$ is $C^{\infty}$ on $\mathbb{R}_{+}$ (for fixed $u",medium |
| 123e082b7918,"Example 1 The table below gives an ""arithmetic array"": |
| where each row and each column are arithmetic sequences, $a_{i j}$ represents the number located at the $i$-th row and $j$-th column. |
| (I) Write down the value of $a_{45}$; |
| (II) Write down the formula for $a_{i i}$; |
| (III) Prove: A positive integer $N$ is in this arithmetic array if and only if $2 N+1$ can be factored into the product of two positive integers that are not 1.",See reasoning trace,medium |
| 70452f58950c,"Example 2 If the function |
| $$ |
| y=\frac{1}{2}\left(x^{2}-100 x+196+\left|x^{2}-100 x+196\right|\right), |
| $$ |
| |
| then when the independent variable $x$ takes the natural numbers $1,2, \cdots, 100$, the sum of the function values is ( ). |
| (A) 540 |
| (B) 390 |
| (C) 194 |
| (D) 97 |
| (1999, National Junior High School Mathematics Competition)",(B),easy |
| 3ec4e89c9d58,"[ Fermat's Little Theorem ] |
| |
| Find the remainder when $3^{102}$ is divided by 101. |
| |
| #",See reasoning trace,easy |
| 7bfa08cce858,"9. (16 points) Given the sequence $\left\{a_{n}\right\}$ satisfies |
| $$ |
| a_{1}=\frac{1}{3}, \frac{a_{n-1}}{a_{n}}=\frac{2 n a_{n-1}+1}{1-a_{n}}(n \geqslant 2) . |
| $$ |
| |
| Find the value of $\sum_{n=2}^{\infty} n\left(a_{n}-a_{n+1}\right)$.",See reasoning trace,medium |
| 65df6c298477,"Let's examine the accuracy of the following approximate construction for the side of a regular nonagon inscribed in circle $k$. |
| |
| The regular hexagon inscribed in $k$ has four consecutive vertices $A, B, C, D$. The intersection point of the circle centered at $B$ with radius $BA$ and the circle centered at $D$ with radius $DB$ inside $k$ is $G$. The intersection point of line $BG$ and line $AC$ is $H$. Then the side of the nonagon is approximated by the length of segment $BH$.",See reasoning trace,medium |
| 39122b7e4e01,"$17 \cdot 81$ A certain triangle has a base of 80, one base angle of $60^{\circ}$, and the sum of the other two sides is 90. Then the shortest side is |
| (A) 45. |
| (B) 40. |
| (C) 36. |
| (D) 17. |
| (E) 12. |
| (10th American High School Mathematics Examination, 1959)",$(D)$,easy |
| 7f3e7aa5b04e,"## |
| |
| Calculate the limit of the function: |
| |
| $\lim _{x \rightarrow 0} \frac{6^{2 x}-7^{-2 x}}{\sin 3 x-2 x}$",See reasoning trace,medium |
| 3bd6681c87ea,"5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid in the form of a circle with a radius of $R=5$ m. The wagon starts from rest, with the jet force having a constant value. What is the maximum speed the wagon will reach after one full circle, if its acceleration over this period should not exceed $a=1 \mathrm{M} / \mathrm{c}^{2} ?$",$\approx 2,medium |
| beb048634f04,"For some positive integer $n$, a coin will be flipped $n$ times to obtain a sequence of $n$ heads and tails. For each flip of the coin, there is probability $p$ of obtaining a head and probability $1-p$ of obtaining a tail, where $0<p<1$ is a rational number. |
| Kim writes all $2^n$ possible sequences of $n$ heads and tails in two columns, with some sequences in the left column and the remaining sequences in the right column. Kim would like the sequence produced by the coin flips to appear in the left column with probability $1/2$. |
| Determine all pairs $(n,p)$ for which this is possible.","(n, p) = (n, \frac{1",medium |
| a18727c26892,"I1.1 Let $a$ be a real number and $\sqrt{a}=\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}}$. Find the value of $a$. |
| I1.2 In Figure 1, the straight line $\ell$ passes though the point $(a, 3)$, and makes an angle $45^{\circ}$ with the $x$-axis. If the equation of $\ell$ is $x+m y+n=0$ and $b=|1+m+n|$, find the value of $b$. |
| |
| I1.3 If $x-b$ is a factor of $x^{3}-6 x^{2}+11 x+c$, find the value of $c$. |
| I1.4 If $\cos x+\sin x=-\frac{c}{5}$ and $d=\tan x+\cot x$, find the value of $d$.",(\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}})^{2} \\ a=7+\sqrt{13}-2 \sqrt{7^{2}-\sqrt{13}^{2}}+7-\sqrt{13,medium |
| 6bebb2a8a24b,"7. Let |
| $$ |
| f(x)=x^{4}-6 x^{3}+26 x^{2}-46 x+65 \text {. } |
| $$ |
| |
| Let the roots of $f(x)$ be $a_{k}+i b_{k}$ for $k=1,2,3,4$. Given that the $a_{k}, b_{k}$ are all integers, find $\left|b_{1}\right|+\left|b_{2}\right|+\left|b_{3}\right|+\left|b_{4}\right|$.",a_{2}$ and $b_{1}=-b_{2} ; a_{3}=a_{4}$ and $b_{3}=-b_{4}$. The constant term of $f(x)$ is the produ,medium |
| a0fb81a1d5ee,"For example, a number in the form of $42 \cdots$ multiplied by 2, with 42 moved to the end, find this number.","42, k=2, c=2, \omega=\frac{x}{10^{2}-2}$ $=\frac{42}{98}=0.42857^{\circ} \mathrm{i}$, therefore, the",easy |
| 7083a595e6de,"23. (IND 2) Let $f$ and $g$ be two integer-valued functions defined on the set of all integers such that |
| (a) $f(m+f(f(n)))=-f(f(m+1)-n$ for all integers $m$ and $n$; |
| (b) $g$ is a polynomial function with integer coefficients and $g(n)=g(f(n))$ for all integers $n$. |
| Determine $f(1991)$ and the most general form of $g$.","g(-n-1)$ for all integers $n$. Since $g$ is a polynomial, it must also satisfy $g(x)=g(-x-1)$ for al",medium |
| a97bbc5904c9,"9. In the figure below, $A$ is the midpoint of $DE$, the areas of $\triangle D C B$ and $\triangle E B C$ satisfy $S_{\triangle D C B}+S_{\triangle E B C}=12$, then the area of $\triangle A B C$ is $\qquad$ .",See reasoning trace,easy |
| fe7c33217106,"Ester goes to a stationery store to buy notebooks and pens. In this stationery store, all notebooks cost $\mathrm{R} \$ 6.00$. If she buys three notebooks, she will have R \$4.00 left. If, instead, her brother lends her an additional $\mathrm{R} \$ 4.00$, she will be able to buy two notebooks and seven pens, all the same. |
| |
| (a) How much does each pen cost? |
| |
| (b) If she buys two notebooks and does not borrow money, how many pens can Ester buy?","22 - 12 = 10$ reais left. Since each pen costs 2 reais, she can buy $10 \div 2 = 5$ pens.",medium |
| b0f1b5606972,Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.,"(1, 1), (1, 2), (2, 3), (2, 1), (3, 2)",medium |
| bc755bf9c964,"## Task B-2.2. |
| |
| In triangle $ABC$, the measures of the angles at vertex $A$ and vertex $C$ are $\alpha=60^{\circ}$ and $\gamma=75^{\circ}$, respectively. Calculate the distance from the orthocenter of triangle $ABC$ to vertex $B$ if $|BC|=8\sqrt{6}$.","\frac{|HD|}{|BH|}$, or $|BH| = \frac{8}{\frac{1}{2}} = 16$.",medium |
| 7efd28d088fa,Example 2. Find the derivative of the function $y=\sqrt{1-x^{2}} \arccos x$.,See reasoning trace,easy |
| 728a00b10023,"The Bank of Oslo produces coins made of aluminum (A) and bronze (B). Martin arranges $2 n$ coins, $n$ of each type, in a line in an arbitrary order. Then he fixes $k$ as an integer between 1 and $2 n$ and applies the following process: he identifies the longest sequence of consecutive coins of the same type that contains the $k$-th coin from the left, and moves all the coins in this sequence to the left of the line. For example, with $n=4, k=4$, we can have the sequence of operations |
| |
| $$ |
| A A B \underline{B} B A B A \rightarrow B B B \underline{A} A A B A \rightarrow A A A \underline{B} B B B A \rightarrow B B B \underline{B} A A A A \rightarrow \ldots |
| $$ |
| |
| Find all pairs $(n, k)$ with $1 \leqslant k \leqslant 2 n$ such that for any initial configuration, the $n$ coins on the left are of the same type after a finite number of steps.",See reasoning trace,medium |
| b7a690b87558,"Example 20 The Fibonacci numbers are defined as |
| $$ |
| a_{0}=0, a_{1}=a_{2}=1, a_{n+1}=a_{n}+a_{n-1} \quad(n \geqslant 1) . |
| $$ |
| |
| Find the greatest common divisor of the 1960th and 1988th terms.",See reasoning trace,medium |
| 6b2b42697f75,"3. (10 points) A barrel of oil, the oil it contains is $\frac{3}{5}$ of the barrel's full capacity. After selling 18 kilograms, 60% of the original oil remains. Therefore, this barrel can hold $\qquad$ kilograms of oil.",This oil drum can hold 75 kilograms of oil,medium |
| 4c260e72e0fa,"2. Solve the equation |
| |
| $$ |
| 2^{\sqrt[12]{x}}+2^{\sqrt[4]{x}}=2 \cdot 2^{\sqrt[6]{x}} |
| $$",0$ and $x_{2}=1$.,medium |
| 5ee4b76c09a6,11.226 The base of the pyramid $SABC$ is a triangle $ABC$ such that $AB=AC=10$ cm and $BC=12$ cm. The face $SBC$ is perpendicular to the base and $SB=SC$. Calculate the radius of the sphere inscribed in the pyramid if the height of the pyramid is 1.4 cm.,$\frac{12}{19}$ cm,medium |
| 8bf4ba7aeefd,"Task B-2.4. In a kite where the lengths of the diagonals are $d_{1}=24 \mathrm{~cm}$ and $d_{2}=8 \mathrm{~cm}$, a rectangle is inscribed such that its sides are parallel to the diagonals of the kite. Determine the dimensions of the inscribed rectangle that has the maximum area.",4 \text{ cm}$ and $y=12 \text{ cm}$.,medium |
| 5a1d66e79b7d,"In a book written in 628 AD, there is also such a",255,easy |
| 8b0ead3ee2be,"4. A student is given a budget of $\$ 10000$ to produce a rectangular banner for a school function. The length and width (in metres) of the banner must be integers. If each metre in length costs $\$ 330$ while each metre in width costs $\$ 450$, what is the maximum area (in $\mathrm{m}^{2}$ ) of the banner that can be produced? |
| (1 mark) |
| 一名學生要為學校的活動製作一幅長方形的宣傳橫額, 開支上限為 $\$ 10000$ 。橫額的長度和闊度（以米為單位）必須是整數。若製作橫額每長一米收費 $\$ 330$, 每闊一米收費 $\$ 450$, 問可製作的橫額的面積最大是多少平方米？",15$ and $y=11$ satisfies all conditions and give an area of $165 \mathrm{~m}^{2}$. Thus the maximum ,medium |
| de53df214801,"5. Given the sets |
| $$ |
| \begin{array}{l} |
| M_{1}=\{x \mid(x+2)(x-1)>0\}, \\ |
| M_{2}=\{x \mid(x-2)(x+1)>0\} . |
| \end{array} |
| $$ |
| |
| Then the set equal to $M_{1} \cup M_{2}$ is ( ). |
| (A) $\left\{x \mid\left(x^{2}-4\right)\left(x^{2}-1\right)>0\right\}$ |
| (B) $\left\{x \mid\left(x^{2}+4\right)\left(x^{2}-1\right)>0\right\}$ |
| (C) $\left\{x \mid\left(x^{2}-4\right)\left(x^{2}+1\right)>0\right\}$ |
| (D) $\left\{x \mid\left(x^{2}+4\right)\left(x^{2}+1\right)>0\right\}$",See reasoning trace,easy |
| 865cf6d90f57,"## Task A-2.1. |
| |
| Determine all pairs of real numbers $(x, y)$ that satisfy the system |
| |
| $$ |
| \begin{aligned} |
| & x+y^{2}=y^{3} \\ |
| & y+x^{2}=x^{3} |
| \end{aligned} |
| $$","1, y=1$ and $x+y=0$.",medium |
| 6452eeccd22e,"Example 3 Let $a \leqslant b<c$ be the side lengths of a right-angled triangle. Find the maximum constant $M$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geqslant \frac{M}{a+b+c}$. |
| (1991 China National Training Team Test)","b=1, c=\sqrt{2}$, $I=(2+\sqrt{2})\left(2+\frac{\sqrt{2}}{2}\right)=5+3 \sqrt{2}$, hence $M=5+3 \sqrt",medium |
| bd1ba878b3e7,"$33 \cdot 72$ If $D(n)$ represents the number of pairs of adjacent different digits in the binary representation of a positive integer $n$. For example, $D(3)=D\left(11_{2}\right)=0, D(21)=D\left(10101_{2}\right)=4$, $D(97)=D\left(1100001_{2}\right)=2$, etc. The number of positive integers $n$ not greater than 97 that satisfy $D(n)=2$ is |
| (A) 16. |
| (B) 20. |
| (C) 26. |
| (D) 30. |
| (E) 35. |
| (48th American High School Mathematics Examination, 1997)",$(C)$,medium |
| 2e4931e103ee,1. Randomly rolling three identical cubic dice. The probability that the minimum value among the three dice is 3 is $\qquad$,\frac{37}{216}$.,easy |
| fb48368c172f,"4. Find all rational numbers $r$ and all integers $k$ that satisfy the equation |
| |
| $$ |
| r(5 k-7 r)=3 |
| $$",See reasoning trace,medium |
| 7a4a3cfa5927,"1. There are 70 apples in two boxes. Taking 6 apples from the first box and putting them into the second box makes the number of apples in both boxes equal. Originally, there were $\qquad$ apples in the first box.",See reasoning trace,easy |
| 150cf43e2e3f,3. The maximum value of the function $f(x)=\lg 2 \cdot \lg 5-\lg 2 x \cdot \lg 5 x$ is,"-\frac{1}{2}$, the maximum value of $f(x)$ is $\frac{1}{4}$.",easy |
| 90351ba037ab,"3. Write $\mathbf{2 0 1 2}$ as the sum of $N$ distinct positive integers, the maximum value of $N$ is $\qquad$","(62+1) \times 62 \div 2=1953<2012$, thus $N=62$",easy |
| e7022a9cd895,"(5) Let the three-digit number $n=\overline{a b c}$, if the lengths of the sides of a triangle can be formed by $a, b, c$ to constitute an isosceles (including equilateral) triangle, then the number of such three-digit numbers $n$ is ( ). |
| (A) 45 |
| (B) 81 |
| (C) 165 |
| (D) 216",C,medium |
| 7a94fe530d98,"In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.",20,medium |
| 13cf01d71c9c,"1. Solve the equation |
| |
| $$ |
| (\sqrt{3-\sqrt{8}})^{x}+(\sqrt{3+\sqrt{8}})^{x}=6 |
| $$",2$ and $x_{2}=-2$.,medium |
| 8cfede3c4faf,"Around a circular table, 18 girls are sitting, 11 dressed in blue and 7 dressed in red. Each of them is asked if the girl to their right is dressed in blue, and each one answers yes or no. It is known that a girl tells the truth only when her two neighbors, the one on the right and the one on the left, are wearing clothes of the same color. How many girls will answer yes? If there is more than one possibility, state all of them.","s yes, and if they both wear red, she answers no, because in these cases she tells the truth",medium |
| 95adcfde52ff,"7. It is known that the number $a$ satisfies the equation param1, and the number $b$ satisfies the equation param2. Find the smallest possible value of the sum $a+b$. |
| |
| | param1 | param2 | | |
| | :---: | :---: | :--- | |
| | $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-6 x^{2}+14 x+2=0$ | | |
| | $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}+6 x^{2}+15 x+27=0$ | | |
| | $x^{3}-6 x^{2}+16 x-28=0$ | $x^{3}+3 x^{2}+7 x+17=0$ | | |
| | $x^{3}+6 x^{2}+17 x+7=0$ | $x^{3}-3 x^{2}+8 x+5=0$ | |",0$ | $x^{3}-3 x^{2}+8 x+5=0$ | -1 |,easy |
| bb64b70a3c3f,"14 Labeled as $1,2, \cdots, 100$, there are some matches in the matchboxes. If each question allows asking about the parity of the sum of matches in any 15 boxes, then to determine the parity of the number of matches in box 1, at least how many questions are needed?",s is the same as the parity of $a_{1}$ (the other boxes each appear exactly twice in the three questions),medium |
| 523db0501e5e,"## 128. Math Puzzle $1 / 76$ |
| |
| A circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given. |
| |
| a) How many program leaflets were printed if one fourth of the visitors bought a leaflet? |
| |
| b) How many Marks were additionally collected from the entrance fees for the animal show if it was visited by half of the visitors and the entrance fee was 0.30 M?","15000$ and $30 \cdot 15000=450000$; from the animal show, €450,000 was earned.",easy |
| d15e1fa861c3,"## |
| |
| Calculate the definite integral: |
| |
| $$ |
| \int_{0}^{1} \frac{(4 \sqrt{1-x}-\sqrt{x+1}) d x}{(\sqrt{x+1}+4 \sqrt{1-x})(x+1)^{2}} |
| $$",See reasoning trace,medium |
| f4e814f9eb27,"10-8. In trapezoid $A B C D$, the bases $A D$ and $B C$ are 8 and 18, respectively. It is known that the circumcircle of triangle $A B D$ is tangent to the lines $B C$ and $C D$. Find the perimeter of the trapezoid.",56,medium |
| 5c2f5e2c9878,"Task 3. (15 points) The bases $AB$ and $CD$ of trapezoid $ABCD$ are equal to 367 and 6, respectively, and its diagonals are perpendicular to each other. Find the scalar product of vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.",. 2202,medium |
| 90ec44f8e421,"At a dance party a group of boys and girls exchange dances as follows: The first boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, then: |
| $\textbf{(A)}\ b = g\qquad \textbf{(B)}\ b = \frac{g}{5}\qquad \textbf{(C)}\ b = g - 4\qquad \textbf{(D)}\ b = g - 5\qquad \\ \textbf{(E)}\ \text{It is impossible to determine a relation between }{b}\text{ and }{g}\text{ without knowing }{b + g.}$",g - 4}$,easy |
| 7814ba7d60d9,"9. (6 points) If $\overline{\mathrm{ab}} \times 65=\overline{48 \mathrm{ab}}$, then $\overline{\mathrm{ab}}=$",: 75,easy |
| 13de92bebcb5,"4. In the Cartesian coordinate system, the curve represented by the equation $m\left(x^{2}+y^{2}+2 y\right.$ $+1)=(x-2 y+3)^{2}$ is an ellipse, then the range of $m$ is . |
| (A) $(0,1)$ |
| (B) $(1,+\infty)$ |
| (C) $(0,5)$ |
| (D) $(5,+\infty)$ |
| (Hunan contribution)","\sqrt{\frac{5}{m}}$. This indicates that the ratio of the distance from $(x, y)$ to the fixed point ",easy |
| 0e5c0fb5dbcb,"1. (2000 Hebei Province Competition Question) In the circle $x^{2}+y^{2}-5 x=0$, there are three chords passing through the point $\left(\frac{5}{2}, \frac{3}{2}\right)$ whose lengths form a geometric sequence. Then the range of the common ratio is ( ). |
| A. $\left[\frac{\sqrt{2}}{\sqrt[4]{5}}, \frac{2}{\sqrt{5}}\right]$ |
| B. $\left[\sqrt[3]{\frac{4}{5}}, \frac{2}{\sqrt{5}}\right]$ |
| C. $\left[\frac{2}{\sqrt{5}}, \frac{\sqrt{5}}{2}\right]$ |
| D. $\left[\frac{2}{\sqrt[3]{5}}, \frac{\sqrt{5}}{2}\right]$","0$ is 5, so the maximum length of the chord passing through the point $\left(\frac{5}{2}, \frac{3}{2",easy |
| c5a05cd11ac4,11 positive numbers each equal to the sum of the squares of the other 10. Determine the numbers.,See reasoning trace,medium |
| 0cddbe2271be,"3. Find all pairs $(a, b)$ of real numbers such that whenever $\alpha$ is a root of $x^{2}+a x+b=0$, $\alpha^{2}-2$ is also a root of the equation.",See reasoning trace,medium |
| 9e8e2a377c05,"The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is: |
| $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$",\textbf{(C),easy |
| 69e01b6204e1,"5. In Rt $\triangle A B C$, $\angle C=90^{\circ}, A C=3$, $B C=4$. Then the value of $\cos (A-B)$ is ( ). |
| (A) $\frac{3}{5}$ |
| (B) $\frac{4}{5}$ |
| (C) $\frac{24}{25}$ |
| (D) $\frac{7}{25}$",See reasoning trace,easy |
| 97e54a26aca4,"Example: $10 f(x)$ is a continuous function defined on the interval $[0,2015]$, and $f(0)=f(2015)$. Find the minimum number of real number pairs $(x, y)$ that satisfy the following conditions: |
| (1) $f(x)=f(y)$; |
| (2) $x-y \in \mathbf{Z}_{+}$. |
| $(2015$, Peking University Mathematics Summer Camp)",2015$.,medium |
| 72efff291118,"All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? |
| $\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$",\textbf{(C),easy |
| e49766cb7bbd,"[ Properties of medians. The centroid of a triangle. ] [ The ratio of the areas of triangles with a common angle ] |
| |
| Given triangle $A B C$, the area of which is 2. On the medians $A K, B L$ and $C N$ of triangle $A B C$, points $P, Q$ and $R$ are taken respectively such that $A P: P K=1, B Q: Q L=1: 2, C R: R N=5: 4$. Find the area of triangle $P Q R$.",$\frac{1}{6}$,medium |
| 3619dda4a921,"$$ |
| \left(\frac{1+x}{1-x}-\frac{1-x}{1+x}\right)\left(\frac{3}{4 x}+\frac{x}{4}-x\right)=? |
| $$",See reasoning trace,easy |
| c3c00eca2d93,"3. $\log _{\sin 1} \cos 1, \log _{\sin 1} \tan 1, \log _{\operatorname{cos1}} \sin 1$, from largest to smallest is",See reasoning trace,easy |
| d4764451f925,"2.2.1. A covered football field of rectangular shape with a length of 90 m and a width of 60 m is being designed, which should be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle, the radius of which is equal to the height at which the spotlight is hanging. It is necessary to find the minimum possible ceiling height at which the following conditions are met: every point on the football field is illuminated by at least one spotlight; the ceiling height must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.).",. 27,medium |
| e6142771457d,Example 9. Find the integral $\int \sqrt{a^{2}-x^{2}} d x(a>0)$.,See reasoning trace,medium |
| d0e3a260f9dd,,68 minutes,medium |
| d20746b6b476,"Question 74, Given $a \geq b \geq c \geq d>0, a^{2}+b^{2}+c^{2}+d^{2}=\frac{(a+b+c+d)^{2}}{3}$, find the maximum value of $\frac{a+c}{b+d}$.",\frac{x}{3-x} \leq \frac{\frac{6+\sqrt{6}}{4}}{3-\frac{6+\sqrt{6}}{4}}=\frac{7+2 \sqrt{6}}{5}$. Equa,medium |
| ae2eb33e08bf,"[ Auxiliary area. The area helps to solve the |
| |
| The sides of the triangle are 13, 14, and 15. Find the radius of the circle that has its center on the middle side and touches the other two sides.",6,medium |
| d58916b8b5ff,"## Task Condition |
| |
| Find the derivative. |
| |
| $$ |
| y=\ln \left(e^{x}+\sqrt{1+e^{2 x}}\right) |
| $$",See reasoning trace,medium |
| 32cf81a4b7a6,"The probability that a purchased light bulb will work is 0.95. |
| |
| How many light bulbs need to be bought so that with a probability of 0.99, there will be at least five working ones among them? |
| |
| #", |
| c698b7043f63,"\section*{Exercise 1 - 041211} |
| |
| From a four-digit table, we obtain the following approximate values: |
| |
| \[ |
| \sqrt[3]{636000} \approx 86.00 \text { and } \sqrt[3]{389000} \approx 73.00 |
| \] |
| |
| Therefore, \(z=\sqrt[3]{636000}-\sqrt[3]{389000} \approx 13\). |
| |
| Without using any further table, it should be decided whether \(z\) is greater, smaller, or equal to 13.","12.9986<13\). Such problems are interesting when both sides are equal (as in Problem 041116), becaus",medium |
| 0e663346ec67,"A store normally sells windows at $$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? |
| $\textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D) } 400\qquad \textbf{(E) } 500$",\textbf{(A),easy |
| 3a8080111edf,"6. Given the line $6 x-5 y-28=0$ and |
| |
| the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a>b>0\right.$, and $\left.a^{2} 、 b \in \mathbf{Z}\right)$ |
| intersect at points $A$ and $C$. Let $B(0, b)$ be the upper vertex of the ellipse, and the centroid of $\triangle A B C$ is the right focus $F_{2}$ of the ellipse. Then the equation of the ellipse is $\qquad$.",1$.,medium |
| bf8dbf4bf3d8,"8. 1b.(TUR 5) Find the smallest positive integer $n$ such that (i) $n$ has exactly 144 distinct positive divisors, and (ii) there are ten consecutive integers among the positive divisors of $n$.","$ $2^{\alpha_{1}} 3^{\alpha_{2}} 5^{\alpha_{3}} 7^{\alpha_{4}} 11^{\alpha_{5}} \cdots$, where $\alph",medium |
| 387f906db9e8,"[7] $\triangle A B C$ is right angled at $A . D$ is a point on $A B$ such that $C D=1 . A E$ is the altitude from $A$ to $B C$. If $B D=B E=1$, what is the length of $A D$ ?",$\sqrt[3]{2}-1$,medium |
| bebc91a19569,"## |
| |
| Calculate the indefinite integral: |
| |
| $$ |
| \int \frac{(\arcsin x)^{2}+1}{\sqrt{1-x^{2}}} d x |
| $$",See reasoning trace,medium |
| 1a425b8074db,"17. The number $M=124563987$ is the smallest number which uses all the non-zero digits once each and which has the property that none of the pairs of its consecutive digits makes a prime number. For example, the 5th and 6th digits of $M$ make the number 63 which is not prime. $N$ is the largest number which uses all the non-zero digits once each and which has the property that none of the pairs of its consecutive digits makes a prime number. |
| What are the 5 th and 6 th digits of $N$ ? |
| A 6 and 3 |
| B 5 and 4 |
| C 5 and 2 |
| D 4 and 8 |
| E 3 and 5",987635421$. It follows that the 5 th and 6th digits of $N$ are 3 and 5 .,medium |
| 8c522b0a637b,,See reasoning trace,medium |
| 2852b41d4876,Task B-4.4. Determine all natural numbers $a$ for which the number $a^{3}+1$ is a power of 3.,"1$, and $a^{3}+1=9$ so $a=2$.",easy |
| 8f0494133188,"Example 4 Let $x=b y+c z, y=c z+a x, z=a x$ $+b y$. Find the value of $\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}$.",See reasoning trace,easy |
| 600e72280079,"A team of several workers can complete a task in 7 full days. The same team, without two people, can complete the same task in several full days, and the same team, without six people, can also complete the same task in several full days. How many workers are in the team? (The productivity of the workers is the same.) |
| |
| #",See reasoning trace,easy |
| 16e2d2bd97c3,Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.,296,medium |
| 93607047619b,"13. The sequence $\left\{a_{n} \mid\right.$ is defined as: $a_{0}=0, a_{1}=a_{2}=1$ $a_{n+1}=a_{n}+a_{n-1}(n \in \mathbf{N})$, find the greatest common divisor of $a_{2002}$ and $a_{1998}$.",See reasoning trace,medium |
| e9c1d486f9ca,4. Calculate $\sec \frac{2 \pi}{9}+\sec \frac{4 \pi}{9}+\sec \frac{6 \pi}{9}+\sec \frac{8 \pi}{9}$.,"\frac{2 z\left(3+z^{3}+z^{6}\right)}{-z^{4}-z^{7}}$. Since $z^{3}+z^{6}=-1$, we have $M=\frac{2 z \c",medium |
| d915268736b2,Each of the three cutlets needs to be fried on a pan for five minutes on each side. Only two cutlets fit on the pan. Can all three cutlets be fried in less than 20 minutes (neglecting the time for flipping and moving the cutlets)?,15 minutes,easy |
| 2ae2be929179,"10. (20 points) Given |
| $$ |
| A=\{1,2, \cdots, 2014\} |
| $$ |
| |
| is a non-empty subset, satisfying that the sum of its elements is a multiple of 5. Find the number of such subsets.",See reasoning trace,medium |
| bc2f539b6d6d,"B1. If the polynomial $p$ is divided by the polynomial $2-x$, the quotient is $2 x^{2}-x+3$. Determine the remainder of this division, given that the product of all the zeros of the polynomial $p$ is equal to $\frac{11}{2}$.",See reasoning trace,medium |
| d97851687777,"Let $A$ be the following. |
| |
| A numerical sequence is defined by the conditions: $a_{1}=1, a_{n+1}=a_{n}+\left[\sqrt{a_{n}}\right]$. |
| |
| How many perfect squares occur among the first terms of this sequence, not exceeding |
| |
| 1000000?",1024^{2}>10^{6}\right)$.,easy |
| ee07aee8ecd4,"Let's determine all the numbers $x$ for which the following equation holds: |
| |
| $$ |
| |x+1| \cdot|x-2| \cdot|x+3| \cdot|x-4|=|x-1| \cdot|x+2| \cdot|x-3| \cdot|x+4| |
| $$","|-(x-1)|=|x-1| \), if we have already plotted the left-hand side for the purpose of a graphical solu",medium |
| 51bd9ae1a665,"We unpack 187 books in a box, which weigh a total of $189 \mathrm{~kg}$. The average weight of large books is $2.75 \mathrm{~kg}$, medium-sized books weigh $1.5 \mathrm{~kg}$ on average, and small books weigh $\frac{1}{3} \mathrm{~kg}$ on average. How many books are there of each size, if the total weight of the large books is the greatest and the total weight of the small books is the smallest?",See reasoning trace,medium |
| 7ec9bff37761,"A point object of mass $m$ is connected to a cylinder of radius $R$ via a massless rope. At time $t = 0$ the object is moving with an initial velocity $v_0$ perpendicular to the rope, the rope has a length $L_0$, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds $T_{max}$. Express your answers in terms of $T_{max}$, $m$, $L_0$, $R$, and $v_0$. [asy] |
| size(200); |
| real L=6; |
| filldraw(CR((0,0),1),gray(0.7),black); |
| path P=nullpath; |
| for(int t=0;t<370;++t) |
| { |
| pair X=dir(180-t)+(L-t/180)*dir(90-t); |
| if(X.y>L) X=(X.x,L); |
| P=P--X; |
| } |
| draw(P,dashed,EndArrow(size=7)); |
| draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); |
| filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the kinetic energy of the object at the instant that the rope breaks? |
| |
| $ \textbf{(A)}\ \frac{mv_0^2}{2} $ |
| |
| $ \textbf{(B)}\ \frac{mv_0^2R}{2L_0} $ |
| |
| $ \textbf{(C)}\ \frac{mv_0^2R^2}{2L_0^2} $ |
| |
| $ \textbf{(D)}\ \frac{mv_0^2L_0^2}{2R^2} $ |
| |
| $ \textbf{(E)}\ \text{none of the above} $",\frac{mv_0^2,medium |
| 5c6dd701fab1,"Folklore |
| |
| In an acute-angled triangle $A B C$, the bisector $A N$, the altitude $B H$, and the line perpendicular to side $A B$ and passing through its midpoint intersect at one point. Find the angle $BAC$. |
| |
| #",$60^{\circ}$,medium |
| 44b89fe00de9,"## |
| |
| Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$. |
| |
| $A(5 ; 3 ;-1)$ |
| |
| $B(0 ; 0 ;-3)$ |
| |
| $C(5 ;-1 ; 0)$",See reasoning trace,easy |
| ed0f6f493dd9,"Exercise 5. In a classroom, there are ten students. Aline writes ten consecutive integers on the board. Each student chooses one of the ten integers written on the board, such that any two students always choose two different integers. Each student then calculates the sum of the nine integers chosen by the other nine students. Each student whose result is a perfect square then receives a gift. |
| |
| What is the maximum number of students who will receive a gift? |
| |
| A perfect square is an integer of the form $n^{2}$, where $n$ is a natural number.",See reasoning trace,medium |
| 72d56492657c,"If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team? |
| |
| $\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$",\frac{1,medium |
| 112d5a516592,"5. A semicircle of radius 1 is drawn inside a semicircle of radius 2, as shown in the diagram, where $O A=O B=2$. |
| A circle is drawn so that it touches each of the semicircles and their common diameter, as shown. |
| What is the radius of the circle?",See reasoning trace,medium |
| 07c4ddb9bdb2,"1. Given that the two roots of the equation $x^{2}+x-1=0$ are $\alpha, \beta$. Then the value of $\frac{\alpha^{3}}{\beta}+\frac{\beta^{3}}{\alpha}$ is $\qquad$",-4-3=-7$.,medium |
| 6ec2800962cf,"# |
| |
| Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions. |
| |
| At some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time. |
| |
| How many minutes does it take for one runner to complete the entire track?",See reasoning trace,medium |
| 387b177f7730,"4 The remainder when $7^{2008}+9^{2008}$ is divided by 64 is |
| (A) 2 |
| (B) 4 |
| (C) 8 |
| (D) 16 |
| (E) 32","(8-1)^{2008}=64 k_{1}+1$ for some integers $k_{1}$. Similarly, we have $9^{2008}=(8+1)^{2008}=64 k_{",easy |
| 875a492a07b0,"Task B-4.3. (20 points) What are the minimum and maximum of the function |
| |
| $$ |
| y=\frac{\sin ^{2} x-\sin x+1}{\sin ^{2} x+\sin x+1} |
| $$ |
| |
| For which $x \in[0,2 \pi]$ does the function take its minimum, and for which its maximum value?","\frac{1}{3}$ into (1), we get the quadratic equation $(\sin x-1)^{2}=0$, i.e., $\sin x=1$, so the mi",medium |
| b4795df23c4a,"Exercise 8. Determine all pairs \((m, n)\) of strictly positive integers such that: |
| |
| $$ |
| 125 \times 2^{n} - 3^{m} = 271 |
| $$",See reasoning trace,medium |
| 911e4734a0bb,"## |
| |
| Calculate the definite integral: |
| |
| $$ |
| \int_{-\pi / 2}^{0} 2^{8} \cdot \cos ^{8} x d x |
| $$",See reasoning trace,medium |
| 82ffde4b1ab2,"6. Let $a, b, c, d, m, n$ all be positive real numbers. |
| $P=\sqrt{a b}+\sqrt{c d}, Q=\sqrt{m a+n c} \cdot \sqrt{\frac{b}{m}+\frac{d}{n}}$, then, |
| (A) $P \geqslant Q$; |
| (B) $P \leqslant Q$; |
| (C) $P < Q$; |
| (D) The relationship between $P$ and $Q$ is uncertain and depends on the sizes of $m$ and $n$. |
| Answer ( )",See reasoning trace,easy |
| 8626144f26e0,"Exercise 8. The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams). Martin and Théodore unearth a treasure composed of precious stones which are either diamonds or crystals, and the total value is $5000000 €$. They cut each precious stone in half, and each takes one half of each stone. The total value of Martin's stones is $2000000 €$. In euros, what was the total initial value of the diamonds contained in the treasure? |
| |
| Only a numerical answer is expected here.","is expected, it is recommended to write down the reasoning to earn some points",medium |
| e307a2fe449a,"Example 8 Try to solve the congruence equation |
| $$x^{2} \equiv 33(\bmod 128)$$",See reasoning trace,easy |
| 997b01908ba4,"$\begin{array}{l}\text { 1. In } \triangle A B C, A B=4, B C=7, C A=5, \\ \text { let } \angle B A C=\alpha. \text { Find } \sin ^{6} \frac{\alpha}{2}+\cos ^{6} \frac{\alpha}{2} \text {. }\end{array}$",See reasoning trace,medium |
| 713c97e35ad5,"6. One mole of an ideal gas was expanded so that during the process, the pressure of the gas turned out to be directly proportional to its volume. In this process, the gas heated up by $\Delta T=100{ }^{\circ} \mathrm{C}$. Determine the work done by the gas in this process. The gas constant $R=8.31$ J/mol$\cdot$K. (15 points)",415,medium |
| 7c716b2ac0a4,,35 or 36,medium |
| 435dc5bbf27c,"9. As shown in Figure 2, in quadrilateral $ABCD$, $AB=BC=CD$, $\angle ABC=78^{\circ}$, $\angle BCD=162^{\circ}$. Let the intersection point of line $AD$ and $BC$ be $E$. Then the size of $\angle AEB$ is",21^{\circ}$.,medium |
| 7f846bff8282,"2. Let $x$ be a positive integer, and $y$ is obtained from $x$ when the first digit of $x$ is moved to the last place. Determine the smallest number $x$ for which $3 x=y$.",See reasoning trace,medium |
| cb6e23721125,"B. As shown in Figure 2, in the square $A B C D$ with side length 1, $E$ and $F$ are points on $B C$ and $C D$ respectively, and $\triangle A E F$ is an equilateral triangle. Then the area of $\triangle A E F$ is",See reasoning trace,medium |
| f1b5268fa729,"Example 3.21. Find the points of discontinuity of the function |
| |
| $$ |
| z=\frac{x y+1}{x^{2}-y} |
| $$","0$ or $y=x^{2}$ is the equation of a parabola. Therefore, the given function has a line of discontin",easy |
| ef593ac9d436,"Example 14 Given a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $M$ and $N$ are the midpoints of $B B_{1}$ and $B_{1} C_{1}$ respectively, and $P$ is the midpoint of line segment $M N$. Find the distance between $D P$ and $A C_{1}$.",\frac{|1 \cdot 1-6 \cdot 0+7 \cdot 0-2|}{\sqrt{1^{2}+\left(-6^{2}\right)+7^{2}}}=\frac{\sqrt{86}}{86,medium |
| 75bc82a8bde4,"The arrows on the two spinners shown below are spun. Let the number $N$ equal $10$ times the number on Spinner $\text{A}$, added to the number on Spinner $\text{B}$. What is the probability that $N$ is a perfect square number? |
| |
| $\textbf{(A)} ~\dfrac{1}{16}\qquad\textbf{(B)} ~\dfrac{1}{8}\qquad\textbf{(C)} ~\dfrac{1}{4}\qquad\textbf{(D)} ~\dfrac{3}{8}\qquad\textbf{(E)} ~\dfrac{1}{2}$",\textbf{(B),easy |
| 9937815355d5,"8,9 |
| |
| In a sphere of radius 9, three equal chords $A A 1, B B 1$, and $C C 1$ are drawn through a point $S$ such that $A S=4, A 1 S=8, B S < B 1 S, C S < C 1 S$. Find the radius of the sphere circumscribed about the pyramid $S A B C$.",7,medium |
| 952ca7d892a5,"4. The area of a rectangle is 180 units $^{2}$ and the perimeter is 54 units. If the length of each side of the rectangle is increased by six units, what is the area of the resulting rectangle?",See reasoning trace,medium |
| e486405a6df9,"Auto: Shapovesov A.B. On the board, four three-digit numbers are written, which sum up to 2012. Only two different digits were used to write all of them. |
| |
| Provide an example of such numbers. |
| |
| #",$2012=353+553+553+553=118+118+888+888=118+188+818+888=188+188+818+818$,medium |
| b2b4583bdf86,"1. Given real numbers $a$, $b$, $c$ satisfy $(a+b)(b+c)(c+a)=0$ and $abc<0$. Then the value of the algebraic expression $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}$ is",See reasoning trace,easy |
| 0c3dd90c4bd7,"2. (1974 American High School Mathematics Exam) A die is rolled six times, the probability of getting at least 5 points at least five times is |
| A. $\frac{13}{729}$ |
| B. $\frac{12}{729}$ |
| C. $\frac{2}{729}$ |
| D. $\frac{3}{729}$ |
| E. None of these",\frac{13}{729}$.,medium |
| 595d280fbcb2,221. $\log x = \log a + \log b - \log c$.,"\log \frac{a b}{c}$, from which $x=\frac{a b}{c}$.",easy |
| f60f33d61cf0,"31. The general term formula of the sequence $\left\{f_{n}\right\}$ is $f_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right], n \in \mathbf{N}^{+}$. Let $S_{n}=\mathrm{C}_{n}^{1} f_{1}+\mathrm{C}_{n}^{2} f_{2}+\cdots+\mathrm{C}_{n}^{n} f_{n}$, find all positive integers $n$ such that $8 \mid S_{n}$.",See reasoning trace,medium |
| 7b78adc8bec3,"3Given points $A(-2 ; 1), B(2 ; 5)$ and $C(4 ;-1)$. Point $D$ lies on the extension of median $A M$ beyond point $M$, and quadrilateral $A B D C$ is a parallelogram. Find the coordinates of point $D$.",$(8 ; 3)$,medium |
| e20b6672e4af,"8. In the figure shown in Figure 3, on both sides of square $P$, there are $a$ and $b$ squares to the left and right, and $c$ and $d$ squares above and below, where $a$, $b$, $c$, and $d$ are positive integers, satisfying |
| $$ |
| (a-b)(c-d)=0 \text {. } |
| $$ |
| |
| The shape formed by these squares is called a ""cross star"". |
| There is a grid table consisting of 2014 squares, forming a $38 \times 53$ grid. Find the number of cross stars in this grid table. |
| (Tao Pingsheng, provided)",See reasoning trace,medium |
| fb2958b78bbb,"19.5. Given 8 objects, one of which is marked. It is required to ask 3 questions, to which only ""yes"" and ""no"" answers are given, and find out which object is marked.","to the $i$-th question is ""yes"", and $\varepsilon_{i}=1$ otherwise",medium |
| 27db92d60f00,"3. The function $f: R \rightarrow R$, for any real numbers $x, y$, as long as $x+y \neq 0$, then $f(x y)=$ $\frac{f(x)+f(y)}{x+y}$ holds, then the odd-even property of the function $f(x)(x \in \mathbf{R})$ is $(\quad)$. |
| A. Odd function |
| B. Even function |
| C. Both odd and even function |
| D. Neither odd nor even function",0(x \in \mathbf{R})$.,easy |
| b5a02230855b,"122. Solve the equation |
| |
| $$ |
| 4 x^{4}+12 x^{3}+5 x^{2}-6 x-15=0 |
| $$","$x_{1}=1, x_{2}=-2",easy |
| a36840f29fc9,"10. $n>10$ teams take part in a soccer tournament. Each team plays every other team exactly once. A win gives two points, a draw one point and a defeat no points. After the tournament, it turns out that each team has won exactly half of its points in the games against the 10 worst teams (in particular, each of these 10 teams has scored half of its points against the 9 remaining teams). Determine all possible values of $n$, and give an example of such a tournament for these values. |
| |
| ## 1st solution |
| |
| We call the 10 worst teams the losers, the $n-10$ best teams the winners. We repeatedly use the following fact: If $k$ teams play against each other, then the total number of points won is exactly $k(k-1)$. We count the total number of points won in two ways. On the one hand, this is exactly $n(n-1)$. On the other hand, the 10 losers in the games among themselves receive exactly $10 \cdot 9=90$ points. By assumption, this is exactly half of the total number of points that these 10 teams have achieved. Consequently, the total number of points scored by the losers is 180. The $n-10$ winners scored a total of $(n-10)(n-11)$ points in the games among themselves. Again, this is half of the total number of points, the latter is therefore equal to $2(n-10)(n-11)$. A comparison yields the equation |
| |
| $$ |
| n(n-1)=180+2(n-10)(n-11) |
| $$ |
| |
| This is equivalent to $n^{2}-41 n+400=0$ and has the solutions $n=16$ and $n=25$. |
| |
| According to the above calculations, it also follows that the average score of the $n-10$ winners is equal to 2( $n-11)$, the average score of the 10 losers is equal to 18. |
| |
| Of course, $2(n-10) \geq 18$ must now apply, i.e. $n \geq 20$. Consequently, $n=16$ is not possible. |
| |
| Finally, we show that such a tournament exists for $n=25$. The 10 losers always play to a draw among themselves, as do the $n-10$ winners among themselves. The following diagram shows the games of the 15 winners $G_{i}$ against the 10 losers $V_{j}$. Where 2 means a win for $G_{i}$, 0$ a win for $V_{j}$ and 1 a draw. |
| |
| | | $G_{1}$ | $G_{2}$ | $G_{3}$ | $G_{4}$ | $G_{5}$ | $G_{6}$ | $G_{7}$ | $G_{8}$ | $G_{9}$ | $G_{10}$ | $G_{11}$ | $G_{12}$ | $G_{13}$ | $G_{14}$ | $G_{15}$ | |
| | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | |
| | $V_{1}$ | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 0 | |
| | $V_{2}$ | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 2 | 2 | 2 | 0 | |
| | $V_{3}$ | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 2 | 2 | 2 | |
| | $V_{4}$ | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | 2 | 2 | |
| | $V_{5}$ | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | 2 | |
| | $V_{6}$ | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | 2 | |
| | $V_{7}$ | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | |
| | $V_{8}$ | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 0 | 1 | 2 | |
| | $V_{9}$ | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 0 | 1 | |
| | $V_{10}$ | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 1 | 1 | 2 | 2 | 2 | 0 | 1 | |
| |
| In total, the final score of each winner is 28, of which exactly 14 points come from games against the losers. Each loser has a final score of 18, half of which again comes from games against the other losers. This shows everything. |
| |
| ## 2nd solution |
| |
| Here is another example of such a tournament. Again, the losers and the winners play each other to a draw. Now assume that a winner never loses to a loser. A winner scores a total of 14 points against the other winners, so he must also score 14 points against the losers. To do this, he must win four times and draw six times. Similarly, each loser draws nine times. We can now divide the $10 \times 15$ table from above into $2 \times 3$ boxes and obtain the following table with the same notations: |
| |
| | | $G_{1}, G_{2}, G_{3}$ | $G_{4}, G_{5}, G_{6}$ | $G_{7}, G_{8}, G_{9}$ | $G_{10}, G_{11}, G_{12}$ | $G_{13}, G_{14}, G_{15}$ | |
| | :---: | :---: | :---: | :---: | :---: | :---: | |
| | $V_{1}, V_{2}$ | 1 | 1 | 1 | 2 | 2 | |
| | $V_{3}, V_{4}$ | 2 | 1 | 1 | 1 | 2 | |
| | $V_{5}, V_{6}$ | 2 | 2 | 1 | 1 | 1 | 1 | |
| | $V_{7}, V_{8}$ | 1 | 2 | 2 | 1 | 1 | |
| | $V_{9}, V_{10}$ | 1 | 1 | 2 | 2 | 1 |",See reasoning trace,easy |
| 131acb674f3d,"3.9. $\left\{\begin{array}{l}x+y+x y=2+3 \sqrt{2}, \\ x^{2}+y^{2}=6 .\end{array}\right.$","x+y$ and $v=x y$. Then $u+v=2+3 \sqrt{2}$ and $u^{2}-2 v=6$, so $u^{2}+2 u=6+2(2+3 \sqrt{2})=10+6 \s",medium |
| 385bedb37147,"B4. $\overline{a b}$ is the notation for the number written with the digits $a$ and $b$, where $a \neq 0$. |
| |
| Give all positive integer values of $K$ for which the following holds: |
| |
| - $K$ is a positive integer |
| - there exists a number $\overline{a b}$ that is not divisible by 9 with $\overline{a b}=K \times(a+b)$. |
| |
| Note: Points will be deducted for incorrect values of $K$!",389&width=415&top_left_y=2107&top_left_x=1343),medium |
| db0a35fd1047,"Three. (Full marks 12 points) Solve the equation: |
| $$ |
| \frac{13 x-x^{2}}{x+1}\left(x+\frac{13-x}{x+1}\right)=42 \text {. } |
| $$","1, x_{2}=6, x_{3}=3+\sqrt{2}, x_{4}=3-\sqrt{2}$.",medium |
| b7ff3ee95c78,"Question 47, Find the maximum value of the function $f(x)=\frac{\sqrt{2} \sin x+\cos x}{\sin x+\sqrt{1-\sin x}}(0 \leq x \leq \pi)$.","Since $(1-\sin x)^{2} \geq 0 \Rightarrow \cos x \leq \sqrt{2} \cdot \sqrt{1-\sin x}$, therefore $f(x)=$ $\frac{\sqrt{2} \sin x+\cos x}{\sin x+\sqrt{1-\sin x}} \leq \frac{\sqrt{2} \sin x+\sqrt{2} \cdot \sqrt{1-\sin x}}{\sin x+\sqrt{1-\sin x}}=\sqrt{2}$, so the maximum value of $f(x)=\frac{\sqrt{2} \sin x+\cos x}{\sin x+\sqrt{1-\sin x}}(0 \leq x \leq \pi)$ is $\sqrt{2}$, which is achieved when $x=\frac{\pi}{2}$",easy |
| a830cde13221,"Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$? |
| $\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{7}{16} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{9}{16} \qquad \textbf{(E) } \frac{2}{3}$",\textbf{(B),medium |
| faf72f0aeec4,"10.243. Inside an angle of $60^{\circ}$, there is a point located at distances of $\sqrt{7}$ and $2 \sqrt{7}$ cm from the sides of the angle. Find the distance from this point to the vertex of the angle.",$\frac{14\sqrt{3}}{3}$ cm,medium |
| 85e5415806a0,"3. In a school, all 300 Secondary 3 students study either Geography, Biology or both Geography and Biology. If $80 \%$ study Geography and $50 \%$ study Biology, how many students study both Geography and biology? |
| (A) 30; |
| (B) 60; |
| (C) 80; |
| (D) 90 ; |
| (E) 150 .",See reasoning trace,easy |
| d3ca416beac7,"G9.3 If the $r^{\text {th }}$ day of May in a year is Friday and the $n^{\text {th }}$ day of May in the same year is Monday, where $15<n<25$, find $n$.",20$,easy |
| 2c0e6ae75e7d,"$$ |
| \log _{a}\left(a b^{2}\right)+\log _{b}\left(b^{2} c^{3}\right)+\log _{c}\left(c^{5} d^{6}\right)+\log _{d}\left(d^{35} a^{36}\right) |
| $$",67,medium |
| b23631ac0380,"4. Find all integers n for which the fraction |
| |
| $$ |
| \frac{n^{3}+2010}{n^{2}+2010} |
| $$ |
| |
| is equal to an integer.",See reasoning trace,medium |
| 50e480f5f0f7,"## Task 2 - 020512 |
| |
| ""It takes exactly one million two hundred and nine thousand six hundred seconds until we meet again,"" says Walter, who likes to calculate with large numbers, to Rolf, as they say goodbye on May 10th at 12:00 PM. |
| |
| When will the two meet again?",See reasoning trace,easy |
| 71d819d58701,"4.9 On a $10 \times 10$ grid paper, there are 11 horizontal grid lines and 11 vertical grid lines. A line segment connecting two adjacent nodes on the same straight line is called a ""link segment"". How many link segments need to be erased at least, so that at most 3 link segments remain at each node?",See reasoning trace,medium |
| 1608f33e91ee,"Solve the equation |
| $$ |
| \frac{x}{2+\frac{x}{2+} \cdot \cdot 2+\frac{x}{1+\sqrt{1+x}}}=1 . |
| $$ |
| (The expression on the left side of the equation contains 1985 twos)","x$, solving which gives $x_{1}=0, x_{2}=3$. $x_{0}$ is obviously not a solution, hence $x=3$.",easy |
| f6f781515a02,"81. Let positive real numbers $x, y, z$ satisfy the condition $2 x y z=3 x^{2}+4 y^{2}+5 z^{2}$, find the minimum value of the expression $P=3 x+2 y+z$.",y=z=6$.,medium |
| 7ab413956a27,"$4 \cdot 206$ There are two forces $f_{1} 、 f_{2}$ acting on the origin $O$ of the coordinate axes, |
| $$ |
| \begin{array}{l} |
| \vec{f}_{1}=\overrightarrow{O A}=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right), \\ |
| \overrightarrow{f_{2}}=\overrightarrow{O B}=2\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right] . |
| \end{array} |
| $$ |
| (1) Find the magnitude and direction of their resultant force; |
| (2) Find the distance between points $A$ and $B$ (accurate to 0.1).",See reasoning trace,medium |
| afdfd2b28458,"1. Let the general term formula of the sequence $\left\{a_{n}\right\}$ be $a_{n}=3^{3-2 n}$, then the sum of the first $n$ terms of the sequence $S_{n}=$","3 \cdot\left(\frac{1}{9}\right)^{n-1}$, so $a_{1}=3, q=\frac{1}{9}$. Therefore, $S_{n}=\frac{3\left[",easy |
| 1398aa8910e7,"The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved.
|
| a) Find $x,y$.
|
| b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$.
|
| c) Give more numbers $g$ such that a) can be solved; give more of them such that a) can't be solved, too.","(x, y) = (98, 32)",medium |
| 70e4e15ee81a,"10. According to the pattern of the following equations, find the sum of the 2018th equation. |
| $$ |
| 2+3,3+7,4+11,5+15,6+19 \ldots |
| $$",10090,easy |
| 400fb22b3a17,"$\left[\begin{array}{l}{[\text { Area of a triangle (using two sides and the angle between them).] }} \\ {[\quad \text { Law of Sines }}\end{array}\right]$ |
| |
| In triangle $A B C$, it is known that $\angle B A C=\alpha, \angle B C A=\gamma, A B=c$. Find the area of triangle $A B C$.",See reasoning trace,medium |
| 4c658c8e4d54,"*2. Let $M, N$ be two points on the line segment $AB$, $\frac{AM}{MB}=\frac{1}{4}, \frac{AN}{NB}=\frac{3}{2}$. Construct any right triangle $\triangle ABC$ with $AB$ as the hypotenuse. Then construct $MD \perp BC$ at $D$, $ME \perp AC$ at $E$, $NF \perp BC$ at $F$, and $NG \perp AC$ at $G$. The maximum possible value of the ratio $y=\frac{MD+ME+NF+NG}{AB+BC+AC}$ is",See reasoning trace,medium |
| 284aad2ede18,Let's find two different natural numbers whose sum of cubes is equal to the square of their sum.,See reasoning trace,medium |
| e203cd90330c,"9. Solving the fractional equation $\frac{2}{x+1}+\frac{5}{1-x}=\frac{m}{x^{2}-1}$ will produce extraneous roots, then $m=$ $\qquad$ . |
| A. -10 or -4 |
| B. 10 or -4 |
| C. 5 or 2 |
| D. 10 or 4",See reasoning trace,easy |
| 0d40ea24f801,"## Task Condition |
| |
| Find the derivative. |
| |
| $$ |
| y=\frac{(1+x) \operatorname{arctg} \sqrt{x}-\sqrt{x}}{x} |
| $$",See reasoning trace,medium |
| 769ccea37137,"3. Solve the equation |
| |
| $$ |
| 3 \sqrt{6 x^{2}+13 x+5}-6 \sqrt{2 x+1}-\sqrt{3 x+5}+2=0 |
| $$",$-1 / 3 ;-4 / 9$,medium |
| aa96673549bc,"9. (12 points) There are 50 candies in a pile, and Xiaoming and Xiaoliang are playing a game. Every time Xiaoming wins, he takes 5 candies, eats 4, and puts the remaining 1 candy in his pocket; every time Xiaoliang wins, he also takes 5 candies, eats 3, and puts the remaining 2 candies in his pocket. When the game ends, all the candies are taken, and at this point, the number of candies in Xiaoliang's pocket is exactly 3 times the number of candies in Xiaoming's pocket. How many candies did the two of them eat in total?","】Solution: If each person wins 5 candies each time, then 50 candies have been distributed $50 \div 5=10$ (times)",medium |
| 6ec2735f4924,"2. There is a sequence of numbers, starting from the 3rd number, each number is the sum of all the numbers before it. If the 1st number of this sequence is 1, and the 2nd number is 2, then the remainder when the 2022nd number is divided by 7 is $\qquad$ -",See reasoning trace,easy |
| fec3be500ad6,"The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? |
| $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$",\textbf{(E),medium |
| 9ecc1f041e6f,"Example 4 As shown in Figure 2, there is a rectangular plot of land $A B C D$, and a rectangular flower bed $E F G H$ is to be built in the center, with its area being half of the area of this plot, and the width of the paths around the flower bed being equal. Now, without any measuring tools, only a sufficiently long rope with equal markings, how can the width of the path be measured?",See reasoning trace,medium |
| 3c7a67895868,"2. |
| $$ |
| a^{2}+b^{2}+c^{2}=1 \quad \text { and } \quad a(2 b-2 a-c) \geq \frac{1}{2} |
| $$",See reasoning trace,medium |
| b9797da3e869,"Example 4. Calculate the integral |
| |
| $$ |
| \int_{1-i}^{2+i}\left(3 z^{2}+2 z\right) d z |
| $$",See reasoning trace,medium |
| e41c03d3238b,"In the diagram, how many paths can be taken to spell ""KARL""? |
| (A) 4 |
| (B) 16 |
| (C) 6 |
| (D) 8 |
| (E) 14 |
| |
| ",(D),medium |
| ae6bae2664f4,"## Task 3 - 140813 |
| |
| Given a circle $k_{1}$ with radius $r_{1}$ and center $M$. Around $M$, a circle $k_{2}$ is to be drawn such that the area of the annulus between $k_{1}$ and $k_{2}$ is three times the area of the circle $k_{1}$. |
| |
| Calculate the radius $r_{2}$ of the circle $k_{2}$!",See reasoning trace,medium |
| 3fc54f2f7f9f,"The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$. |
| |
| [i]Proposed by Christopher Cheng[/i] |
| |
| [hide=Solution][i]Solution. [/i] $\boxed{6}$ |
| Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]",6,medium |
| 1e2fba5b09ca,"For a finite sequence $A=(a_1,a_2,...,a_n)$ of numbers, the Cesáro sum of A is defined to be |
| $\frac{S_1+\cdots+S_n}{n}$ , where $S_k=a_1+\cdots+a_k$ and $1\leq k\leq n$. If the Cesáro sum of |
| the 99-term sequence $(a_1,...,a_{99})$ is 1000, what is the Cesáro sum of the 100-term sequence |
| $(1,a_1,...,a_{99})$? |
| $\text{(A) } 991\quad \text{(B) } 999\quad \text{(C) } 1000\quad \text{(D) } 1001\quad \text{(E) } 1009$","991, A",medium |
| 032447deb336,30th IMO 1989 shortlist,"270 mutually visible pairs. Suppose bird P is at A and bird Q is at B, where A and B are distinct bu",medium |
| 534d30e4f247,"$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$ |
| $\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$",\text{D,easy |
| e8d97fc81806,"2. For a real number $a$, let $[a]$ denote the greatest integer not greater than $a$. Find all integers $y$ for which there exists a real number $x$ such that $\left[\frac{x+23}{8}\right]=[\sqrt{x}]=y$.","4$, they receive 1 point.",medium |
| b3c71f84c3e7,"For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when |
| \[\sum_{n=1}^{2017} d_n\]is divided by $1000$.",069,easy |
| 70afff560226,"2. A nonahedron was formed by gluing a cube and a regular quadrilateral pyramid. On each face of this nonahedron, a number is written. Their sum is 3003. For each face $S$ of the considered nonahedron, we sum the numbers on all faces that share exactly one edge with $S$. This gives us nine identical sums. Determine all the numbers written on the faces of the nonahedron.","a_{1}, a_{4}=a_{2}$ and 1 point for the equalities $b_{3}=b_{1}, b_{4}=b_{2}$. If these symmetries a",medium |
| 1a60cdc0023c,"3. Given points $A(1,2)$ and $B(3,4)$, there is a point $P$ on the coordinate axis, and $P A + P B$ is minimized. Then the coordinates of point $P$ are $\qquad$.",See reasoning trace,easy |
| 221ad1088785,"13. (3 points) As shown in the figure: In parallelogram $A B C D$, $O E=E F=F D$. The area of the parallelogram is 240 square centimeters, and the area of the shaded part is $\qquad$ square centimeters.",: 20,medium |
| 6192a627ba9e,"## Task 1 - 090731 |
| |
| Imagine all natural numbers from 1 to 2555, each written exactly once. Determine the total number of the digit 9 that would need to be written!",705$.,medium |
| 91e9be09dc3a,"If $2 x+3 x+4 x=12+9+6$, then $x$ equals |
| (A) 6 |
| (B) 3 |
| (C) 1 |
| (D) $\frac{1}{3}$ |
| (E) $10 \frac{1}{2}$",(B),easy |
| 7a3e7d4dfa12,"11. Given $\frac{x y}{x+y}=2, \frac{x z}{x+z}=3, \frac{y z}{y+z}=4$. Find the value of $7 x+5 y-2 z$.",0$.,easy |
| 3c858a629196,"7. Put 48 chess pieces into 9 boxes, with at least 1 piece in each box, and the number of pieces in each box is different. The box with the most pieces can contain $\qquad$ pieces at most. |
| |
| Put 48 chess pieces into 9 boxes, with at least 1 piece in each box, and the number of pieces in each box is different. The box with the most pieces can contain $\qquad$ pieces at most.","36$ pieces; therefore, the box with the most chess pieces can contain at most $48-36=12$ pieces.",medium |
| ce78410f6082,"Determine all integers $a>0$ for which there exist strictly positive integers $n, s, m_{1}, \cdots, m_{n}, k_{1}, \cdots, k_{s}$ such that |
| |
| $$ |
| \left(a^{m_{1}}-1\right) \cdots\left(a^{m_{n}}-1\right)=\left(a^{k_{1}}+1\right) \cdots\left(a^{k_{s}}+1\right) |
| $$",See reasoning trace,medium |
| ba0abcf02295,5. $\tan \frac{\pi}{9} \cdot \tan \frac{2 \pi}{9} \cdot \tan \frac{4 \pi}{9}=$ $\qquad$,See reasoning trace,medium |
| 9159005a4da4,110. Find the general solution of the equation $y^{\prime \prime}=4 x$.,See reasoning trace,medium |
| 53b046f8a237,"Transform the following expression into a product: |
| |
| $$ |
| x^{7}+x^{6} y+x^{5} y^{2}+x^{4} y^{3}+x^{3} y^{4}+x^{2} y^{5}+x y^{6}+y^{7} |
| $$","1$, this is the 1st problem of the 1941 Eötvös competition.",medium |
| e26ec3c149c2,"How many three-digit numbers exist in which the digits 1, 2, 3 appear exactly once each? |
| |
| #",$3!=6$ numbers,easy |
| ff9ba2fe63ef,"8. The selling price of an item is obtained by increasing the actual amount by a certain percentage, called VAT (a tax, which is then paid to the tax office). In a store, the selling price of a sweater is $61.00 €$, including VAT at $22 \%$. If the VAT were to increase to $24 \%$, what would the new selling price of the sweater be? |
| (A) $62.25 €$ |
| (B) $62.22 €$ |
| (C) $63.00 €$ |
| (D) $62.00 €$ |
| (E) $61.50 €$",is $(D)$,medium |
| e7d68deb9aa7,"## Task B-4.1. |
| |
| Solve the system of equations |
| |
| $$ |
| \left\{\begin{aligned} |
| \sin \frac{\pi x}{2022}-\sin \frac{\pi y}{2022} & =1 \\ |
| x-y & =2022 |
| \end{aligned}\right. |
| $$ |
| |
| if $|x| \leqslant 2022$ and $|y| \leqslant 2022$.",See reasoning trace,medium |
| 0bf0db940fe2,"11.2. Several married couples came to the New Year's Eve party, each of whom had from 1 to 10 children. Santa Claus chose one child, one mother, and one father from three different families and took them for a ride in his sleigh. It turned out that he had exactly 3630 ways to choose the required trio of people. How many children could there be in total at this party? |
|
|
| (S. Volchonkov)",33,medium |
| f5e332414f93,"$23 \cdot 31$ In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $P, M, N$ are the midpoints of edges $A A_{1}$, $B C, C C_{1}$ respectively. The figure formed by the intersection lines of the plane determined by these three points with the surface of the cube is |
| (A) a rhombus. |
| (B) a regular pentagon. |
| (C) a regular hexagon. |
| (D) a hexagon with only equal sides. |
| (2nd ""Hope Cup"" National Mathematics Invitational Competition, 1991)",$(C)$,medium |
| 2ff2dfccd9a3,"5. Let the equation |
| $$ |
| 1+x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots-\frac{x^{2018}}{2018}=0 |
| $$ |
|
|
| have all its real roots within the interval $[a, b](a, b \in \mathbf{Z}$, $a<b)$. Then the minimum value of $b-a$ is ( ). |
| (A) 1 |
| (B) 2 |
| (C) 3 |
| (D) 4","-1, b=2$, the equality holds. Hence, the minimum value of $b-a$ is 3.",medium |
| d1b86f444841,"[ equations in integers ] |
|
|
| A combination ( $x, y, z$ ) of three natural numbers, lying in the range from 10 to 20 inclusive, is an unlocking combination for a code lock if |
|
|
| $3 x^{2}-y^{2}-7 z=99$. Find all the unlocking combinations.",At $h = 1$,medium |
| 3cc5a5f124e8,"\section*{ |
|
|
| If \(n\) is a natural number greater than 1, then on a line segment \(A B\), points \(P_{1}, P_{2}, P_{3}, \ldots, P_{2 n-1}\) are placed in this order such that they divide the line segment \(A B\) into \(2 n\) equal parts. |
|
|
| a) Give (as a function of \(n\)) the probability that two points \(P_{k}, P_{m}\) chosen from the points \(P_{1}, P_{2}, P_{3}, \ldots, P_{2 n-1}\) with \(0 < k < m < 2 n\) divide the line segment \(A B\) in such a way that a triangle can be constructed from the three segments \(A P_{k}, P_{k} P_{m}, P_{m} B\). |
|
|
| b) Investigate whether this probability converges to a limit as \(n \rightarrow \infty\), and determine this limit if it exists. |
|
|
| Note: The probability sought in a) is defined as follows: Each selection of two points \(P_{k}, P_{m}\) with \(0 < k < m < 2 n\) is referred to as a ""case."" |
|
|
| A ""case"" is called a ""favorable case"" if \(P_{k}\) and \(P_{m}\) are chosen such that a triangle can be formed from the segments \(A P_{k}, P_{k} P_{m}\), and \(P_{m} B\). |
|
|
| If \(z\) is the total number of possible ""cases"" and \(z_{1}\) is the number of ""favorable cases,"" then the probability is defined as the quotient \(\frac{z_{1}}{z}\).",\frac{1}{4}\) follows.,medium |
| e8b85b220c1e,"Example 6 Solve the system of equations $\left\{\begin{array}{l}x+y+z=3, \\ x^{2}+y^{2}+z^{2}=3,(\text{Example 4 in [1]}) \\ x^{5}+y^{5}+z^{5}=3 .\end{array}\right.$",y=z=\frac{1}{3}$.,medium |
| 057d83c193cf,"3. [5 points] Solve the system of equations |
|
|
| $$ |
| \left\{\begin{array}{l} |
| \left(\frac{x^{4}}{y^{2}}\right)^{\lg y}=(-x)^{\lg (-x y)} \\ |
| 2 y^{2}-x y-x^{2}-4 x-8 y=0 |
| \end{array}\right. |
| $$","$(-4 ; 2),(-2 ; 2),\left(\frac{\sqrt{17}-9}{2} ; \frac{\sqrt{17}-1}{2}\right)$",medium |
| 83f8f328e704,"Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes. |
| \[ 2^m p^2 + 1 = q^5 \]","(1, 11, 3)",medium |
| 4751157e6a81,"$10 \cdot 10$ Given a four-digit number that satisfies the following conditions: |
| (1) If the units digit and the hundreds digit, as well as the tens digit and the thousands digit, are simultaneously swapped, then its value increases by 5940; |
| (2) When divided by 9, the remainder is 8. |
| Find the smallest odd number among these four-digit numbers. |
| (Shandong Province, China Mathematics Competition, 1979)","9$. Therefore, the four-digit number we are looking for is 1979.",medium |
| 40f55edb18fb,"2. Given $f(x)=|1-2 x|, x \in[0,1]$, then the number of solutions to the equation |
| $$ |
| f\left(f(f(x))=\frac{1}{2} x\right. |
| $$ |
|
|
| is","f(f(f(x)))$ intersects the line $w=\frac{x}{2}$ at 8 points, i.e., the number of solutions is 8.",medium |
| 796c1c8c6d90,"Determine the number of pairs $(x, y)$ of positive integers for which $0<x<y$ and $2 x+3 y=80$.",5,medium |
| c3b263125cc9,,18,medium |
| b0582f4f7e90,"13. In $\triangle A B C$, $\angle B=\frac{\pi}{4}, \angle C=\frac{5 \pi}{12}, A C$ $=2 \sqrt{6}, A C$'s midpoint is $D$. If a line segment $P Q$ of length 3 (point $P$ to the left of point $Q$) slides on line $B C$, then the minimum value of $A P+D Q$ is . $\qquad$",See reasoning trace,easy |
| bd24fdb72701,"Points $( \sqrt{\pi} , a)$ and $( \sqrt{\pi} , b)$ are distinct points on the graph of $y^2 + x^4 = 2x^2 y + 1$. What is $|a-b|$? |
| $\textbf{(A)}\ 1 \qquad\textbf{(B)} \ \frac{\pi}{2} \qquad\textbf{(C)} \ 2 \qquad\textbf{(D)} \ \sqrt{1+\pi} \qquad\textbf{(E)} \ 1 + \sqrt{\pi}$",\textbf{(C),medium |
| c9fe9052152c,"One. (20 points) Given the equation in terms of $x$ |
| $$ |
| k x^{2}-\left(k^{2}+6 k+6\right) x+6 k+36=0 |
| $$ |
| |
| the roots of which are the side lengths of a certain isosceles right triangle. Find the value of $k$.",See reasoning trace,medium |
| 3d6497f7145f,"9. If $\sin \theta+\cos \theta$ $=\frac{7}{5}$, and $\tan \theta<1$, then $\sin \theta=$ $\qquad$",See reasoning trace,easy |
| 269be838e150,"A triangle has side lengths $10$, $10$, and $12$. A rectangle has width $4$ and area equal to the |
| area of the triangle. What is the perimeter of this rectangle? |
| $\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36$",\textbf{(D),medium |
| aec91eacdc8a,"5. The giants were prepared 813 burgers, among which are cheeseburgers, hamburgers, fishburgers, and chickenburgers. If three of them start eating cheeseburgers, then in that time two giants will eat all the hamburgers. If five take on eating hamburgers, then in that time six giants will eat all the fishburgers. If seven start eating fishburgers, then in that time one giant can eat all the chickenburgers. How many burgers of each type were prepared for the giants? (The time it takes for one giant to eat one burger does not depend on the type of burger, and all giants eat at the same speed.)","252 fishburgers, 36 chickenburgers, 210 hamburgers, and 315 cheeseburgers",medium |
| c372d722476f,"Given $0<a<1,0<b<1$, and $a b=\frac{1}{36}$. Find the minimum value of $u=\frac{1}{1-a}+\frac{1}{1-b}$.",$ $\frac{4}{2-(a+b)} \geqslant \frac{4}{2-2 \sqrt{a b}}=\frac{4}{2-\frac{1}{3}}=\frac{12}{5}$. When ,easy |
| c800082a9868,,See reasoning trace,medium |
| 72e34630b62b,"Example 1. In $\triangle A B C$, $\angle C=3 \angle A$, $a=27, c=48$. What is $b=$ ? |
| (A) 33 ; |
| (B) 35; |
| (C) 37 ; |
| (D) 39 ; |
| (E) The value of $b$ is not unique. |
| |
| [36th American High School Mathematics Examination (February 26, 1985, Beijing), Question 28] |
| |
| Your publication provided a trigonometric solution to this |
| |
| 保留源文本的换行和格式，直接输出翻译结果。","35$, hence (B) is true.",medium |
| 242b52ee3a8d,"4. The height to the base of an isosceles triangle has a length of $\sqrt{2+\sqrt{2}}$. If the measure of the angle opposite the base is $45^{\circ}$, what is the length of the height to the leg of the triangle?",See reasoning trace,medium |
| 59647a1f4273,"## Task 5 - 301235 |
| |
| Investigate whether the sequence $\left(x_{n}\right)$ defined by |
| |
| $$ |
| x_{1}=1, \quad x_{n+1}=\frac{1}{x_{n}+1} \quad(n=1,2,3, \ldots) |
| $$ |
| |
| is convergent, and determine its limit if it is convergent.",s all the questions here,medium |
| 01c11d4833eb,"1. There is a quadratic equation, whose two roots are two-digit numbers formed by the digits 1, 9, 8, and 4. Let the difference between these two roots be $\mathrm{x}$, which makes $\sqrt{1984 \mathrm{x}}$ an integer. Try to find this equation and its two roots.","0$. The two roots are $\mathrm{y}_{1}=49, \mathrm{y}_{2}=18$.",easy |
| 7b1a899ec06d,"7. At the conference. $85 \%$ of the delegates at the conference know English, and $75 \%$ know Spanish. What fraction of the delegates know both languages?","160 \%$, which exceeds the total number of conference delegates by $60 \%$. The excess is due to tho",easy |
| a42ea9828e18,"3. It is known that the ages of A, B, and C |
| |
| are all positive integers. A's age is twice B's age, B is 7 years younger than C. If the sum of the ages of these three people is a prime number less than 70, and the sum of the digits of this prime number is 13, then the ages of A, B, and C are","30, y=15, z=22$.",easy |
| 651f505bd08e,"6. Given the set |
| $$ |
| A=\left\{x \mid x=a_{0}+a_{1} \times 7+a_{2} \times 7^{2}+a_{3} \times 7^{3}\right\} \text {, } |
| $$ |
| |
| where, $a_{i} \in\{0,1, \cdots, 6\}(i=0,1,2,3)$, and $a_{3} \neq 0$. |
| If positive integers $m 、 n \in A$, and $m+n=2010(m>n)$, then the number of positive integers $m$ that satisfy the condition is $\qquad$.",See reasoning trace,easy |
| 5c7371711754,34th CanMO 2002,"3 and at most 8 + 9 = 17. There are only 15 numbers at least 3 and at most 17, so each of them must ",easy |
| bc7739129062,Example 1. Find $P_{n}=\prod_{k=1}^{n}\left(2 \cos 2^{k-1} \theta-1\right)$.,2 \cos 2^{k-1} \theta-1 \\ & =\frac{\left(2 \cos 2^{k-1} \theta-1\right)\left(2 \cos 2^{\mathbf{k}-1,medium |
| 7422c5f5c374,"13.223. Point $C$ is located 12 km downstream from point $B$. A fisherman set out from point $A$, located upstream from point $B$, to point $C$. After 4 hours, he arrived at $C$, and the return trip took 6 hours. On another occasion, the fisherman used a motorboat, thereby tripling his own speed relative to the water, and reached from $A$ to $B$ in 45 minutes. It is required to determine the speed of the current, assuming it is constant.",1 km/h,easy |
| 6010cbe6bdfd,"When I went to receive the gold medal I won in the OBMEP, the following information appeared on the passenger cabin screens of my flight to Recife: |
| |
| $$ |
| \begin{aligned} |
| \text { Average speed: } & 864 \mathrm{~km} / \mathrm{h} \\ |
| \text { Distance from the departure location: } & 1222 \mathrm{~km} \\ |
| \text { Arrival time in Recife: } & 1 \mathrm{~h} 20 \mathrm{~min} |
| \end{aligned} |
| $$ |
| |
| If the plane maintained the same speed, then what is the distance, approximately, in kilometers, between Recife and the city where my flight started? |
| (a) 2300 |
| (b) 2400 |
| (c) 2500 |
| (d) 2600 |
| (e) 2700","1152$ $\mathrm{km}$. Since we were $1222 \mathrm{~km}$ from the departure city, the distance between",easy |
| 83d9294906c6,1. Solve the equation $x^{\log _{5}(0.008 x)}=\frac{125}{x^{5}}$.,"$x=5, x=\frac{1}{125}$",easy |
| fc90afba32db,"Example 1 Find all $n$ such that there exist $a$ and $b$ satisfying |
| $$ |
| S(a)=S(b)=S(a+b)=n . |
| $$",See reasoning trace,easy |
| 582c6ef1778d,"3. Given are positive numbers $a, b$ and $c$. Determine all triples of positive numbers $(x, y, z)$ such that |
| |
| $$ |
| x+y+z=a+b+c \quad \text { and } \quad 4 x y z-a^{2} x-b^{2} y-c^{2} z=a b c . |
| $$",4 y z-a^{2}-\lambda$ and so on. It follows that $4 y z-a^{2}=4 z x-b^{2}=4 x y-c^{2}=\lambda$ and $x,medium |
| dc133a1a584f,"5. In trapezoid $A B C D$, $A B / / C D$, the base angles $\angle D A B=36^{\circ}, \angle C B A=54^{\circ}, M$ and $N$ are the midpoints of sides $A B$ and $C D$, respectively. If the lower base $A B$ is exactly 2008 units longer than the upper base $C D$, then the line segment $M N=$ $\qquad$",\frac{1}{2} S T=1004$.,medium |
| 9aef0347ee6e,2. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which $f(x+y)=f(x-y)+2 f(y) \cos x$ holds for all real numbers $x$ and $y$.,a \cdot \sin x$ are solutions (this point is also recognized if the contestant guesses the solution ,medium |
| 8b28b89d87c6,"$C D$ is the median of triangle $A B C$. The circles inscribed in triangles $A C D$ and $B C D$ touch the segment $C D$ at points $M$ and $N$. Find $M N$, if $A C - B C = 2$.",1,easy |
| 2f2613582baa,"4-224 A unit square is divided into 9 equal parts by lines parallel to the sides, and the central part is removed. The remaining 8 smaller squares are each divided into 9 equal parts by lines parallel to the sides, and the central part is removed. Then, a similar process is applied to each of the remaining squares. If this process is repeated $n$ times, try to find: |
| （1）How many squares with side length $\frac{1}{3^{n}}$ are there? |
| （2）What is the limit of the sum of the areas of the squares removed when $n$ is infinitely increased? |
| |
| （1）How many squares with side length $\frac{1}{3^{n}}$ are there? |
| （2）What is the limit of the sum of the areas of the squares removed when $n$ is infinitely increased?",See reasoning trace,medium |
| 9f122151d8c1,"A3. Which of the following statements is not true for the function $f$ defined by $f(x)=\frac{1}{2}-\frac{1}{2} \cos x$? |
| (A) The range of the function $f$ is $[0,1]$. |
| (B) The fundamental period of the function $f$ is $2 \pi$. |
| (C) The function $f$ is even. |
| (D) The zeros of the function $f$ are $x=k \pi, k \in \mathbb{Z}$. |
| (E) The function $f$ achieves its maximum value for $x=\pi+2 k \pi, k \in \mathbb{Z}$.","0$, thus $x=2 k \pi, k \in \mathbb{Z}$, and not $x=k \pi, k \in \mathbb{Z}$.",easy |
| 5809d7ac36b3,"5. Given the set $A=\left\{x \mid x^{2}-2 x-3=0\right\}, B=\{x \mid a x=1\}$. If $B \subseteq A$, then the product of all possible values of the real number $a$ is ( ). |
| A. -.1 |
| B. $-\frac{1}{3}$ |
| C. 0 |
| D. None of the above","0$, $B=\varnothing \subseteq A$; when $a \neq 0$, $B=\left\{\frac{1}{a}\right\}$, then $\frac{1}{a}=",easy |
| f0846afc4738,"Determine all integers $ n>1$ for which the inequality \[ x_1^2\plus{}x_2^2\plus{}\ldots\plus{}x_n^2\ge(x_1\plus{}x_2\plus{}\ldots\plus{}x_{n\minus{}1})x_n\] holds for all real $ x_1,x_2,\ldots,x_n$.","n \in \{2, 3, 4, 5\",medium |
| d5c7938f1e53,"19. Given that $x, y$ are positive integers, and satisfy |
| $$ |
| x y-(x+y)=2 p+q \text{, } |
| $$ |
| |
| where $p, q$ are the greatest common divisor and the least common multiple of $x$ and $y$, respectively. Find all such pairs $(x, y)(x \geqslant y)$.",See reasoning trace,medium |
| 9358e7c29173,"## Task Condition |
| |
| Find the derivative. |
| |
| $y=\frac{\sqrt{2 x+3}(x-2)}{x^{2}}$",See reasoning trace,medium |
| 09f5535fec2a,"30. Given that $[x]$ represents the greatest integer not exceeding $x$, if $[x+0.1]+[x+0.2]+\ldots[x+0.9]=104$, then the minimum value of $x$ is ( ) . |
| A. 9.5 |
| B. 10.5 |
| C. 11.5 |
| D. 12.5",C,easy |
| 38d05192ce88,8.5. Given a point $A$ and a circle $S$. Draw a line through point $A$ such that the chord cut off by the circle $S$ on this line has a given length $d$.,"O Q^{2}-M Q^{2}=R^{2}-d^{2} / 4$. Therefore, the desired line is tangent to the circle of radius $\s",easy |
| 50bfc0464780,"$17 \cdot 134$ In an isosceles $\triangle A B C$, the length of the altitude from one of the equal sides is 1. This altitude forms a $45^{\circ}$ angle with the base. Then the area of $\triangle A B C$ is |
| (A) 1 . |
| (B) 0.5 . |
| (C) 0.25 . |
| (D) $\sqrt{3}$. |
| (E) None of the above answers is correct. |
| (China Beijing Junior High School Mathematics Competition, 1983)",$(B)$,easy |
| 500eeac58f9f,"## |
| |
| Write the decomposition of vector $x$ in terms of vectors $p, q, r$: |
| |
| $x=\{-1 ; 7 ; 0\}$ |
| |
| $p=\{0 ; 3 ; 1\}$ |
| |
| $q=\{1 ;-1 ; 2\}$ |
| |
| $r=\{2 ;-1 ; 0\}$",See reasoning trace,medium |
| c1699af0834a,"## Task 6 - 190936 |
| |
| For suitable natural numbers $n$, there are polyhedra with $n$ vertices and fewer than $n$ faces. For example, for $n=8$, a cuboid is such a polyhedron, as it has exactly 8 vertices and is bounded by exactly 6 planar faces (rectangles). |
| |
| Investigate whether there exists a natural number $N$ such that for every natural number $n \geq N$, there is a polyhedron with $n$ vertices that is bounded by fewer than $n$ planar faces! |
| |
| If this is the case, determine the smallest natural number $N$ with this property!",6$ is indeed the smallest such value.,medium |
| ca7ddcbc9b82,"30. 20 identical balls are placed into 4 identical boxes, with no box left empty. How many ways are there to do this? If empty boxes are allowed, how many ways are there?",See reasoning trace,medium |
| 450f48f44ee3,"Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties: |
| |
| (a) for any integer $n$, $f(n)$ is an integer; |
| (b) the degree of $f(x)$ is less than $187$. |
| |
| Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns. |
| |
| [i]Proposed by YaWNeeT[/i]",187,medium |
| 07200d21862c,"2. Find the value of the fraction |
| |
| $$ |
| \frac{2 \cdot 2020}{1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+3+\ldots+2020}} |
| $$",2021,medium |
| 31037ce1f9a7,"The natural numbers from 1 to 2100 are entered sequentially in 7 columns, with the first 3 rows as shown. The number 2002 occurs in column $m$ and row $n$. The value of $m+n$ is |
| |
| | | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | Column 6 | Column 7 | |
| | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | |
| | Row 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| | Row 2 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |
| | Row 3 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | |
| | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
| |
| (A) 290 |
| (B) 291 |
| (C) 292 |
| (D) 293 |
| (E) 294","7(286)$. Since there are 7 natural numbers in each row, and the last entry in each row is the multip",easy |
| c94aa6e86047,"For real numbers $a$ and $b$, define |
| $$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$ |
| Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$",4 \sqrt{2018,medium |
| eb5f70d352a1,"2. As shown in the figure, $\odot O$ is tangent to the sides $A B, A D$ of the square $A B C D$ at points $L, K$, respectively, and intersects side $B C$ at points $M, P$, with $B M=8$ cm, $M C=17$ cm. Then the area of $\odot O$ is $\qquad$ square cm.","As shown in the figure, the side length of square $ABCD$ is $25, BM=8$",easy |
| 63c38741f8b5,6. What is the last digit of $1^{1}+2^{2}+3^{3}+\cdots+100^{100}$ ?,"0, L(1, n)=1, L(5, n)=5, L(6, n)=6$. All numbers ending in odd digits in this series are raised to o",medium |
| 251b65e411fe,"If $3^{x}=5$, the value of $3^{x+2}$ is |
| (A) 10 |
| (B) 25 |
| (C) 2187 |
| (D) 14 |
| (E) 45 |
| |
| Part B: Each correct answer is worth 6.",(E),easy |
| 8e8e6ee7975f,"1. Let real numbers $x, y$ satisfy the equation |
| $$ |
| 2 x^{2}+3 y^{2}=4 x \text {. } |
| $$ |
| |
| Then the minimum value of $x+y$ is ( ). |
| (A) $1+\frac{\sqrt{15}}{3}$ |
| (B) $1-\frac{\sqrt{15}}{3}$ |
| (C) 0 |
| (D) None of the above",See reasoning trace,easy |
| c20b6fe692e6,"(solved by Alice Héliou). Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: |
| |
| $$ |
| f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y)) |
| $$ |
| |
| for all real numbers $x$ and $y$.","(x+1)(f(x)-f(1))$, we derive $f(x)=f(1) x$. Therefore, any function satisfying the equation is a lin",medium |
| 54662d14db23,12. The sum of all real roots of the equation $\left|x^{2}-3 x+2\right|+\left|x^{2}+2 x-3\right|=11$ is $\qquad$,See reasoning trace,medium |
| 3c0a416cf569,"Example 3 Let $a b c \neq 0, a+b+c=a^{2}+b^{2}+c^{2}=2$. Find the value of the algebraic expression $\frac{(1-a)^{2}}{b c}+\frac{(1-b)^{2}}{c a}+\frac{(1-c)^{2}}{a b}$.",3$.,easy |
| cf0b3f2b594f,5.085 Six boxes of different materials are delivered to eight floors of a construction site. In how many ways can the materials be distributed across the floors? In how many of these ways will at least two materials be delivered to the eighth floor?,$8^{6} ; 8^{6} - 13 \cdot 7^{5}$,medium |
| 3f2eb9af5d0d,"1. If the sum of positive integers $a$ and $b$ is $n$, then $n$ can be transformed into $a b$. Can this method be used several times to change 22 into 2001?",3 \times 667$ from $3+667=670$; $670=10 \times 67$ from $10+67=77$; $77=7 \times 11$ from $7+11=18$.,easy |
| 20396783cc30,"10,11 |
| |
| The lower base of a truncated quadrilateral pyramid is a rhombus $ABCD$, where $AB=4$ and $\angle BAD=60^{\circ}$. $AA1, BB1, CC1, DD1$ are the lateral edges of the truncated pyramid, edge $A1B1=2$, edge $CC1$ is perpendicular to the base plane and equals 2. A point $M$ is taken on edge $BC$ such that $BM=3$, and a plane is drawn through points $B1$, $M$, and the center of the rhombus $ABCD$. Find the dihedral angle between this plane and the plane $AA1C1C$.",$\arccos \frac{9}{\sqrt{93}}=\operatorname{arctg} \frac{2}{3\sqrt{3}}$,medium |
| f5a4ecc9f9de,"5. For the circle $x^{2}+y^{2}-5 x=0$, passing through the point $\left(\frac{5}{2}, \frac{3}{2}\right)$, if $d \in\left(\frac{1}{6}, \frac{1}{3}\right]$, then the set of values for $n$ is ( ). |
| (A) $\{4,5,6\}$ |
| (B) $\{6,7,8,9\}$ |
| (C) $\{3,4,5\}$ |
| (D) $\{3,4,5,6\}$","\frac{4}{13}\left(\frac{1}{6},-\frac{1}{3}\right]$, which also meets the requirements, in this case,",medium |
| d5865d5d104a,"3. Let $f(x)$ be a monotonic function defined on $(0,+\infty)$. If for any $x \in(0,+\infty)$, we have $f\left[f(x)-2 \log _{2} x\right]=4$, then the solution set of the inequality $f(x)<6$ is $\qquad$.",See reasoning trace,easy |
| 3db2e8341e72,"(a) (2 points) Find $x_{1}+x_{3}+x_{5}$. |
| |
| (b) (2 points) What is the smallest degree that $G(x)$ can have?",(a) -24,medium |
| ee9d8576c63f,"Three. (20 points) There are $m$ regular $n$-sided polygons, and the sum of the interior angles of these $m$ regular polygons can be divided by 8. Find the minimum value of $m+n$.",See reasoning trace,easy |
| ab18f3e422bc,"12. The teacher is buying souvenirs for the students. There are three different types of souvenirs in the store (souvenirs of the same type are identical), priced at 1 yuan, 2 yuan, and 4 yuan respectively. The teacher plans to spend 101 yuan, and at least one of each type of souvenir must be purchased. There are $\qquad$ different purchasing schemes.","101$. When $z=1$, $x+2 y=97, y=1,2, \cdots, 48 ;$ when $z=2$, $x+2 y=93, y=1,2, \cdots, 46$; when $z",easy |
| b0c2b329b0cd,"Example 11 Let $x>0, y>0, \sqrt{x}(\sqrt{x}+2 \sqrt{y})$ $=\sqrt{y}(6 \sqrt{x}+5 \sqrt{y})$. Find the value of $\frac{x+\sqrt{x y}-y}{2 x+\sqrt{x y}+3 y}$.",\frac{1}{2}$.,easy |
| 1a47945582ab,"[ Right triangle with an angle in ] |
| |
| The hypotenuse $AB$ of the right triangle $ABC$ is 2 and is a chord of a certain circle. The leg $AC$ is 1 and lies inside the circle, and its extension intersects the circle at point $D$, with $CD=3$. Find the radius of the circle.",2,easy |
| 40faee1dc6ae,"Yelena recites $P, Q, R, S, T, U$ repeatedly (e.g. $P, Q, R, S, T, U, P, Q, R, \ldots$ ). Zeno recites $1,2,3,4$ repeatedly (e.g. $1,2,3,4,1,2, \ldots$ ). If Yelena and Zeno begin at the same time and recite at the same rate, which combination will not be said? |
| (A) $T 1$ |
| (B) $U 2$ |
| (C) $Q 4$ |
| (D) $R 2$ |
| (E) $T 3$",s with the 12 possibilities given in the table,medium |
| ce6b5b038e64,"11.4 If $\sqrt{3-2 \sqrt{2}}=\sqrt{c}-\sqrt{d}$, find the value of $d$.",See reasoning trace,easy |
| ec1512239d8a,"Example 7 A shipping company has a ship leaving Harvard for New York every noon, and at the same time every day, a ship also leaves New York for Harvard. It takes seven days and seven nights for the ships to complete their journeys in both directions, and they all sail on the same route. How many ships of the same company will the ship leaving Harvard at noon today encounter on its way to New York?",See reasoning trace,medium |
| 2dc31d457cb1,"1. If $x=\frac{a-b}{a+b}$, and $a \neq 0$, then $\frac{b}{a}$ equals ( ). |
| (A) $\frac{1-x}{1+x}$ |
| (B) $\frac{1+x}{1-x}$ |
| (C) $\frac{x-1}{x+1}$ |
| (D) $\frac{x+1}{x-1}$",\frac{1-x}{1+x}$.,easy |
| 269386b573cc,"19. In $\triangle A B C$, $A B=A C, \angle A=100^{\circ}, I$ is the incenter, $D$ is a point on $A B$ such that $B D=B I$. Find the measure of $\angle B C D$.",.,medium |
| 4f1cdcdecb9b,"Esquecinaldo has a terrible memory for remembering numbers, but excellent for remembering sequences of operations. Therefore, to remember his 5-digit bank code, he can remember that none of the digits are zero, the first two digits form a power of 5, the last two digits form a power of 2, the middle digit is a multiple of 3, and the sum of all the digits is an odd number. Now he no longer needs to memorize the number because he knows it is the largest number that satisfies these conditions and has no repeated digits. What is this code?",See reasoning trace,medium |
| 2628ca836ef5,2. (10 points) Calculate: $1+2+4+5+7+8+10+11+13+14+16+17+19+20=$,: 147,easy |
| b993f66b75c4,"2. Consider the sequence of natural numbers $3,10,17,24,31, \ldots$. |
| |
| a) Determine the 2014th term of the sequence. |
| |
| b) Determine the numbers $x$ and $y$ knowing that they are consecutive terms of the sequence and $x<608<y$. |
| |
| Ionuţ Mazalu, Brăila",7 \cdot 86+6-3=605$ and $y=7 \cdot 87+3=612$ $2 p$,easy |
| d00983412a26,"Queenie and Horst play a game on a \(20 \times 20\) chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move. Find the maximal positive \(K\) such that, regardless of the strategy of Queenie, Horst can put at least \(K\) knights on the board. (Armenia)",See reasoning trace,medium |
| c3bcf7db8c19,"21. In the Kangaroo republic each month consists of 40 days, numbered 1 to 40 . Any day whose number is divisible by 6 is a holiday, and any day whose number is a prime is a holiday. How many times in a month does a single working day occur between two holidays? |
| A 1 |
| B 2 |
| C 3 |
| D 4 |
| E 5",See reasoning trace,medium |
| 19e551527a51,"40. Pete bought eight rice and cabbage buns and paid 1 ruble for them. Vasya bought nine buns and paid 1 ruble 1 kopek. How much does a rice bun cost, if it is known that it is more expensive than a cabbage bun and the buns cost more than 1 kopek?",". A rice-filled bun costs 13 kopecks, while a cabbage-filled bun costs 9 or 11 kopecks",easy |
| 82aa1b78b9d3,"Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity |
| $$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$ |
| holds for all $x,y \in \mathbb N$",f(f(n)) = f(n+1),medium |
| 8497c5cf8c4e,"B2 The point $S$ lies on the chord $A B$ of a circle such that $S A=3$ and $S B=5$. The radius of the circle from the center $M$ through $S$ intersects the circle at $C$. Given $C S=1$. |
| |
| Calculate the length of the radius of the circle. |
| |
| ",329&width=466&top_left_y=2194&top_left_x=1229),easy |
| 0410a899dd04,"Let $C$ be the [graph] of $xy = 1$, and denote by $C^*$ the [reflection] of $C$ in the line $y = 2x$. Let the [equation] of $C^*$ be written in the form |
| \[12x^2 + bxy + cy^2 + d = 0.\] |
| Find the product $bc$.",084,medium |
| 07f84f7985de,"5.35 (1) Find the possible minimum value of the polynomial $P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}$. |
| (2) Prove that this polynomial cannot be expressed as a sum of squares of polynomials in variables $x, y$.","p(0, y)=4$, the polynomials $g_{i}(x, y)$ cannot have monomials of the form $a x^{k}$ and $b y^{\pri",medium |
| 842626065c60,"6. Let the complex number $z$ satisfy $|z|=1$. Then |
| $$ |
| |(z+1)+\mathrm{i}(7-z)| |
| $$ |
| |
| cannot be ( ). |
| (A) $4 \sqrt{2}$ |
| (B) $4 \sqrt{3}$ |
| (C) $5 \sqrt{2}$ |
| (D) $5 \sqrt{3}$",See reasoning trace,easy |
| 474e48670e4c,"7. It is known that for some natural numbers $a, b$, the number $N=\frac{a^{2}+b^{2}}{a b-1}$ is also natural. Find all possible values of $N$. |
| |
| --- |
| |
| The provided text has been translated into English while preserving the original formatting and structure.",5,medium |
| c633cc83c992,"4. Vasya remembers that his friend Petya lives on Kurchatovskaya Street, house number 8, but he forgot the apartment number. In response to a request to clarify the address, Petya replied: “My apartment number is a three-digit number. If you rearrange the digits, you can get five other three-digit numbers. So, the sum of these five numbers is exactly 2017.” Help Vasya remember Petya's apartment number.",425,medium |
| 6fc123c612aa,7. A rectangular sheet of iron was divided into 2 parts such that the first part was 4 times larger than the second. What is the area of the entire sheet if the first part is $2208 \mathrm{~cm}^{2}$ larger than the second?,the area of the rectangle is $3680 \mathrm{~cm}^{2}$,medium |
| 6bdb8df06268,"$11 \cdot 90$ For any positive integer $k$, try to find the smallest positive integer $f(k)$, such that there exist 5 sets $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$, satisfying the following conditions: |
| (1) $\left|S_{i}\right|=k, i=1,2,3,4,5$; |
| (2) $S_{i} \cap S_{i+1}=\varnothing\left(S_{6}=S_{1}\right), i=1,2,3,4,5$; |
| (3) $\left|\bigcup_{i=1}^{5} S_{i}\right|=f(k)$. |
|
|
| Also, ask what the result is when the number of sets is a positive integer $n(n \geqslant 3)$.","\varnothing$. Therefore, we have proved that the equality in (1) holds.",medium |
| e2a07beff171,"6. If $f(x)=2 \sin \omega x(0<\omega<1)$ has a maximum value of $\sqrt{2}$ on the interval $\left[0, \frac{\pi}{3}\right]$, then $\omega=$ $\qquad$","2 \sin \left(\frac{\pi}{3} \omega\right)=\sqrt{2}$, so $\omega=\frac{3}{4}$",easy |
| 98dd91b5f600," |
|
|
| Fritz wants to draw four points $A, B, C, D$ in this order on a straight line. The following conditions must be met: |
|
|
| (1) The length of the segment $A D$ should be $15 \mathrm{~cm}$. |
|
|
| (2) The segment $B C$ should be $3 \mathrm{~cm}$ longer than the segment $A B$. |
|
|
| (3) The segment $C D$ should be twice as long as the segment $A C$. |
|
|
| Investigate whether these conditions can be met! If this is the case, determine all the length specifications for the segments $A B, B C$, and $C D$ that satisfy these conditions!","A B+B C=5 \mathrm{~cm}$, the segment $C D$ is twice as long as the segment $A C$, and thus condition",medium |
| b32f25bb940e,"12. Chris planned a $210 \mathrm{~km}$ bike ride. However, he rode $5 \mathrm{~km} / \mathrm{h}$ faster than he planned and finished his ride 1 hour earlier than he planned. His average speed for the ride was $x \mathrm{~km} / \mathrm{h}$. What is the value of $x$ ?","x(x-5)$ and hence $1050=x^{2}-5 x$. Thus $x^{2}-5 x-1050=0$ and hence $(x-35)(x+30)=0$. Therefore, s",medium |
| fd911a413afa,"An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus?
|
|
|
| $ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$
|
| [asy]unitsize(1.4cm); |
| defaultpen(linewidth(.8pt)); |
| dotfactor=3; |
|
|
| real r1=1.0, r2=1.8; |
| pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); |
| pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0]; |
| pair[] points={X,O,Y,Z}; |
|
|
| filldraw(Circle(O,r2),mediumgray,black); |
| filldraw(Circle(O,r1),white,black); |
|
|
| dot(points); |
| draw(X--Y--O--cycle--Z); |
|
|
| label(""$O$"",O,SSW,fontsize(10pt)); |
| label(""$Z$"",Z,SW,fontsize(10pt)); |
| label(""$Y$"",Y,N,fontsize(10pt)); |
| label(""$X$"",X,NE,fontsize(10pt)); |
|
|
| defaultpen(fontsize(8pt)); |
|
|
| label(""$c$"",midpoint(O--Z),W); |
| label(""$d$"",midpoint(Z--Y),W); |
| label(""$e$"",midpoint(X--Y),NE); |
| label(""$a$"",midpoint(X--Z),N); |
| label(""$b$"",midpoint(O--X),SE);[/asy]",\pi a^2,medium |
| 715c8a71ec56,"For a right-angled triangle with acute angles $\alpha$ and $\beta$: |
|
|
| $$ |
| \operatorname{tg} \alpha+\operatorname{tg} \beta+\operatorname{tg}^{2} \alpha+\operatorname{tg}^{2} \beta+\operatorname{tg}^{3} \alpha+\operatorname{tg}^{3} \beta=70 |
| $$ |
|
|
| Determine the angles.",See reasoning trace,medium |
| c1b53234a819,"11. The sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$, satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$ and $a_{0}=3$, then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$ $\qquad$ .",See reasoning trace,medium |
| 8e7b186e535c,"[Bezout's Theorem. Factorization] |
| |
| For what values of the parameter $a$ does the polynomial $P(x)=x^{n}+a x^{n-2}(n \geq 2)$ divide by $x-2$?",For $a=-4$,easy |
| 1fe892191bdf,"6. A rectangular piece of land enclosed by fences has a length and width of $52 \mathrm{~m}$ and $24 \mathrm{~m}$, respectively. An agricultural science technician wants to divide this land into several congruent square test plots. The land must be fully divided, and the sides of the squares must be parallel to the boundaries of the land. There are $2002 \mathrm{~m}$ of fencing available. The land can be divided into a maximum of $\qquad$ square test plots.","3$. At this point, the total number of squares is $m n=702$ (pieces).",medium |
| 52f51bd03414,"If $50 \%$ of $N$ is 16 , then $75 \%$ of $N$ is |
| (A) 12 |
| (B) 6 |
| (C) 20 |
| (D) 24 |
| (E) 40",(D),easy |
| 148fd5357ab8,"A1. Samo wrote a 3-digit odd natural number on a piece of paper and told Peter only the last digit of this number. Peter immediately realized that the number Samo wrote on the paper is not a prime number. Which digit did Samo tell Peter? |
| (A) 1 |
| (B) 3 |
| (C) 5 |
| (D) 7 |
| (E) 9",See reasoning trace,easy |
| e28165149107,Consider a right-angled triangle with an area of $t=84 \mathrm{~cm}^{2}$ and a perimeter of $k=56 \mathrm{~cm}$. Calculate the lengths of the sides without using the Pythagorean theorem.,See reasoning trace,medium |
| 4bcb3a94feea,"## Task B-3.2. |
| |
| The first digit of a four-digit number is one greater than its third digit, and the second digit is equal to the sum of the remaining digits. The last digit of this number is five less than the first digit. Determine this four-digit number.","3 \cdot 5-6=9, c=5-1=4, d=5-5=0$. The desired four-digit number is 5940.",medium |
| 6d7388b3818b,"$\mathrm{Az}$ |
| |
| $$ |
| \frac{x^{2}+p}{x}=-\frac{1}{4} |
| $$ |
| |
| equation, whose roots are $x_{1}$ and $x_{2}$, determine $p$ such that |
| a) $\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=-\frac{9}{4}$, |
| |
| b) one root is 1 less than the square of the other root.",See reasoning trace,medium |
| 9179a3d91f2a,"F9 (27-5, UK) Let $f$ be a function defined on the set of non-negative real numbers and taking values in the same set. Find all functions $f$ that satisfy the following conditions: |
| (1) $f(x f(y)) f(y)=f(x+y)$; |
| (2) $f(2)=0$; |
| (3) $f(x) \neq 0$, when $0 \leqslant x<2$.",See reasoning trace,medium |
| d79b31690480,"11. (15 points) Given the function $f(x)=-2 x^{2}+b x+c$ has a maximum value of 1 at $x=1$, and $0<m<n$. When $x \in [m, n]$, the range of $f(x)$ is $\left[\frac{1}{n}, \frac{1}{m}\right]$. Find the values of $m$ and $n$.","1, n=\frac{1+\sqrt{3}}{2}$.",medium |
| 90f055c9b6b4,"3. (8 points) When Tongtong was calculating a division with a remainder, she mistakenly read the dividend 472 as 427, resulting in a quotient that was 5 less than the original, but the remainder was exactly the same. What is this remainder? |
| A. 4 |
| B. 5 |
| C. 6 |
| D. 7",The remainder is 4,easy |
| 986c7cac8bb7,"Example 5 Let $n$ be a positive integer. How many solutions does $x^{2}-\left[x^{2}\right]=$ $(x-[x])^{2}$ have in $1 \leqslant x \leqslant n$? |
| $",See reasoning trace,medium |
| 15e52228d720,We write a rotational cone around a unit radius sphere. What is the minimum surface area of the cone?,See reasoning trace,medium |
| 7e86475b2b1b,"A circle of radius $4$ is inscribed in a triangle $ABC$. We call $D$ the touchpoint between the circle and side BC. Let $CD =8$, $DB= 10$. What is the length of the sides $AB$ and $AC$?",12.5,medium |
| 6ec292f053b7,"## Task 5 - 211235 |
| |
| 37 cards, each of which is colored red on one side and blue on the other, are laid on a table so that exactly 9 of them show their blue side on top. |
| |
| Now, in work steps, cards are to be flipped, and in each individual work step, exactly 20 of the 37 cards are to be flipped. |
| |
| Investigate whether it is possible to achieve with a finite number of work steps that all 37 cards |
| |
| a) show their red side on top, |
| |
| b) show their blue side on top. |
| |
| If this is possible, determine the smallest number of work steps required for each case!",See reasoning trace,medium |
| c3f2964d5372,"3. Calculate the angle $\gamma$ in triangle $ABC$, if |
| |
| $$ |
| \frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c} |
| $$ |
| |
| where $a, b, c$ are the lengths of the sides of triangle $ABC$ opposite to vertices $A, B, C$ respectively, and $\gamma$ is the angle at vertex $C$.","\frac{1}{2}$, from which we get $\gamma=60^{\circ}$.",medium |
| 6f72f0a005fb,"20.50 Line segment $AB$ is both the diameter of a circle with radius 1 and a side of equilateral $\triangle ABC$. This circle intersects $AC$ and $BC$ at points $D$ and $E$. The length of $AE$ is |
| (A) $\frac{3}{2}$. |
| (B) $\frac{5}{3}$. |
| (C) $\frac{\sqrt{3}}{2}$. |
| (D) $\sqrt{3}$. |
| (E) $\frac{2+\sqrt{3}}{2}$. |
| (34th American High School Mathematics Examination, 1983)",$(D)$,easy |
| b7832781483f,"A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.",272,medium |
| c020560d9f71,"6. Several cells on a $14 \times 14$ board are marked. It is known that no two of the marked cells are in the same row and the same column, and also that a knight can, starting from any marked cell, reach any other marked cell via marked cells. What is the maximum possible number of marked cells?",14,easy |
| 2f24657b06dd,Example 2. Find the third-order derivative of the function $y=x^{2}+3 x+2$.,See reasoning trace,easy |
| f87f315bada1,"Example 12 Given that $x, y, z$ are 3 non-negative rational numbers, and satisfy $3x+2y+z=5, x+y-z=2$. If $s=2x+y-z$, then what is the sum of the maximum and minimum values of $s$? |
| (1996, Tianjin Junior High School Mathematics Competition)",See reasoning trace,medium |
| 940cca7e3698,"Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of $n$.",n \equiv 1 \pmod{2,medium |
| 79a54913e69a,"Example 2 Find the maximum positive integer $n$, such that there exist $n$ distinct real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying: for any $1 \leqslant i<$ $j \leqslant n$, we have $\left(1+x_{i} x_{j}\right)^{2} \leqslant 0.99\left(1+x_{i}^{2}\right)\left(1+x_{j}^{2}\right)$.",See reasoning trace,medium |
| cfd174ad357e,"7.042. $x(\lg 5-1)=\lg \left(2^{x}+1\right)-\lg 6$. |
| |
| Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. |
|
|
| 7.042. $x(\lg 5-1)=\lg \left(2^{x}+1\right)-\lg 6$.",1,easy |
| a0891f813c03,"5-6. A rectangular table of size $x$ cm $\times 80$ cm is covered with identical sheets of paper of size 5 cm $\times 8$ cm. The first sheet touches the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length $x$ in centimeters? |
|
|
| ",77,medium |
| 57f8b7389a62,"Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying for all $x, y \in \mathbb{R}$, |
|
|
| $$ |
| f(x+y)=f(x-y) |
| $$",See reasoning trace,medium |
| 8d02758fa1ab,"Example 8 Find the sum: |
| $$ |
| \begin{array}{l} |
| \left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{60}\right)+\left(\frac{2}{3}+\frac{2}{4}+\cdots+\frac{2}{60}\right)+ \\ |
| \left(\frac{3}{4}+\frac{3}{5}+\cdots+\frac{3}{60}\right)+\cdots+\left(\frac{58}{59}+\frac{59}{60}\right) |
| \end{array} |
| $$",See reasoning trace,easy |
| 4db484b64119,"The bisector of an angle is a ray originating from the vertex of the angle that divides it into two other congruent angles. For example, in the drawing below, the ray $O C$ is the bisector of the angle $\angle A O B$. |
|
|
|  |
|
|
| a) The difference between two consecutive but non-adjacent angles is $100^{\circ}$. Determine the angle formed by their bisectors. |
|
|
| Note: Remember that two angles are consecutive if they share the same vertex and at least one side, and that two angles are adjacent if they do not share any interior points. |
|
|
| b) In the drawing below, $D A$ is the bisector of the angle $\angle C A B$. Determine the value of the angle $\angle D A E$ given that $\angle C A B + \angle E A B = 120^{\circ}$ and $\angle C A B - \angle E A B = 80^{\circ}$. |
|
|
| ",See reasoning trace,medium |
| 0f8b9ff2fbeb,"Two points $B$ and $C$ are in a plane. Let $S$ be the set of all points $A$ in the plane for which $\triangle ABC$ has area $1.$ Which of the following describes $S?$ |
| $\textbf{(A) } \text{two parallel lines} \qquad\textbf{(B) } \text{a parabola} \qquad\textbf{(C) } \text{a circle} \qquad\textbf{(D) } \text{a line segment} \qquad\textbf{(E) } \text{two points}$",\mathrm{(A) \,easy |
| 1d56cf88d778,(4) The smallest positive integer $a$ that makes the inequality $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}<a-2007 \frac{1}{3}$ hold for all positive integers $n$ is $\qquad$.,"\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}$. Clearly, $f(n)$ is monotonically decreasing. Th",easy |
| 98dbdbcca82e,"Solve the following equation: |
|
|
| $$ |
| \sqrt[4]{16+x}+\sqrt[4]{16-x}=4 |
| $$","a_{2}=\ldots a_{n}$. In the solution, we proved this inequality for $n=2, \alpha=1 / 4$ and $\beta=1",medium |
| 7432e4adabc9,"Example 7 There is a sequence of numbers: $1,3,4,7,11,18, \cdots$, starting from the third number, each number is the sum of the two preceding numbers. |
| (1) What is the remainder when the 1991st number is divided by 6? |
| (2) Group the above sequence as follows: |
| $(1),(3,4),(7,11,18), \cdots$, |
|
|
| where the $n$-th group has exactly $n$ numbers. What is the remainder when the sum of the numbers in the 1991st group is divided by 6? |
| (6th Spring Festival Cup Mathematics Competition)",See reasoning trace,medium |
| d391b0a5c8e7," |
|
|
| In the electrical circuit shown in the diagram, the resistances of the resistors are $R_{1}=10$ Ohms and $R_{2}=20$ Ohms. A current |
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|  |
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| 9aefaf8c047d,"## Task Condition |
| |
| Find the differential $d y$. |
| |
| $$ |
| y=\operatorname{arctg} \frac{x^{2}-1}{x} |
| $$",See reasoning trace,medium |
| 7a317a34c9d0,"In the expansion of $(x+y)^{n}$ using the binomial theorem, the second term is 240, the third term is 720, and the fourth term is 1080. Find $x, y$, and $n$. |
| |
| #","$x=2, y=3, n=5$",easy |
| 5a0c3c09ef3d,"7. We have 9 numbers: $-6,-4,-2,-1,1,2,3,4,6$. It is known that the sum of some of these numbers is -8. Write down these numbers.",See reasoning trace,easy |
| 1eaf1b229646,"Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials |
| \(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\). |
| |
| [list='a'] |
| [*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\). |
| [*]Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\). |
| [/list]",1008,medium |
| dc4cc458cdcc,4. The range of the function $y=x+\sqrt{x^{2}-3 x+2}$ is,"(y-x)^{2} \\ y \geqslant x\end{array}\right.$ having a solution, which means $\left\{\begin{array}{l",easy |
| aa9c34293b21,"The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is |
| |
| $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $",13,medium |
| 669923858bc0,"As shown in Figure 2, given a square $ABCD$, extend $BC$ and $DC$ to $M$ and $N$ respectively, such that $S_{\triangle QMN} = S_{\text{square } ABCD}$. Determine the degree measure of $\angle MAN$.",See reasoning trace,medium |
| c186b9c426fa,"# |
| |
| From a natural number, its largest digit was subtracted. Then, from the resulting number, its largest digit was also subtracted, and so on. After five such operations, the number 200 was obtained. What number could have been written initially? List all the options and prove that there are no others.",See reasoning trace,medium |
| 6242a7a96fff,12. A. Do there exist prime numbers $p$ and $q$ such that the quadratic equation $p x^{2}-q x+p=0$ has rational roots?,See reasoning trace,medium |
| 37e353421f51,"For a journey of $25 \mathrm{~km}$, she spends 3 liters, so to travel $100 \mathrm{~km}$, Maria will spend $4 \times 3=12$ liters. Therefore, to travel $600 \mathrm{~km}$, the car will consume $6 \times 12=72$ liters. Since each liter costs 0.75 reais, then 72 liters will cost $0.75 \times 72=54$ reais.","25 \times 24$, the car will consume $24 \times 3=72$ liters.",easy |
| 93d08176cfcd,"An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] |
| the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? |
| |
| |
| [i]2021 CCA Math Bonanza Individual Round #11[/i]",107,medium |
| 268aca397ab8,"4. (10 points) The bell of 2013 has rung, and Brother Xiaoming sighed: This is the first time in my life that I will go through a year with no repeated digits. It is known that Brother Xiaoming was born in a year that is a multiple of 19. Therefore, in 2013, Brother Xiaoming's age is ( ) years old. |
| A. 16 |
| B. 18 |
| C. 20 |
| D. 22",See reasoning trace,easy |
| 55774e6756c2,"1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is","BE = \frac{1}{2}, AF = \frac{\sqrt{3}}{2}$ $\Rightarrow EF = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^{",medium |
| 6943156b71f8,"1. Let $n$ be a natural number, $a, b$ be positive real numbers, and satisfy the condition $a+b=2$, then the minimum value of $\frac{1}{1+a^{n}}+\frac{1}{1+b^{n}}$ is $\qquad$.",\frac{1+a^{n}+1+b^{n}}{1+a^{n}+b^{n}+a^{n} b^{n}} \geqslant \frac{1+a^{n}+b^{n}+1}{1+a^{n}+b^{n}+1}=,medium |
| 7617e2533ca2,Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$.,251,medium |
| ed0d1684fa39,"Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says: |
| |
| - Alina, your age is now double the age of Valentina. |
| |
| a) Show that this could never have happened before and that, after the next birthday of Valentina, it will never happen again. |
| |
| Then, Alina, who was very clever, exclaimed: |
| |
| - Dad, exactly 10 years ago, the age of one of us three was double the age of another, and in 10 years, the same fact will happen again! |
| |
| b) It is known that the oldest of the daughters is over 30 years old. How old is Civela?",See reasoning trace,medium |
| 87be2b89a2b5,"6. (15 points) The radar received the signal reflected from the target after 6 microseconds. Determine the distance to the target, given that the speed of the radar signal is $300000 \mathrm{k} \nu / \mathrm{c}$. Note that one microsecond is one millionth of a second.",See reasoning trace,easy |
| cc33d6ee17ba,"Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$",\frac{1 \pm \sqrt{5,medium |
| 0b781e8d244d,"The real numbers $a_1,a_2,a_3$ and $b{}$ are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots, $c_1,c_2,c_3.$ Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\][i]Proposed by A. Antropov and K. Sukhov[/i]","-a_1, -a_2, -a_3",medium |
| 4ff8f4495a5b,6.143. $\frac{1}{x^{2}}+\frac{1}{(x+2)^{2}}=\frac{10}{9}$.,"$x_{1}=-3, x_{2}=1$",medium |
| d05c506b52cd,"In triangle $ABC$, points $D$ and $E$ lie on side $AB$, with $D$ being closer to $A$. $AD=100 \mathrm{~m}, EB=200 \mathrm{~m}$, $\angle ACD=30^{\circ}, \angle DCE=50^{\circ}$, and $\angle ECB=35^{\circ}$. What is the length of segment $DE$?",See reasoning trace,medium |
| 487bcff87622,"In a circle with center $O$ and radius $r$, let $A B$ be a chord smaller than the diameter. The radius of the circle inscribed in the smaller sector determined by the radii $A O$ and $B O$ is $\varrho$. Express $A B$ in terms of $r$ and $\varrho$.",277&width=460&top_left_y=428&top_left_x=846),medium |
| 2775650d80ac,"Task 2. Let's call a year interesting if a person turns as many years old as the sum of the digits of their birth year in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century. What is the difference in their ages? |
| |
| Note. For convenience, we assume they were born on the same day, and all calculations are done in whole years.",18 years,medium |
| 5ee30004f85d,"Given $c \in\left(\frac{1}{2}, 1\right)$. Find the smallest constant $M$, such that for any integer $n \geqslant 2$ and real numbers $0<a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, if $\frac{1}{n} \sum_{k=1}^{n} k a_{k}=c \sum_{k=1}^{n} a_{k}$, then $\sum_{k=1}^{n} a_{k} \leqslant M \sum_{k=1}^{m} a_{k}$, where $m=[c n]$ denotes the greatest integer not exceeding $c n$.",See reasoning trace,medium |
| d15d272e5640,"Two unit-radius circles intersect at points $A$ and $B$. One of their common tangents touches the circles at points $E$ and $F$. What can be the radius of a circle that passes through points $E, F$, and $A$?",See reasoning trace,medium |
| 93cb61d7163c,"One, (20 points) Given real numbers $a, b, c, d$ satisfy $2a^2 + 3c^2 = 2b^2 + 3d^2 = (ad - bc)^2 = 6$. Find the value of $\left(a^2 + \dot{b}^2\right)\left(c^2 + d^2\right)$.",See reasoning trace,medium |
| 1a661796cf1b,"$$ |
| d(3 n+1,1)+d(3 n+2,2)+\cdots+d(4 n, n)=2006 |
| $$ |
| |
| Ivan Landjev",708$.,medium |
| 28acd5aabcfa,"31. Given the difference of squares formula: $a^{2}-b^{2}=(a+b) \times(a-b)$. Calculate: |
| $$ |
| 99^{2}+97^{2}-95^{2}-93^{2}+91^{2}+89^{2}-87^{2}-85^{2}+\cdots+11^{2}+9^{2}-7^{2}-5^{2}= |
| $$ |
| $\qquad$",9984,easy |
| 33b779a549e0,"5.6. Find the unit vector collinear with the vector directed along the bisector of angle $B A C$ of triangle $A B C$, if its vertices are given: $A(1 ; 1 ; 1), B(3 ; 0 ; 1), C(0 ; 3 ; 1)$.",$\left(\frac{1}{\sqrt{2}} ; \frac{1}{\sqrt{2}} ; 0\right)$,medium |
| f4ae26b95ac4,"$36 \cdot 28$ There are 7 boys and 13 girls standing in a row, let $S$ be the number of positions where a boy and a girl are adjacent, for example, the arrangement $G B B G G G B G B G G G B G B G G B G G$ has $S=12$, then the average value of $S$ is closest to (considering all 20 positions) |
| (A) 9 . |
| (B) 10 . |
| (C) 11 . |
| (D) 12 . |
| (E) 13 . |
| (40th American High School Mathematics Examination, 1989)",$(A)$,medium |
| 368ea45fd996,"Which is the three-digit (integer) number that, when increased or decreased by the sum of its digits, results in a number consisting of the same digit repeated?","105$. This indeed fits, because the sum of its digits is 6 and $105+6=111,105-6=99$.",medium |
| 170b19c0b438,"25. The cube shown has sides of length 2 units. Holes in the shape of a hemisphere are carved into each face of the cube. The six hemispheres are identical and their centres are at the centres of the faces of the cube. The holes are just large enough to touch the hole on each neighbouring face. What is the diameter of each hole? |
| A 1 |
| B $\sqrt{2}$ |
| C $2-\sqrt{2}$ |
| D $3-\sqrt{2}$ |
| E $3-\sqrt{3}$",M Q=1$. Also $M P Q$ is a right-angled triangle since the two faces are perpendicular. By Pythagoras,easy |
| a33ab83a1c40,"17. Find all positive integers $(x, y, z, n)$, such that |
| |
| $$ |
| x^{3}+y^{3}+z^{3}=n x^{2} y^{2} z^{2} |
| $$",See reasoning trace,medium |
| fe1e8adc3066,"## Task B-2.5. |
| |
| The math test consisted of three |
| |
| The first and second |
| |
| It is known that there are 24 students in the class and that no student failed to solve any of the",See reasoning trace,medium |
| d6153674993e,"69. This year, the sum of Dan Dan's, her father's, and her mother's ages is 100 years. If 6 years ago, her mother's age was 4 times Dan Dan's age, and 13 years ago, her father's age was 15 times Dan Dan's age, then Dan Dan is $\qquad$ years old this year.",15,easy |
| 0dcb7c9c6692,"19 Let $p$ and $q$ represent two consecutive prime numbers. For some fixed integer $n$, the set $\{n-1,3 n-19,38-5 n, 7 n-45\}$ represents $\{p, 2 p, q, 2 q\}$, but not necessarily in that order. Find the value of $n$.","6 n-27, p+q=2 n-9$ which is odd. So $p=2, q=3$ and $n=7$.",easy |
| abe2c4156162,"8. Given that $a, b, c$ are the lengths of the 3 sides of a right triangle, and for a natural number $n$ greater than 2, $\left(a^{n} + b^{n} + c^{n}\right)^{2} = 2\left(a^{2 n} + b^{2 n} + c^{2 n}\right)$ holds, then $n=$ $\qquad$ .","2, n=4$.",medium |
| a0700f510b20,"## Task 4. |
| |
| Determine all triples $(a, b, c)$ of natural numbers for which |
| |
| $$ |
| a \mid (b+1), \quad b \mid (c+1) \quad \text{and} \quad c \mid (a+1) |
| $$","a$ and $c=a+1$, and in the second case, we consider the possibilities $b=a$, $b=a+1$, and $b=a+2$.",medium |
| 20475189de71,"2. Function $f$ is defined on the set of integers, satisfying |
| $$ |
| f(n)=\left\{\begin{array}{ll} |
| n-3 & n \geqslant 1000 \\ |
| f[f(n+5)] & n<1000 |
| \end{array},\right. |
| $$ |
| |
| Find $f(84)$.",See reasoning trace,easy |
| 5b3f672b4dce,"Alice sends a secret message to Bob using her RSA public key $n=400000001$. Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n$. Eve knows that $n=pq$, a product of two prime factors. Find $p$ and $q$.",19801 \text{ and,medium |
| 3e4c8bf85da5,"1. (8 points) Today is December 19, 2010, welcome to the 27th ""Mathematical",The integer part is 16,easy |
| 071d270ccef4,"Let's determine $m$ such that the expression |
| |
| $$ |
| m x^{2}+(m-1) x+m-1 |
| $$ |
| |
| is negative for all values of $x$. |
| |
| --- |
| |
| Determine $m$ so that the expression |
| |
| $$ |
| m x^{2}+(m-1) x+m-1 |
| $$ |
| |
| is negative for all values of $x$.",See reasoning trace,medium |
| 11d7246968e2,"There are $2020\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions. |
| ・For any square, there are at least two pieces that are on the diagonals containing that square. |
| |
| Note : We say the square $(a,b)$ is on the diagonals containing the square $(c,d)$ when $|a-c|=|b-d|$.",2020,medium |
| 1929ea1a031d,"Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together? |
| $\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880$",\textbf{(C),medium |
| 57d144bfbfb5,"5. Find the greatest common divisor and the least common multiple of the following expressions: |
| |
| $$ |
| 4^{x}-9^{x}, \quad 4^{x}+2 \cdot 6^{x}+9^{x}, \quad 4^{x}+3 \cdot 6^{x}+2 \cdot 9^{x}, \quad 8^{x}+27^{x} |
| $$ |
| |
| Solve each |
| |
| Do not sign the sheets, write only your code. |
| |
| Write your answers with a pen clearly and neatly. Each |
| |
| 4th National Mathematics Competition for Students of Secondary Technical and Vocational Schools |
| |
| Maribor, April 17, 2004 |
| |
| ##",See reasoning trace,medium |
| dbfa6f1d7467,"Task B-3.2. Calculate the area of a rhombus whose shorter diagonal is $8 \mathrm{~cm}$ long, and the measure of the obtuse angle is $150^{\circ}$.","32 \operatorname{tg} 75^{\circ}, P=\frac{32}{\operatorname{tg} 15^{\circ}}, P=\frac{8}{\sin ^{2} 15^",medium |
| 02453b2ea843,"If we extend the internal angle bisectors of the triangle $ABC$ to the circumference of the circumcircle of the triangle, we obtain the triangle $A^{\prime} B^{\prime} C^{\prime}$; the sides of this triangle intersect the angle bisectors of the original triangle at points $A_{1}, B_{1}$, and $C_{1}$; the angle bisectors of the triangle $A_{1} B_{1} C_{1}$ intersect the circumference of the circumcircle of this triangle at points $A^{\prime \prime}, B^{\prime \prime}, C^{\prime \prime}$; the sides of the triangle $A^{\prime \prime}, B^{\prime \prime}, C^{\prime \prime}$ intersect the angle bisectors of the triangle $A_{1} B_{1} C_{1}$ at points $A_{2} B_{2} C_{2}$, and so on. Continuing this process to infinity, calculate the sum of the perimeters and areas of the triangles |
| |
| $$ |
| A B C, \quad A_{1} B_{1} C_{1}, \quad A_{2} B_{2} C_{2}, \ldots |
| $$ |
| |
| if the perimeter of the triangle $ABC$ is $K$ and its area is $T$. Furthermore, show that |
| |
| $$ |
| T: T^{\prime}=8 \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \sin \frac{\gamma}{2} |
| $$ |
| |
| and |
| |
| $$ |
| r: r^{\prime}=a b c\left(a^{\prime}+b^{\prime}+c^{\prime}\right): a^{\prime} b^{\prime} c^{\prime}(a+b+c), |
| $$ |
| |
| where $T^{\prime}$ is the area of the triangle $A^{\prime} B^{\prime} C^{\prime}$, and $r$ and $r^{\prime}$ are the radii of the inscribed circles of the respective triangles.",See reasoning trace,medium |
| 579870a1d61a,"1. Find all values of $p$, for each of which the numbers $-p-12, 2 \cdot \sqrt{p}$, and $p-5$ are respectively the first, second, and third terms of some geometric progression.",$p=4$,easy |
| ae0f1efd3212,33. It is known that there is only one pair of positive integers $a$ and $b$ such that $a \leq b$ and $a^{2}+b^{2}+8 a b=2010$. Find the value of $a+b$.,a^{2}+b^{2}+8 a b \geq 1+b^{2}+8 b . b^{2}+8 b-2009 \leq 0$. However $b^{2}+8 b-2009=0$ has an integ,easy |
| 5c5b8e6923d7,9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?,See reasoning trace,medium |
| c9e764068a31,Example 3. Solve the equation $y^{\prime \prime}+y=\frac{1}{\cos x}$.,See reasoning trace,medium |
| 7a482552de68,3.15. Form the equation of a circle with center at point $P_{0}(1; -2)$ and radius $R=3$.,3^{2}=9$.,easy |
| e4966c1940ce,"Let $BCDE$ be a trapezoid with $BE\parallel CD$, $BE = 20$, $BC = 2\sqrt{34}$, $CD = 8$, $DE = 2\sqrt{10}$. Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$, and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$, and let $X$ be the intersection of $AM$ and $BE$. If $BX$ can be written as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers, find $a + b$",203,medium |
| 3515804017c1,"II. (25 points) Given the quadratic function |
| $$ |
| y=x^{2}+b x-c |
| $$ |
| |
| the graph passes through three points |
| $$ |
| P(1, a), Q(2,3 a)(a \geqslant 3), R\left(x_{0}, y_{0}\right) . |
| $$ |
| |
| If the centroid of $\triangle P Q R$ is on the $y$-axis, find the minimum perimeter of $\triangle P Q R$.",See reasoning trace,medium |
| c2566adfbad5,"## Task 3 - 150523 |
| |
| When a pioneer group reported on their trips taken over the past few years, the following was revealed: |
| |
| (1) Exactly 13 members of this group have been to the Baltic Sea. |
| |
| (2) Exactly 15 pioneers have been to the Harz. |
| |
| (3) Exactly 6 pioneers have been to both the Baltic Sea and the Harz. |
| |
| (4) Exactly 4 pioneers have never been to either the Baltic Sea or the Harz. |
| |
| Determine the total number of pioneers who belong to this group!",26$.,easy |
| 8100da96bbd2,"9. Let the function $f(x)=\frac{a x}{2 x+3}$. If $f(f(x))=x$ always holds, then the value of the real number $a$ is $\qquad$ .",See reasoning trace,easy |
| 87d13326b1df,"For which of the following values of $x$ is $x^{3}<x^{2}$ ? |
| (A) $x=\frac{5}{3}$ |
| (B) $x=\frac{3}{4}$ |
| (C) $x=1$ |
| (D) $x=\frac{3}{2}$ |
| (E) $x=\frac{21}{20}$",(B),medium |
| 44aea608e122,"A4. Two sides of a triangle are $7a-4b$ and $11a-3b$ units long, where $a$ and $b$ are natural numbers. What is the third side if the perimeter is $21a+2b$ units? |
| (A) $15a+16b$ |
| (B) $2a-21b$ |
| (C) $3a+9b$ |
| (D) $2a+5b$ |
| (E) $3a-9b$","a+b+c$, substitute the given data $21a+2b=7a-4b+11a-3b+c$. We calculate that $c=3a+9b$.",easy |
| f3665d1dafab,8. Find the value using multiple methods: $\cos \frac{\pi}{7}-\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}$.,See reasoning trace,medium |
| b523e0310ebe,"Given an integer $\mathrm{n} \geq 3$, let $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots, \mathrm{~A}_{2 \mathrm{n}}$ be pairwise distinct non-empty subsets of the set $\{1,2, \ldots, \mathrm{n}\}$, and let $A_{2 n+1}=A_{1}$. Find the maximum value of $\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}$.",1}^{2 n} \frac{\left|A_{1} \cap A_{i+1}\right|}{\left|A_{1}\right| \cdot \left|A_{i+1}\right|}$ is $,medium |
| 681e3c9f072d,"SG. 1 If $a * b=a b+1$, and $s=(2 * 4) * 2$, find $s$. |
| SG. 2 If the $n^{\text {th }}$ prime number is $s$, find $n$. |
| SG. 3 If $K=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right) \cdots\left(1-\frac{1}{50}\right)$, find $K$ in the simplest fractional form. |
| SG. 4 If $A$ is the area of a square inscribed in a circle of radius 10 , find $A$.",See reasoning trace,medium |
| e8c72585421c,"14.3. 29 * Let $m$ and $n$ be natural numbers, how many different prime factors can the natural number |
| $$ |
| m(n+9)\left(m+2 n^{2}+3\right) |
| $$ |
| |
| have at least?",2,easy |
| ef30d4a552a5,"Example 2-61 $n$ ones and $n$ zeros form a $2n$-bit binary number. It is required that when scanning from left to right, the cumulative number of 1s is not less than the cumulative number of 0s. Try to find how many numbers satisfy this condition.",See reasoning trace,medium |
| a16b0a8e3ff5,How does the volume of a regular tetrahedron relate to that of a regular octahedron with the same edge length?,See reasoning trace,medium |
| 32cb65915b82,"One, (20 points) On the Cartesian plane, it is known that the line $y=x+a(-1<a<1)$ intersects the parabola $y=1-x^{2}$ at points $A$ and $B$, and the coordinates of point $C$ are $(1,0)$. Question: For what value of $a$ is the area of $\triangle A B C$ maximized? Find the maximum area of $\triangle A B C$.","\frac{1}{2}$. Therefore, the maximum area of $\triangle A B C$ is $\frac{3 \sqrt{3}}{4}$.",medium |
| 78599eb83a5b,"For example, $5 x, y, z$ are real numbers, and satisfy $x+y+z=0, xyz=2$. Find the minimum value of $|x|+|y|+|z|$. |
| (1990, Beijing Junior High School Mathematics Competition).",See reasoning trace,medium |
| 24ac62bacf07,"12. If the integer part of $\frac{1}{\sqrt{17-12 \sqrt{2}}}$ is $a$, and the fractional part is $b$, then, $a^{2}-$ $a b+b^{2}$ is $\qquad$ .",(a+b)^{2}-3 a b$ $=17+12 \sqrt{2}-30(\sqrt{2}-1)=47-18 \sqrt{2}$.,easy |
| a08d43f189f6,"Three. (25 points) Given that $a$ is a positive integer not greater than 2005, and $b, c$ are integers, the parabola $y=a x^{2}+b x+c$ is above the $x$-axis and passes through points $A(-1,4 a+7)$ and $B(3,4 a-1)$. Find the minimum value of $a-b+c$. |
| |
| The parabola $y=a x^{2}+b x+c$ passes through points $A(-1,4 a+7)$ and $B(3,4 a-1)$. |
| |
| Substituting these points into the equation of the parabola, we get: |
| 1. For point $A(-1,4 a+7)$: |
| \[ 4a + 7 = a(-1)^2 + b(-1) + c \] |
| \[ 4a + 7 = a - b + c \] |
| \[ 3a + 7 = -b + c \] |
| \[ c - b = 3a + 7 \quad \text{(1)} \] |
| |
| 2. For point $B(3,4 a-1)$: |
| \[ 4a - 1 = a(3)^2 + b(3) + c \] |
| \[ 4a - 1 = 9a + 3b + c \] |
| \[ -5a - 1 = 3b + c \] |
| \[ c + 3b = -5a - 1 \quad \text{(2)} \] |
| |
| We now have the system of linear equations: |
| \[ c - b = 3a + 7 \quad \text{(1)} \] |
| \[ c + 3b = -5a - 1 \quad \text{(2)} \] |
| |
| Subtract equation (1) from equation (2): |
| \[ (c + 3b) - (c - b) = (-5a - 1) - (3a + 7) \] |
| \[ 4b = -8a - 8 \] |
| \[ b = -2a - 2 \] |
| |
| Substitute \( b = -2a - 2 \) into equation (1): |
| \[ c - (-2a - 2) = 3a + 7 \] |
| \[ c + 2a + 2 = 3a + 7 \] |
| \[ c = a + 5 \] |
| |
| Now, we need to find \( a - b + c \): |
| \[ a - b + c = a - (-2a - 2) + (a + 5) \] |
| \[ a + 2a + 2 + a + 5 \] |
| \[ 4a + 7 \] |
| |
| Since \( a \) is a positive integer not greater than 2005, the minimum value of \( 4a + 7 \) occurs when \( a = 1 \): |
| \[ 4(1) + 7 = 11 \] |
| |
| Thus, the minimum value of \( a - b + c \) is: |
| \[ \boxed{11} \]",See reasoning trace,medium |
| 1ed9834376b3,"6. Given $a, b, c \in [0,1]$. Then |
| $$ |
| \frac{a}{bc+1}+\frac{b}{ca+1}+\frac{c}{ab+1} |
| $$ |
| |
| the range of values is",See reasoning trace,medium |
| 9344373c8adf,"Let $*$ denote an operation, assigning a real number $a * b$ to each pair of real numbers ( $a, b)$ (e.g., $a * b=$ $a+b^{2}-17$ ). Devise an equation which is true (for all possible values of variables) provided the operation $*$ is commutative or associative and which can be false otherwise.",(x * x) * x$ which is obviously true if $*$ is any commutative or associative operation but does not,easy |
| 3e9737af191d,"9. Vremyankin and Puteykin simultaneously set out from Utrenneye to Vechernoye. The first of them walked half of the time spent on the journey at a speed of $5 \mathrm{km} / \mathrm{h}$, and then at a speed of $4 \mathrm{km} /$ h. The second, however, walked the first half of the distance at a speed of 4 km/h, and then at a speed of 5 km/h. Who arrived in Vechernoye earlier?",See reasoning trace,easy |
| 9baba25f67df,"13. Given that $p$, $q$, $\frac{2 q-1}{p}$, and $\frac{2 p-1}{q}$ are all integers, and $p>1$, $q>1$. Find the value of $p+q$.","3$, and simultaneously $p=2 q-1=5$, hence $p+q=8$.",medium |
| c7b44bf4228a,"7.251. $x^{2-\lg ^{2} x-\lg x^{2}}-\frac{1}{x}=0$. |
| |
| 7.251. $x^{2-\log ^{2} x-\log x^{2}}-\frac{1}{x}=0$.",0,medium |
| 1048ea3ca80d,"13. (6 points) Given the five-digit number $\overline{54 \mathrm{a} 7 \mathrm{~b}}$ can be simultaneously divisible by 3 and 5, the number of such five-digit numbers is $\qquad$.",: 7,medium |
| 9165e43fec67,"## Task 5 - 120735 |
| |
| Determine all non-negative rational numbers $x$ that satisfy the equation $x+|x-1|=1$!",See reasoning trace,medium |
| 05473675e8a2,"One of five brothers baked a pie for their mother. Andrey said: ""It's Vitya or Tolya"". Vitya said: ""It wasn't me or Yura"". Tolya said: ""You are both joking"". Dima said: ""No, one of them is telling the truth, and the other is lying"". Yura said: ""No, Dima, you are wrong"". Mother knows that three of her sons always tell the truth. Who baked the pie?",See reasoning trace,medium |
| 5c6f41d26c02,"* Find all prime numbers $p, q$ and $r$ that satisfy the following equation: $p^{q}+q^{p}=r$.",See reasoning trace,medium |
| e83abab761aa,"At the edge of a circular lake, there are stones numbered from 1 to 10, in a clockwise direction. Frog starts from stone 1 and jumps only on these 10 stones in a clockwise direction. |
| |
| a) If Frog jumps 2 stones at a time, that is, from stone 1 to stone 3, from stone 3 to stone 5, and so on, on which stone will Frog be after 100 jumps? |
| |
| b) If on the first jump, Frog goes to stone 2, on the second jump to stone 4, on the third jump to stone 7, that is, in each jump he jumps one more stone than in the previous jump. On which stone will Frog be after 100 jumps?","$ 5,050. Since every 10 movements, he returns to stone 1 and 5,050 is a multiple of 10, after 100 ju",medium |
| 072cde96e95c,"12. (5 points) A group of students participates in a tree-planting activity. If 1 girl and 2 boys form a group, there are 15 boys left over; if 1 girl and 3 boys form a group, there are 6 girls left over. Therefore, the number of boys participating in the tree-planting activity is $\qquad$, and the number of girls is $\qquad$.",There are 81 boys and 33 girls,medium |
| de3af0cad059,"For integers $a, b, c$ and $d$, it holds that $a>b>c>d$ and |
| |
| $$ |
| (1-a)(1-b)(1-c)(1-d)=10 |
| $$ |
| |
| What values can the expression $a+b-c-d$ take?","9$, and in the second case, $6+2-0-(-1)=9$, so the only possibility is $a+b-c-d=9$.",medium |
| f1d32428b352,"$13 \cdot 10$ On the coordinate plane, a point with both coordinates as integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Find the value of $f(1)+f(2)+\cdots+f(1990)$. |
| (China High School Mathematics League, 1990)",2 \cdot\left[\frac{1990}{3}\right]=1326$.,medium |
| 66c8f5cd93ab,9.070. $\log _{1.5} \frac{2 x-8}{x-2}<0$.,$\quad x \in(4 ; 6)$,easy |
| 248810395166,"Example 5 (1993 National High School League Question) Let the sequence of positive integers $a_{0}, a_{1}, a_{2}, \cdots$ satisfy $a_{0}=a_{1}=1$ and $\sqrt{a_{n} \cdot a_{n-2}}-\sqrt{a_{n-1} \cdot a_{n-2}}=2 a_{n-1}(n=2,3, \cdots)$, find the general term formula of this sequence.",See reasoning trace,medium |
| d10b0e66ea11,"8. (10 points) In the inscribed quadrilateral $A B C D$, the degree measures of the angles are in the ratio $\angle A: \angle B: \angle C=2: 3: 4$. Find the length of $A C$, if $C D=8, B C=7.5 \sqrt{3}-4$.",See reasoning trace,medium |
| 73e2b8af7770,"3. (BUL) Solve the equation $\cos ^{n} x-\sin ^{n} x=1$, where $n$ is a given positive integer. ## Second Day",See reasoning trace,medium |
| 35af979e702e,"7.107 There are two grasshoppers at the two endpoints of the line segment $[0,1]$, and some points are marked within the line segment. Each grasshopper can jump over the marked points such that the positions before and after the jump are symmetric with respect to the marked point, and they must not jump out of the range of the line segment $[0,1]$. Each grasshopper can independently jump once or stay in place, which counts as one step. How many steps are needed at minimum to ensure that the two grasshoppers can always jump to the same small segment divided by the marked points on $[0,1]$?",See reasoning trace,medium |
| ba603209c471,"The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$? |
| $\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$",502,easy |
| 4894534d29ad,"The sequence $\mathrm{Az}\left(a_{i}\right)$ is defined as follows: $a_{1}=0, a_{2}=2, a_{3}=3, a_{n}=\max _{1<d<n}\left\{a_{d} \cdot a_{n-d}\right\}(n=4,5,6, \ldots)$. Determine the value of $a_{1998}$.",See reasoning trace,medium |
| 468402d88c9b,"$1.3,4$ ** Find all real numbers $a$ such that for any real number $x$, the value of the function $f(x)=x^{2}-2 x-|x-1-a|-|x-2|+4$ is a non-negative real number.",See reasoning trace,medium |
| 0fc1c5da3cdd,"9. [6] I have four distinct rings that I want to wear on my right hand hand (five distinct fingers.) One of these rings is a Canadian ring that must be worn on a finger by itself, the rest I can arrange however I want. If I have two or more rings on the same finger, then I consider different orders of rings along the same finger to be different arrangements. How many different ways can I wear the rings on my fingers?",600,easy |
| 969e1f0fb541,"12. $A B C D$ is a square of side length 1 . $P$ is a point on $A C$ and the circumcircle of $\triangle B P C$ cuts $C D$ at $Q$. If the area of $\triangle C P Q$ is $\frac{6}{25}$, find $C Q$. |
| $A B C D$ 是個邊長為 1 的正方形。 $P$ 是 $A C$ 上的一點, 且 $\triangle B P C$ 的外接圓交 $C D$於 $Q$ 。若 $\triangle C P Q$ 的面積為 $\frac{6}{25}$, 求 $C Q$ 。","\sqrt{2} P B=\sqrt{2} P Q$ and $B C=1$, this simplifies to $Q C+1=\sqrt{2} P C$. Let $C Q=y$. Then $",medium |
| 79fdd47f4eb9,2. The sum of the terms of an infinite geometric series is 2 and the sum of the squares of the terms of this series is 6 . Find the sum of the cubes of the terms of this series.,See reasoning trace,medium |
| 12b33f1aa1ec,"Points $A$ and $B$ are on a circle of radius $5$ and $AB = 6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$? |
| $\textbf{(A)}\ \sqrt {10} \qquad \textbf{(B)}\ \frac {7}{2} \qquad \textbf{(C)}\ \sqrt {14} \qquad \textbf{(D)}\ \sqrt {15} \qquad \textbf{(E)}\ 4$",\text{A,medium |
| 5da1d4c52f8a,"## |
| |
| Calculate the limit of the function: |
| |
| $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\ln (\sin x)}{(2 x-\pi)^{2}}$",See reasoning trace,medium |
| 6c45ce70ca8b,"Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.",a > n^4 - n^3,medium |
| 047130f3b906,Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.,"(x, y) = (11, 5) \text{ or",medium |
| 31ef8a2dd81f,"# Task 10. Game Theory |
| |
| |
| |
| Note. The answer is the planned amount of cubic meters of liquefied and natural gas to be extracted and supplied.",See reasoning trace,medium |
| c49d0535c2f2,"5. Mr. Patrick is the math teacher of 15 students. After a test, he found that the average score of the rest of the students, excluding Peyton, was 80 points, and the average score of the entire class, including Peyton, was 81 points. What was Peyton's score in this test? ( ) points. |
| (A) 81 |
| (B) 85 |
| (C) 91 |
| (D) 94 |
| (E) 95",See reasoning trace,easy |
| a3d242b62ce0,"Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of
|
|
|
| $ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$",1004,medium |
| e02c969e09b9,"$26 \cdot 26$ The number of common points of the curves $x^{2}+4 y^{2}=1$ and $4 x^{2}+y^{2}=4$ is |
| (A)0. |
| (B) 1 . |
| (C) 2 . |
| (D) 3 . |
| (E) 4 . |
| (17th American High School Mathematics Examination, 1966)",$(C)$,easy |
| d8804f878c39,"One, (50 points) The product of all elements in a finite set $S$ is called the ""product number"" of the set $S$. Given the set $M=$ $\left\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \cdots, \frac{1}{99}, \frac{1}{100}\right\}$, determine the sum of the ""product numbers"" of all even-numbered (2, 4, $\cdots$, 98) element subsets of $M$.",See reasoning trace,medium |
| bc47f964baf3,"12. The system of equations $\left\{\begin{array}{l}x+3 y=3, \\ || x|-| y||=1\end{array}\right.$ has ( ) solutions. |
| (A) 1 |
| (B) 2 |
| (C) 3 |
| (D) 4 |
| (E) 8","3,|| x|-| y||=1$ (as shown in Figure 3), the two functions have three intersection points: $(-3,2)$,",easy |
| 2dae87ea1b43,"A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end? |
| $\text{(A)}\ \text{10:00 p.m.} \qquad \text{(B)}\ \text{midnight} \qquad \text{(C)}\ \text{2:30 a.m.} \qquad \text{(D)}\ \text{4:40 a.m.} \qquad \text{(E)}\ \text{6:40 a.m.}$",\text{D,medium |
| 0e3765f18825,"II. (40 points) Let $k \geqslant 1$. Find the largest real number $\lambda$ such that for any real numbers $x_{i}>0$ $(i=1,2, \cdots, n)$ satisfying $\sum_{i=1}^{n} \frac{1}{x_{i}}=2016$, we have |
| $$ |
| \lambda \sum_{i=1}^{n} \frac{x_{i}}{1+x_{i}} \leqslant\left[\sum_{i=1}^{n} \frac{1}{x_{i}^{k}\left(1+x_{i}\right)}\right]\left(\sum_{i=1}^{n} x_{i}^{k}\right) . |
| $$",2016$.,medium |
| 7dc848123d7c,"4. Given a regular tetrahedron $P-ABC$ with a base edge length of 1 and a height of $\sqrt{2}$, then its inscribed sphere radius is $\qquad$",\frac{\sqrt{3}}{6} \Rightarrow PD=\sqrt{\left(\frac{\sqrt{3}}{6}\right)^{2}+2}=\frac{5 \sqrt{3}}{6}$,medium |
| 694167289df0,"4. The numbers 1 and 2 are written on the board. It is allowed to add new numbers in the following way: If the numbers $a$ and $b$ are written on the board, then the number $a b+a+b$ can be written. Can the following numbers be obtained in this way: |
| a) 13121, |
| b) 12131.",See reasoning trace,medium |
| fc98c6636429,11. The common divisors of two numbers that are not both 0 are called their common divisors. Find all the common divisors of 26019 and 354.,See reasoning trace,easy |
| 4aa3761966a0,"(11) (25 points) The sum of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ is 2009. If these $n$ numbers can be divided into 41 groups with equal sums and also into 49 groups with equal sums, find the minimum value of $n$.","\cdots = x_{41} = 41, x_{42} = \cdots = x_{75} = 8, x_{76} = \cdots = x_{79} = 7, x_{80} = \cdots = ",medium |
| 40e7ec767daa,"5. When $1 \leqslant x \leqslant 2$, simplify |
| $$ |
| \sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}}= |
| $$ |
| $\qquad$ .",See reasoning trace,easy |
| 3d29fd9791f5,"Ada has a set of identical cubes. She makes solids by gluing together 4 of these cubes. When cube faces are glued together, they must coincide. Each of the 4 cubes must have a face that coincides with a face of at least one of the other 3 cubes. One such solid is shown. The number of unique solids that Ada can make using 4 cubes is |
| (A) 5 |
| (B) 6 |
| (C) 7 |
| (D) 8 |
| (E) 10 |
| |
| ",(D),medium |
| 4acfec9f249c,Task 1. Boys and girls formed a circle in such a way that the number of children whose right neighbor is of the same gender is equal to the number of children whose right neighbor is of a different gender. What could be the total number of children in the circle?,. Any natural number that is a multiple of four,medium |
| fffbd0e52e4f,"9 Real numbers $x, y$ satisfy $\left\{\begin{array}{l}x+\sin y=2008, \\ x+2008 \cos y=2007,\end{array}\right.$ where $0 \leqslant y \leqslant \frac{\pi}{2}$, then $x+y=$",See reasoning trace,easy |
| 3a9c41351f7d,"Three police officers need to be placed at some intersections so that at least one police officer is on each of the 8 streets. Which three intersections should the police officers be placed at? It is sufficient to provide at least one suitable arrangement. |
| |
| All streets run along straight lines. |
| |
| Horizontal streets: $A-B-C-D, E-F-G, H-I-J-K$. |
| |
| Vertical streets: $A-E-H, B-F-I, D-G-J$. |
| |
| Diagonal streets: $H-F-C, C-G-K$. |
| |
| ","$B, G, H$",medium |
| 34c93ea3987e,7. (10 points) The value of the expression $\frac{1-\frac{2}{7}}{0.25+3 \times \frac{1}{4}}+\frac{2 \times 0.3}{1.3-0.4}$ is,The value of $m+n$ is $1 \frac{8}{21}$,medium |
| d45632b7c2c6,"Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$, and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac 19$. There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6}$, $\overline{PA_7}$, and the minor arc $\widehat{A_6A_7}$ is equal to $\tfrac18 - \tfrac{\sqrt 2}n$. Find $n$.",504,medium |
| 1b09d5db40a2,"If $a, b$, and $c$ are positive real numbers with $2 a+4 b+8 c=16$, what is the largest possible value of $a b c$ ?","4 b=8 c$, since that is the equality case of AM-GM, which means $2 a=4 b=8 c=\frac{16}{3}$, so $a=\f",easy |
| 7edf02b928d1,"86. Using the equality $x^{3}-1=(x-1)\left(x^{2}+x+1\right)$, solve the equation |
| |
| $$ |
| x^{3}-1=0 |
| $$ |
| |
| Applying the obtained formulas, find the three values of $\sqrt[3]{1}$ in 103 -arithmetic.",See reasoning trace,easy |
| 50d477db9367,"5. Let integer $n \geqslant 2$, |
| $$ |
| A_{n}=\sum_{k=1}^{n} \frac{3 k}{1+k^{2}+k^{4}}, B_{n}=\prod_{k=2}^{n} \frac{k^{3}+1}{k^{3}-1} \text {. } |
| $$ |
| |
| Then the size relationship between $A_{n}$ and $B_{n}$ is",B_{n}$.,medium |
| 2dd33b471af3,"Example 5 Find the constant $c$, such that $f(x)=\arctan \frac{2-2 x}{1+4 x}+c$ is an odd function in the interval $\left(-\frac{1}{4}, \frac{1}{4}\right)$.","0$, i.e., $c=$ $-\arctan 2$. When $c=-\arctan 2$, $\tan \left(\arctan \frac{2-2 x}{1+4 x}-\arctan 2\",easy |
| 5f60e53685fa,"After distributing 2020 candies, he left. How many children did not receive any candies?",36,medium |
| 0aee24791547,10.231. A circle of radius $r$ is inscribed in a rectangular trapezoid. Find the sides of the trapezoid if its smaller base is equal to $4 r / 3$.,"$4 r, \frac{10 r}{3}, 2 r$",easy |
| ace3eafc387d,"(4) Given that $\vec{a}$ and $\vec{b}$ are non-zero and non-collinear vectors, let condition $M: \vec{b} \perp (\vec{a}-\vec{b})$; |
| condition $N$: for all $x \in \mathbf{R}$, the inequality $|\vec{a}-x \vec{b}| \geqslant |\vec{a}-\vec{b}|$ always holds. Then $M$ is ( ) of $N$. |
| (A) a necessary but not sufficient condition |
| (B) a sufficient but not necessary condition |
| (C) a sufficient and necessary condition |
| (D) neither a sufficient nor a necessary condition",C,medium |
| bd87b3ae775f,"Example 1. Find the mass of the plate $D$ with surface density $\mu=16 x+9 y^{2} / 2$, bounded by the curves |
| |
| $$ |
| x=\frac{1}{4}, \quad y=0, \quad y^{2}=16 x \quad(y \geq 0) |
| $$",. $m=2$ units of mass,medium |
| 5e3e68ad8ecc,"7. Let the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, where $a_{1}=p, b_{1}=q$. It is known that $a_{n}=p a_{n-1}, b_{n}=q a_{n-1}+r b_{n-1}(p, q, r$ are constants, and $q>0, p>r>0, n \geqslant 2)$, then the general term formula of the sequence $\left\{b_{n}\right\}$ is $b_{n}=$ $\qquad$.",See reasoning trace,medium |
| 7642e9b035bc,"4. In tetrahedron $ABCD$, $\angle ADB = \angle BDC = \angle CDA = 60^{\circ}$, the areas of $\triangle ADB$, $\triangle BDC$, and $\triangle CDA$ are $\frac{\sqrt{3}}{2}$, $2$, and $1$, respectively. Then the volume of the tetrahedron is . $\qquad$","$ $\frac{P A_{1} \cdot P B_{1} \cdot P C_{1}}{P A \cdot P B \cdot P C}$. This is because, if $\trian",medium |
| 8b681be0897a,"Solve the following system of equations: |
| |
| $$ |
| \begin{gathered} |
| x + xy = 19 - y \\ |
| \frac{84}{xy} - y = x |
| \end{gathered} |
| $$",See reasoning trace,medium |
| 672726749213,A rectangle has a perimeter of $124 \mathrm{~cm}$. The perimeter of the rhombus determined by the midpoints of the sides is $100 \mathrm{~cm}$. What are the lengths of the sides of the rectangle?,See reasoning trace,easy |
| d1741a22bcf5,"3. Given the complex sequence $\left\{a_{n}\right\}$ with the general term $a_{n}=(1+\mathrm{i}) \cdot\left(1+\frac{\mathrm{i}}{\sqrt{2}}\right) \cdots\left(1+\frac{\mathrm{i}}{\sqrt{n}}\right)$, then $\left|a_{n}-a_{n+1}\right|=$",(1+\mathrm{i}) \cdot\left(1+\frac{\mathrm{i}}{\sqrt{2}}\right) \cdot \cdots \cdot\left(1+\frac{\math,medium |
| 714ec2c1f494,"Task 3. Find the smallest possible value of |
| |
| $$ |
| x y+y z+z x+\frac{1}{x}+\frac{2}{y}+\frac{5}{z} |
| $$ |
| |
| for positive real numbers $x, y$ and $z$.",the smallest possible value is $3 \sqrt[3]{36}$,medium |
| 44c7ff68d2df,"2. Determine the maximum possible value of the expression |
| |
| $$ |
| \frac{a_{1} a_{2} \ldots a_{n}}{\left(1+a_{1}\right)\left(a_{1}+a_{2}\right) \ldots\left(a_{n-1}+a_{n}\right)\left(a_{n}+2^{n+1}\right)} |
| $$ |
| |
| where $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are positive real numbers.",x_{2}=\ldots=x_{n}=x_{n+1}=2$. This means that the maximum possible value of the expression (*) is $,medium |
| 93434031b5a4,"11. Among the positive integers $1,2, \cdots, 20210418$, how many numbers have a digit sum of 8?","8$, such as $|\bigcirc| O O|O \bigcirc||\bigcirc 0| \bigcirc$ representing 122021; The number of 8-d",medium |
| e2b400a80384,"7. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities |
| |
| $$ |
| \left\{\begin{array}{l} |
| |x|+|4-x| \leqslant 4 \\ |
| \frac{x^{2}-4 x-2 y+2}{y-x+3} \geqslant 0 |
| \end{array}\right. |
| $$ |
| |
| Sketch the figure $M$ and find its area.",4,medium |
| dd70bcaf8342,"7. Let $[a]$ denote the greatest integer not exceeding $a$, for example: $[8]=8, [3.6]=3$. |
| Some natural numbers can be expressed in the form $[x]+[2 x]+[3 x]$, such as 6 and 3: |
| $$ |
| \begin{array}{c} |
| 6=\left[\frac{5}{4}\right]+\left[2 \times \frac{5}{4}\right]+\left[3 \times \frac{5}{4}\right], \\ |
| 3=[0.8]+[2 \times 0.8]+[3 \times 0.8] . |
| \end{array} |
| $$ |
| |
| Among the 2020 natural numbers from 1 to 2020, there are $\qquad$ numbers that can be expressed in the above form.",See reasoning trace,easy |
| 7be7eb83d506,"6. The set of all real number pairs $(x, y)$ that satisfy $\left\{\begin{array}{l}4^{-x}+27^{-y}=\frac{5}{6} \\ \log _{27} y-\log _{4} x \geqslant \frac{1}{6} \text { is } \\ 27^{y}-4^{x} \leqslant 1\end{array}\right.$ $\qquad$ . $\qquad$","4^{x}, b=27^{y}$, then $\frac{1}{a}+\frac{1}{b}=\frac{5}{6}, a>\frac{6}{5}, b>\frac{6}{5}$. From $\f",medium |
| 67856cdfd29a,"One, (50 points) Try to find all real-coefficient polynomials $f(x)$, such that for all real numbers $a, b, c$ satisfying $a b + b c + c a = 0$, the following holds: |
| $$ |
| f(a-b) + f(b-c) + f(c-a) = 2 f(a+b+c). |
| $$",See reasoning trace,medium |
| a62c9ffcd7b3,"Task 7. The line $c$ is given by the equation $y=2 x$. Points $A$ and $B$ have coordinates $A(2 ; 2)$ and $B(6 ; 2)$. On the line $c$, find the point $C$ from which the segment $A B$ is seen at the largest angle.",$(2 ; 4)$,medium |
| b1d7422bcbed,"Petya is playing a shooting game. If he scores less than 1000 points, the computer will add $20 \%$ of his score. If he scores from 1000 to 2000 points, the computer will add $20 \%$ of the first thousand points and $30 \%$ of the remaining points. If Petya scores more than 2000 points, the computer will add $20 \%$ of the first thousand points, $30 \%$ of the second thousand, and $50 \%$ of the remaining points. How many bonus points did Petya receive if he had 2370 points at the end of the game? |
| |
| #",See reasoning trace,easy |
| d4d5c883ca03,"Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.",114,easy |
| 704e3a4e985b,"## Task 22/62 |
| |
| The ellipse and the hyperbola with the following properties are sought: |
| |
| 1. The linear eccentricity is \( e = 20 \). |
| 2. The perpendicular focal rays \( l_{1} \) and \( l_{2} \) are in the ratio \( l_{1}: l_{2} = 4: 3 \). |
| |
| a) The lengths of the focal rays \( l_{1} \) and \( l_{2} \) are to be determined. |
| b) The equations of the conic sections are to be established.",See reasoning trace,medium |
| 6cebf62f9b52,4. For what values of $a$ is the system $\left\{\begin{array}{l}3 x^{2}-x-a-10=0 \\ (a+4) x+a+12=0\end{array}\right.$ consistent? Solve the system for all permissible $a$.,"-8$, $x=1$; when $a=4$, $x=-2$; when $a=-10$, $x=\frac{1}{3}$.",easy |
| 176e9604bd4a,"## Task Condition |
| |
| Find the derivative. |
| |
| $y=(\sin x)^{5 e^{x}}$",See reasoning trace,medium |
| 5d8a2d1a0549,"Let $n \geqslant 2$ be a positive integer, and let $a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers and $b_{1}, b_{2}, \cdots, b_{n}$ be non-negative real numbers satisfying |
| (a) $a_{1}+a_{2}+\cdots+a_{n}+b_{1}+b_{2}+\cdots+b_{n}=n$; |
| (b) $a_{1} a_{2} \cdots a_{n}+b_{1} b_{2} \cdots b_{n}=\frac{1}{2}$. |
| |
| Find the maximum value of $a_{1} a_{2} \cdots a_{n}\left(\frac{b_{1}}{a_{1}}+\frac{b_{2}}{a_{2}}+\cdots+\frac{b_{n}}{a_{n}}\right)$. |
| (Tian Kaibin, Chu Xiaoguang, Pan Chenghua)",See reasoning trace,medium |
| 9e5a0a65532c,"If 8 is added to the square of 5 the result is divisible by |
| (A) 5 |
| (B) 2 |
| (C) 8 |
| (D) 23 |
| (E) 11",(E),easy |
| 9ec495e1892e,"Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)",n \geq 3,medium |
| 7dfa730cc1df,"## 104. Math Puzzle $1 / 74$ |
| |
| Determine the smallest natural number $z$ that ends in 4. If you remove the 4 at the end and place it at the front, you get four times $z$.",16$. The 6 is then written instead of $y$. The next partial multiplication is then $4 \cdot 6=24$ pl,medium |
| 3c3b979f5749,Find all positive integers $n$ such that $2^{n}+3$ is a perfect square. The same question with $2^{n}+1$.,"x^{2}$. Then $(x-1)(x+1)=2^{n}$. Therefore, $x-1=2^{k}$ and $x+1=2^{n-k}$. Hence, $2^{k}+1=2^{n-k}-1",medium |
| e2917e0810a9,"299. When dividing the polynomial $x^{1051}-1$ by $x^{4}+x^{3}+2 x^{2}+x+1$, a quotient and a remainder are obtained. Find the coefficient of $x^{14}$ in the quotient.",See reasoning trace,medium |
| 036ac113cb5c,"Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth? |
| $\textbf{(A)}\ \textdollar 1.15\qquad\textbf{(B)}\ \textdollar 1.20\qquad\textbf{(C)}\ \textdollar 1.25\qquad\textbf{(D)}\ \textdollar 1.30\qquad\textbf{(E)}\ \textdollar 1.35$",\mathrm{(A)\,easy |
| 68e558b0e8ad,"## Task A-4.2. |
| |
| The initial term of the sequence $\left(a_{n}\right)$ is $a_{0}=2022$. For each $n \in \mathbb{N}$, the number $a_{n}$ is equal to the sum of the number $a_{n-1}$ and its largest divisor smaller than itself. Determine $a_{2022}$.",a_{3 \cdot 674}=3^{674} \cdot a_{0}=3^{674} \cdot 2022$.,medium |
| 806f3191bae4,"## Task B-4.7. |
| |
| Determine all natural numbers $x$ that are solutions to the inequality |
| |
| $$ |
| \log _{x}^{4} 2017+6 \cdot \log _{x}^{2} 2017>4 \cdot \log _{x}^{3} 2017+4 \cdot \log _{x} 2017 |
| $$",See reasoning trace,medium |
| 8377f607d4f3,118. Compose the equation of a circle with center $O(3; -2)$ and radius $r=5$.,"3, b=-2$ and $r=5$ into equation (1), we get $(x-3)^{2}+(y+2)^{2}=25$.",easy |
| fc5c4d7cb85e,Let $AB = 10$ be a diameter of circle $P$. Pick point $C$ on the circle such that $AC = 8$. Let the circle with center $O$ be the incircle of $\vartriangle ABC$. Extend line $AO$ to intersect circle $P$ again at $D$. Find the length of $BD$.,\sqrt{10,medium |
| 6867b0855edf,"5. Inside an isosceles triangle $A B C (A B=A C)$, a point $K$ is marked. Point $L$ is the midpoint of segment $B K$. It turns out that $\angle A K B=\angle A L C=90^{\circ}, A K=C L$. Find the angles of triangle $A B C$.",the triangle is equilateral,easy |
| 9e51232297a9,"## Task Condition |
| |
| Approximately calculate using the differential. |
| |
| $y=\sqrt[5]{x^{2}}, x=1.03$",See reasoning trace,medium |
| 7936b24bb290,"Example 5. Let $O$ be the center of the base $\triangle ABC$ of a regular tetrahedron $P-ABC$. A moving plane through $O$ intersects the three lateral edges or their extensions of the tetrahedron at points $Q$, $R$, and $S$ respectively. Then the sum $\frac{1}{PQ}+\frac{1}{PR}+\frac{1}{PS}(\quad)$. |
| (A) has a maximum value but no minimum value |
| (B) has a minimum value but no maximum value |
| (C) has both a maximum value and a minimum value, and they are not equal |
| (D) is a constant independent of the position of plane $QRS$",to this problem should be (D),medium |
| cb67ba1a7d84,"The graph shown at the right indicates the time taken by five people to travel various distances. On average, which person travelled the fastest? |
| (A) Alison |
| (D) Daniel |
| (B) Bina |
| (E) Emily |
| (C) Curtis |
| |
| ",(E),medium |
| 931d51d2f967,"In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$? |
| $\textbf{(A) }100\qquad\textbf{(B) }420\qquad\textbf{(C) }500\qquad\textbf{(D) }609\qquad \textbf{(E) }701$",\textbf{(B),medium |
| f00d0aae3c86,9.157. $\log _{0.5}(x+3)<\log _{0.25}(x+15)$.,$x \in (1 ; \infty)$,medium |
| 8850322d21f1,"6. The sequence $\left\{x_{n}\right\}: 1,3,3,3,5,5,5,5,5, \cdots$ is formed by arranging all positive odd numbers in ascending order, and each odd number $k$ appears consecutively $k$ times, $k=1,3,5, \cdots$. If the general term formula of this sequence is $x_{n}=a[\sqrt{b n+c}]+d$, then $a+b+c$ $+d=$","[\sqrt{n-1}]$, so $x_{n}=2[\sqrt{n-1}]+1$, thus, $(a, b, c, d)=(2,1,-1,1), a+b+c+d=3$.",easy |
| 9fc5c5440c31,,"2,3$ or 4. All three cases are possible: $k=2$ for $a=\frac{3}{8}(x=6$ and 7$); k=3$ for $a=\frac{1}",medium |
| d1888a18ea24,"5. On one of two parallel lines, there are 8 points. How many points are on the other line if all the points together determine 640 triangles? |
| |
| Each task is scored out of 10 points. |
| |
| The use of a pocket calculator or any reference materials is not allowed.",See reasoning trace,easy |
| d3a902b87bd3,"Let $n$ be a fixed integer, $n \geqslant 2$. |
| a) Determine the smallest constant $c$ such that the inequality |
| $$ |
| \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4} |
| $$ |
| |
| holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n} \geqslant 0$; |
| b) For this constant $c$, determine the necessary and sufficient conditions for equality to hold. |
| This article provides a simple solution. |
| The notation $\sum_{1 \leq i<j \leq \leqslant} f\left(x_{i}, x_{j}\right)$ represents the sum of all terms $f\left(x_{i}, x_{j}\right)$ for which the indices satisfy $1 \leqslant i<j \leqslant n$, and in the following text, it is simply denoted as $\sum f\left(x_{i}, x_{j}\right)$.",See reasoning trace,medium |
| 76b0a550c666,"7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, the probability that a car will turn off the highway onto each of these roads is $\frac{1}{n}$. The road by which a car leaves the highway is independent of the road by which it entered the highway. |
| |
|  |
| which the car leaves the highway, is independent of the road by which it entered the highway. |
| |
| Find the probability that a random car that has entered the highway will pass the $k$-th intersection (either drive through it or turn at it).",$\frac{2 k n-2 k^{2}+2 k-1}{n^{2}}$,easy |
| c1e4c6a353ac,Evaluate $\frac{1}{2}\left(\frac{1}{\frac{1}{9}}+\frac{1}{\frac{1}{6}}-\frac{1}{\frac{1}{5}}\right)$.,\frac{1}{2}(9+6-5)=\frac{1}{2}(10)=5$.,easy |
| 5764d3936e2d,"6. Given the function $f(x)=|| x-1|-1|$, if the equation $f(x)=m(m \in \mathbf{R})$ has exactly 4 distinct real roots $x_{1}, x_{2}, x_{3}, x_{4}$, then the range of $x_{1} x_{2} x_{3} x_{4}$ is $\qquad$ .",See reasoning trace,easy |
| 00297d949b91,"6. Team A and Team B each draw 7 players to participate in a Go relay match according to a pre-arranged order. Both sides start with their No. 1 players competing. The loser is eliminated, and the winner then competes with the No. 2 player of the losing side, $\cdots$, until all players of one side are eliminated, and the other side wins, forming a match process. The total number of all possible match processes is $\qquad$ (1988 National High School League Question)","1,2, \cdots, 7)$, then $x_{1}+x_{2}+x_{3}+$ $x_{4}+x_{5}+x_{6}+x_{7}=7$, and the match process where",medium |
| e455de894539,"5. Ari is watching several frogs jump straight towards him from the same place. Some frogs are larger, and some are smaller. When the larger frog makes 7 equal jumps, it is still $3 \mathrm{~cm}$ away from Ari. When the smaller frog makes 10 equal jumps, it is still $1 \mathrm{~cm}$ away from Ari. If the frogs are initially more than $2 \mathrm{~m}$ away from Ari, what is the smallest possible length of the jump of the smaller frogs? The length of each jump in centimeters is a natural number. |
| |
| No pocket calculators or any reference materials are allowed. |
| |
| ## Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society |
| |
| ## County Competition in Mathematics February 26, 2024. 5th grade - elementary school |
| |
| Each task is worth 10 points. In addition to the final result, the procedure is also scored. To earn all points, it is necessary to find all solutions and confirm that there are no others, write down the procedure, and justify your conclusions.",See reasoning trace,medium |
| 999a3f99f872,"4. As shown in Figure 2, a chord $BC$ (not a diameter) of the semicircle $\odot O$ is used as the axis of symmetry to fold the arc $\overparen{BC}$, intersecting the diameter at point $D$. If $AD=4, BD=6$, then $\tan B=$ ( . |
| (A) $\frac{1}{2}$ |
| (B) $\frac{2}{3}$ |
| (C) $\frac{3}{4}$ |
| (D) $\frac{4}{5}$",\frac{A C}{B C}=\frac{1}{2}$.,medium |
| 7eea889ba674,"Five students wrote a quiz with a maximum score of 50 . The scores of four of the students were $42,43,46$, and 49 . The score of the fifth student was $N$. The average (mean) of the five students' scores was the same as the median of the five students' scores. The number of values of $N$ which are possible is |
| (A) 3 |
| (B) 4 |
| (C) 1 |
| (D) 0 |
| (E) 2",(A),medium |
| 59532b53da23,The first term of a certain arithmetic and geometric progression is 5; the second term of the arithmetic progression is 2 less than the second term of the geometric progression; the third term of the geometric progression is equal to the sixth term of the arithmetic progression. What progressions satisfy the conditions given here?,See reasoning trace,medium |
| 9218fe156a56,"7. Given the function $f(x)=x^{3}-2 x^{2}-3 x+4$, if $f(a)=f(b)=f(c)$, where $a<b<c$, then $a^{2}+b^{2}+c^{2}=$","f(b)=f(c)=k \Rightarrow a, b, c$ are the roots of the equation $f(x)-k=x^{3}-2 x^{2}-3 x+4-k=0$, $\l",easy |
| bf1d3f87c074,"1.121 A wheel has a rubber tire with an outer diameter of 25 inches. When the radius is reduced by $\frac{1}{4}$ inch, the number of revolutions in one mile will |
| (A) increase by about $2 \%$. |
| (B) increase by about $1 \%$. |
| (C) increase by about $20 \%$. |
| (D) increase by $\frac{1}{2} \%$. |
| (E) remain unchanged. |
| (7th American High School Mathematics Examination, 1956)",$(A)$,easy |
| dc4358c6117a,"2 Let the medians of $\triangle A B C$ corresponding to sides $a, b, c$ be $m_{a}, m_{b}, m_{c}$, and the angle bisectors be $w_{a}, w_{b}, w_{c}$. Suppose $w_{a} \cap m_{b}=P, w_{b} \cap m_{c}=Q, w_{c} \cap m_{a}=R$. Let the area of $\triangle P Q R$ be $\delta$, and the area of $\triangle A B C$ be $F$. Find the smallest positive constant $\lambda$ such that the inequality $\frac{\delta}{F}<\lambda$ holds.",See reasoning trace,medium |
| 22d741587325,"$12 \cdot 37$ Let $i=\sqrt{-1}$, the complex sequence $z_{1}=0, n \geqslant 1$ when, $\approx_{n+1}=$ $z_{n}^{2}+i$. Then the modulus of $z_{111}$ in the complex plane is |
| (A) 1 . |
| (B) $\sqrt{2}$. |
| (C) $\sqrt{3}$. |
| (D) $\sqrt{110}$. |
| (E) $\sqrt{2^{55}}$. |
| (43rd American High School Mathematics Examination, 1992)",$(B)$,easy |
| 973ed44df6e5,7.222. $3 \cdot 16^{x}+2 \cdot 81^{x}=5 \cdot 36^{x}$.,$0 ; \frac{1}{2}$,easy |
| 91491d572683,"7. Given a moving point $P(x, y)$ satisfies the quadratic equation $10 x-2 x y-2 y+1=0$. Then the eccentricity of this quadratic curve is |
| $\qquad$ -",See reasoning trace,easy |
| c51ad5b3fd42,"Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation |
| \[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]",f(x) = \tan(ax),medium |
| ed89ac73106c,"3. Given that |
| $$ |
| x=\left\lfloor 1^{1 / 3}\right\rfloor+\left\lfloor 2^{1 / 3}\right\rfloor+\left\lfloor 3^{1 / 3}\right\rfloor+\cdots+\left\lfloor 7999^{1 / 3}\right\rfloor \text {, } |
| $$ |
| find the value of $\left\lfloor\frac{x}{100}\right\rfloor$, where $\lfloor y\rfloor$ denotes the greatest integer less than or equal to $y$. (For example, $\lfloor 2.1\rfloor=2,\lfloor 30\rfloor=30,\lfloor-10.5\rfloor=-11$.)",See reasoning trace,medium |
| 089438565e43,"## Task B-1.2. |
| |
| Two circles $K_{1}$ and $K_{2}$, shaded in color, intersect such that $10 \%$ of the area of circle $K_{1}$ and $60 \%$ of the area of circle $K_{2}$ lie outside their intersection. Calculate the ratio of the radii of the circles $K_{1}$ and $K_{2}$. What is the sum of these radii if the total shaded area is $94 \pi$? |
| |
| ",See reasoning trace,medium |
| d169fb65846b,"Task B-1.2. The sum of the digits of the natural number $x$ is $y$, and the sum of the digits of the number $y$ is $z$. Determine all numbers $x$ for which |
| |
| $$ |
| x+y+z=60 |
| $$",See reasoning trace,medium |
| eea87f3a539b,"Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$? |
| |
| $\text{(A)}\ \sqrt{10} \qquad \text{(B)}\ \sqrt{12} \qquad \text{(C)}\ \sqrt{13} \qquad \text{(D)}\ \sqrt{14} \qquad \text{(E)}\ \sqrt{15}$",\text{(C),easy |
| d683e4503dde,"10. (This question is worth 20 points) The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{\pi}{6}, a_{n+1}=\arctan \left(\sec a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$. Find the positive integer $m$, such that |
| $$ |
| \sin a_{1} \cdot \sin a_{2} \cdots \cdot \sin a_{m}=\frac{1}{100} . |
| $$","\frac{1}{100}$, we get $m=3333$. $\qquad$",medium |
| 4c2e9226b00c,"Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
|
|
|
| $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$",119,medium |
| d64316af3752,"A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.",7,medium |
| 014e197f4f5b,Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.,222,medium |
| 6f7b317aad74,"4. Let $z_{1}, z_{2}$ be complex numbers, and $\left|z_{1}\right|=3,\left|z_{2}\right|=5,\left|z_{1}+z_{2}\right|=7$, then the value of $\arg \left(\frac{z_{2}}{z_{1}}\right)^{3}$ is . $\qquad$",\pi$.,medium |
| 218aa6682bd6,"8. Given the sequence $\left\{a_{n}\right\}$ with the first term being 2, and satisfying $6 S_{n}=3 a_{n+1}+4^{n}-1$. Then the maximum value of $S_{n}$ is",See reasoning trace,medium |
| 6befba7ec3f7,"10. $[\mathbf{8}]$ Let $f(n)=\sum_{k=1}^{n} \frac{1}{k}$. Then there exists constants $\gamma, c$, and $d$ such that |
| $$ |
| f(n)=\ln (n)+\gamma+\frac{c}{n}+\frac{d}{n^{2}}+O\left(\frac{1}{n^{3}}\right), |
| $$ |
| where the $O\left(\frac{1}{n^{3}}\right)$ means terms of order $\frac{1}{n^{3}}$ or lower. Compute the ordered pair $(c, d)$.","$\left(\frac{1}{2},-\frac{1}{12}\right)$ From the given formula, we pull out the term $\frac{k}{n^{3}}$ from $O\left(\frac{1}{n^{4}}\right)$, making $f(n)=$ $\log (n)+\gamma+\frac{c}{n}+\frac{d}{n^{2}}+\frac{k}{n^{3}}+O\left(\frac{1}{n^{4}}\right)$",medium |
| 777b682f8c57,"3. Let $A=\{1,2,3,4,5\}$. Then the number of mappings $f: A \rightarrow A$ that satisfy the condition $f(f(x))$ $=f(x)$ is $\qquad$ (answer with a number)","a$. Therefore, we can classify $f$ based on the number of elements in its range. The number of $f$ w",medium |
| 4f73dc0b00fe,21.2.10 ** Divide a circle with a circumference of 24 into 24 equal segments. Choose 8 points from the 24 points such that the arc length between any two points is not equal to 3 or 8. How many different ways are there to select such a group of 8 points?,"8, m=3$ in problem 21.2.8. Therefore, by the conclusion of problem 21.2.8, the number of different 8",medium |
| b5457e5b7742,6. Find the sum of all even numbers from 10 to 31. Calculate in different ways.,The sum of all even numbers from 10 to 31 is 220,medium |
| 8d3926406325,"## Task $9 / 89$ |
| |
| Determine all non-negative real numbers $x$ that satisfy the equation: |
| |
| $$ |
| \sqrt[3]{x}+\sqrt[4]{x}=6 \sqrt[6]{x} |
| $$","-0.5 \pm 2.5$. Since $\sqrt[12]{x}>0$, it follows that $\sqrt[12]{x}=2$, and thus $x=2^{12}=4096$. T",easy |
| fe8ccb7c2651,"2. (8 points) Shuaishuai finished memorizing English words in five days. It is known that in the first three days, he memorized $\frac{1}{2}$ of all the words, and in the last three days, he memorized $\frac{2}{3}$ of all the words, and he memorized 120 fewer words in the first three days than in the last three days. Therefore, Shuaishuai memorized $\qquad$ English words on the third day.",Shuai Shuai memorized 120 English words on the third day,easy |
| c66931a3612c,"In triangle $A B C$, the lengths of two sides are given: $A B=6, B C=16$. Additionally, it is known that the center of the circle passing through vertex $B$ and the midpoints of sides $A B$ and $A C$ lies on the bisector of angle $C$. Find $A C$.",18,medium |
| f90c0a29ba5e,"Let X and Y be the following sums of arithmetic sequences: |
| \begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end{eqnarray*} |
| What is the value of $Y - X?$ |
| $\textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112$",92 \ \mathbf{(A),easy |
| 71ae786563c9,"## Task 6 - 020716 |
| |
| In a house with 28 windows, some missing shutters need to be procured so that each window has 2 shutters. Some windows still have 2 shutters, the same number of windows are missing both, and the rest have one shutter each. |
| |
| How many new shutters are needed? Justify your answer!",See reasoning trace,easy |
| 33f28b3cd136,"Write in ascending order the numbers |
| |
| $$ |
| \sqrt{121}, \quad \sqrt[3]{729} \quad \text { and } \quad \sqrt[4]{38416} |
| $$",See reasoning trace,easy |
| 269f562f9147,"4. If for any $x \in(0,+\infty)$, the inequality $a \mathrm{e}^{a e x+a} \geqslant \ln (\mathrm{e} x+1)$ always holds, then the range of values for $a$ is $\qquad$","\mathrm{e} x+1$, then $a \mathrm{e}^{a u} \geqslant \ln u \Rightarrow a u \mathrm{e}^{a u} \geqslant",medium |
| 3a8637077154,"2. 55 In decimal, find the smallest natural number: its square number starts with 19 and ends with 89","3$, we similarly examine the two possibilities, and so on. Among the $x$ values we obtained, the sma",medium |
| 7a1143320e75,17. What is the value of the sum: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{1024}$ ?,$1 \frac{1023}{1024}$,easy |
| 4b873aef485f,"4-4. A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ lying on the circle to the line $A B$ is 8. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 30 less than the other.",34,easy |
| 16f5f2cb55fc,"Variant 9.2.3. On the sides $AB$ and $AD$ of rectangle $ABCD$, points $M$ and $N$ are marked, respectively. It is known that $AN=3$, $NC=39$, $AM=10$, $MB=5$. |
| |
| (a) (1 point) Find the area of rectangle $ABCD$. |
| |
| (b) (3 points) Find the area of triangle $MNC$. |
| |
| ",(a) 585,easy |
| edbcc04cc190,"4. Let the edge length of a regular octahedron be 1, then the distance between its two parallel faces is $\qquad$","As shown in the figure, the regular octahedron $P-ABCD-Q$, it is easy to know that $PQ=\sqrt{2}$",easy |
| 4061ea1b83b2,"### 3.34. Compute the integral |
| |
| $$ |
| \int_{L} \frac{\sin z}{z\left(z-\frac{\pi}{2}\right)} d z |
| $$ |
| |
| where $L-$ is a rectangle bounded by the following lines: $x=2, x=-1, y=2, y=-1$.",See reasoning trace,hard |
| bded06b02206,"[ Dirichlet's Principle (continued).] |
| |
| Every day, from Monday to Friday, the old man went to the blue sea and cast his net into the sea. Each day, the net caught no more fish than the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the smallest total number of fish he could have caught over the three days - Monday, Wednesday, and Friday?",50 fish,easy |
| 129e70237af1,"46. Write the system of equations of the line passing through the point $(3, -2, 1)$ and perpendicular to the lines given by the systems: |
| |
| $$ |
| \frac{x-1}{2}=\frac{y+2}{3}=\frac{z-3}{-1} \quad \text{and} \quad \frac{x+3}{4}=\frac{y+1}{-1}=\frac{z+3}{3} |
| $$",See reasoning trace,easy |
| c7b6b4397963,"## 29. Leonie and Cats |
| |
| When old lady Leonie is asked how many cats she has, she melancholically replies: “Four fifths of my cats plus four fifths of a cat.” How many cats does she have? |
| |
| ",See reasoning trace,easy |
| 542a991b4e89,"\section*{ |
| |
| a) Give three integers \(x, y\) and \(z\) such that: |
| |
| \[ |
| x^{2}+y^{2}+z^{2}-4 x+12 y-14 z-57=0 |
| \] |
| |
| b) Determine the number of all triples \((x, y, z)\) of integers \(x, y, z\) that satisfy equation (1)!","a^{2}+b^{2}+c^{2}\) has exactly \(144+24+24=192\) different integer solution triples, so this is als",medium |
| 2afb14b6b6ec,1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ .,"\sin x, y=\arctan x$ are increasing functions on $[-1,1]$, therefore, $f(x)$ is an increasing functi",medium |
| ce3163c65975,"7.4. In grandmother's garden, apples have ripened: Antonovka, Grushovka, and White Naliv. If there were three times as many Antonovka apples, the total number of apples would increase by $70 \%$. If there were three times as many Grushovka apples, it would increase by $50 \%$. By what percentage would the total number of apples change if there were three times as many White Naliv apples?",increased by $80 \%$,medium |
| 7af4b1d544f9,1. Let $\{x\}$ denote the fractional part of the real number $x$. Given $a=(5 \sqrt{2}+7)^{2017}$. Then $a\{a\}=$ $\qquad$ .,"(a-b)+b(a-b \in \mathbf{Z}, 0<b<1)$, it follows that $b=\{a\} \Rightarrow a\{a\}=a b=1$.",medium |
| 59d7f546d2b8,"Mr. Teacher asked Adam and Eve to calculate the perimeter of a trapezoid, whose longer base measured $30 \mathrm{~cm}$, height $24 \mathrm{~cm}$, and the non-parallel sides $25 \mathrm{~cm}$ and $30 \mathrm{~cm}$. Adam got a different perimeter than Eve, yet Mr. Teacher praised both for their correct solutions. |
| |
| Determine the results of Adam and Eve. |
| |
| (L. Hozová)",See reasoning trace,medium |
| d142d6e64db5,"2. Given that $[x]$ represents the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then the number of positive numbers $x$ that satisfy |
| $$ |
| 20\{x\}+1.1[x]=2011 |
| $$ |
| |
| is ( ). |
| (A) 17 |
| (B) 18 |
| (C) 19 |
| (D) 20","1,2, \cdots, 18)$, and accordingly $\{x\}$ takes $1-\frac{11 k}{200}$.",easy |
| 1ef23bf8dbb6,"## Task B-3.4. |
| |
| For the lengths of the legs of a right triangle $a$ and $b$, the following equality holds: |
| |
| $$ |
| \log (a+b)=\frac{1}{2} \cdot \log b+\frac{1}{2} \cdot \log (a+3 b) |
| $$ |
| |
| Calculate the measure of the angle opposite the leg of length $a$.",See reasoning trace,medium |
| 30b33a2c4a75,"7. Let $x, y, z > 0$, and $x + 2y + 3z = 6$, then the maximum value of $xyz$ is $\qquad$","x+2y+3z \geqslant 3 \cdot \sqrt[3]{x \cdot 2y \cdot 3z}$, so $xyz \leqslant \frac{4}{3}$, hence the ",easy |
| 8d81864657b2,"There are $ n$ sets having $ 4$ elements each. The difference set of any two of the sets is equal to one of the $ n$ sets. $ n$ can be at most ? (A difference set of $A$ and $B$ is $ (A\setminus B)\cup(B\setminus A) $) |
| |
| $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None}$",7,medium |
| fd2fead0e285,"10. The ellipse $x^{2}+4 y^{2}=a^{2}$ (where $a$ is a constant) has a tangent line that intersects the $x$ and $y$ axes at points $A$ and $B$, respectively. Then the minimum value of $S_{\triangle A O B}$ is |
| (A) $a^{2}$ |
| (B) $\frac{\sqrt{2}}{2} a^{2}$ |
| (C) $2 a^{2}$ |
| (D) $\frac{1}{2} a^{2}$",See reasoning trace,easy |
| af11ae70386b,"Condition of the |
| |
| Find the differential $d y$. |
| |
| $y=\operatorname{arctg}(\operatorname{sh} x)+(\operatorname{sh} x) \ln (\operatorname{ch} x)$",See reasoning trace,medium |
| 9618e377c0cf,"4. One angle of a right triangle is $50^{\circ}$. Let $X, Y$ and $Z$ be the points where the inscribed circle touches the sides of the triangle. Calculate the sizes of the angles of triangle $\triangle X Y Z$.",See reasoning trace,medium |
| b31a43c35556,1. How many digits does the number $\left[1.125 \cdot\left(10^{9}\right)^{5}\right]:\left[\frac{3}{32} \cdot 10^{-4}\right]$ have?,See reasoning trace,medium |
| 6369bde1ad2d,"24. Given $a, b$ are real numbers, and $a b=1$, let $M=\frac{a}{a+1}+\frac{b}{b+1}, N=\frac{n}{a+1}+\frac{n}{b+1}$, when $M=N$, $n=$ . $\qquad$",See reasoning trace,easy |
| 450bbce1718b,"6. Let $a=\lg x+\lg \left[(y z)^{-1}-1\right], b=\lg x^{-1}+$ $\lg (x y z+1), c=\lg y+\lg \left[(x y z)^{-1}+1\right]$, and let the maximum of $a, b, c$ be $M$. Then the minimum value of $M$ is $\qquad$ |
| (Fujian contribution)","y=z=1$, $u=2$, so the minimum value of $u$ is 2, hence the minimum value of $M$ is $\lg 2$.",medium |
| 18a573b758cf,"4. Given the quadratic equation in $x$, $x^{2}-2(p+1)$ - $x+p^{2}+8=0$, the absolute value of the difference of its roots is 2. Then the value of $p$ is ( ). |
| (A) 2 |
| (B) 4 |
| (C) 6 |
| (D) 8","\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}$, then apply Vieta's formulas. )",easy |
| 74efdfd2de8e,"$17 \cdot 143$ In $\triangle A B C$, $E$ is the midpoint of side $B C$, and $D$ is on side $A C$. If the length of $A C$ is $1, \angle B A C=60^{\circ}, \angle A B C=100^{\circ}, \angle A C B=20^{\circ}, \angle D E C=80^{\circ}$, then the area of $\triangle A B C$ plus twice the area of $\triangle C D E$ equals |
| (A) $\frac{1}{4} \cos 10^{\circ}$. |
| (B) $\frac{\sqrt{3}}{8}$. |
| (C) $\frac{1}{4} \cos 40^{\circ}$. |
| (D) $\frac{1}{4} \cos 50^{\circ}$. |
| (E) $\frac{1}{8}$.",$(B)$,medium |
| 640a3c78a812,"5. Given $a, b \in \mathbf{R}$, the function $f(x)=a x-b$. If for any $x \in[-1,1]$, we have $0 \leqslant f(x) \leqslant 1$, then the range of $\frac{3 a+b+1}{a+2 b-2}$ is |
| A. $\left[-\frac{1}{2}, 0\right]$ |
| B. $\left[-\frac{4}{5}, 0\right]$ |
| C. $\left[-\frac{1}{2}, \frac{2}{7}\right]$ |
| D. $\left[-\frac{4}{5}, \frac{2}{7}\right]$",$D$,medium |
| fa88b0e76d60,"[ Area of a quadrilateral ] [ Quadrilateral (inequalities) ] |
| |
| The diagonals of a convex quadrilateral are equal to $d_{1}$ and $d_{2}$. What is the maximum value that its area can have?",See reasoning trace,medium |
| c56be7c94041,"2、D Teacher has five vases, these five vases are arranged in a row from shortest to tallest, the height difference between adjacent vases is 2 centimeters, and the tallest vase is exactly equal to the sum of the heights of the two shortest vases, then the total height of the five vases is _ centimeters","x-2+x-4, x=10$, the sum is $5 x=50$",easy |
| c8b8cdf056ae,"12.1. Find the smallest natural number consisting of identical digits and divisible by 18. |
| |
| $$ |
| (5-6 \text { cl.) } |
| $$",See reasoning trace,medium |
| 80bca06465fc,1. A three-digit number is 29 times the sum of its digits. Then this three-digit number is $\qquad$,"2, b=6$.",easy |
| e5c90c4db13d,"# Task 9.2 |
| |
| Factorize $x^{4}+2021 x^{2}+2020 x+2021$. |
| |
| ## Number of points 7 |
| |
| #",See reasoning trace,easy |
| 09b42cdcc951,"2. If $a, b, c$ are three arbitrary integers, then, $\frac{a+b}{2}$, $\frac{b+c}{2}, \frac{c+a}{2}(\quad)$. |
| (A) None of them are integers |
| (B) At least two of them are integers |
| (C) At least one of them is an integer |
| (D) All of them are integers",See reasoning trace,easy |
| f165f7c2a06d,"11.2. Find all parameters $b$, for which the system of equations $\left\{\begin{array}{l}x^{2}-2 x+y^{2}=0 \\ a x+y=a b\end{array}\right.$ has a solution for any $a$.",$b \in[0 ; 2]$,medium |
| 626ae6baa954,"5. Find the number of lattice points in the plane region (excluding the boundary) bounded by the parabola $x^{2}=2 y$, the $x$-axis, and the line $x=21$.","2 k(k=1,2,3, \cdots, 10)$, $0<x<21$, the number of integer $y$ that satisfies $0<y<\frac{x^{2}}{2}=2",medium |
| 6bd82a265c73,"22. A graduating class of 56 people is taking a photo, requiring each row to have 1 more person than the row in front of it, and not to stand in just 1 row. In this case, the first row should have $\qquad$ people.",See reasoning trace,easy |
| 18e1eb7c4ca3,"Find all pairs of positive integers $(a, b)$ such that $\left|3^{a}-2^{b}\right|=1$","1$. We then have, if $a \geqslant 1,(-1)^{b} \equiv 1 \pmod{3}$, so $b=2c$, thus, $2^{2c}-3^{a}=1$. ",medium |
| 99f27c60b33a,"7. (10 points) There are 11, 12, and 17 balls of red, yellow, and blue colors respectively. Each operation can replace 2 balls of different colors with 2 balls of the third color. During the operation process, the maximum number of red balls can be $\qquad$.",】Solution: The remainders when the number of the three types of balls is divided by 3 are $2,medium |
| 1e5fea25a2d7,"1. If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\qquad$","\frac{4 \sqrt{3}}{3}, y = \frac{\sqrt{3}}{3}$, $u = \sqrt{3}$.",medium |
| bb26f082f9c1,"4. For the positive integer $n$, define $a_{n}$ as the unit digit of $n^{(n+1)^{n+2}}$. Then $\sum_{n=1}^{2010} a_{n}=$ $\qquad$ .",1}^{2010} a_{n}=201 \times 29=5829$.,medium |
| 5b6b8cdb06eb,"1. Given the set $M=\{2,0,1,9\}, A$ is a subset of $M$, and the sum of the elements in $A$ is a multiple of 3. Then the number of subsets $A$ that satisfy this condition is ( ). |
| (A) 8 |
| (B) 7 |
| (C) 6 |
| (D) 5",See reasoning trace,easy |
| 07a39e7c02a4,"## Task 4 - 110624 |
| |
| If one third of Rainer's savings is added to one fifth of this savings, then the sum is exactly 7 marks more than half of his savings. |
| |
| How many marks has Rainer saved in total?",See reasoning trace,easy |
| 4188f6372ed2,"18.19 In the quadrilateral formed by connecting the midpoints of the sides of the following quadrilaterals, which one is centrally symmetric but not axially symmetric? |
| (A) A quadrilateral with perpendicular diagonals. |
| (B) A quadrilateral with equal diagonals. |
| (C) A quadrilateral with perpendicular and equal diagonals. |
| (D) A quadrilateral with neither perpendicular nor equal diagonals. |
| (5th ""Jinyun Cup"" Junior High School Mathematics Invitational, 1988)",(D),medium |
| 9fdc46227ca7,4. There is a math competition,See reasoning trace,medium |
| cedd4981262c,"4. Let's call a rectangular parallelepiped typical if all its dimensions (length, width, and height) are different. What is the smallest number of typical parallelepipeds into which a cube can be cut? Don't forget to prove that this is indeed the smallest number.",into 4 typical parallelepipeds,medium |
| ede28f2da9d6,"Example 6 For all $a, b, c \in \mathbf{R}_{+}$, and $abc=1$, find the minimum value of $S=\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}$.",See reasoning trace,medium |
| 2a415fd09cde,40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers.,See reasoning trace,medium |
| 2e2823e4d0e6,9.145. $\sqrt{3} \cos^{-2} x < 4 \tan x$.,See reasoning trace,easy |
| d4a1dee20c5e,"G9.3 If the lines $2 y+x+3=0$ and $3 y+c x+2=0$ are perpendicular, find the value of $c$.",-1 \\ -\frac{1}{2} \times\left(-\frac{c}{3}\right)=-1 \\ c=-6\end{array}$,easy |
| c64b97722ffb,"5. On the plane, there are two points $P$ and $Q$. The number of triangles that can be drawn with $P$ as the circumcenter and $Q$ as the incenter is ( ). |
| (A) Only 1 can be drawn |
| (B) 2 can be drawn |
| (C) At most 3 can be drawn |
| (D) Infinitely many can be drawn",See reasoning trace,easy |
| 9457e1909924,"3. Find the smallest natural number that has exactly 70 natural divisors (including 1 and the number itself). |
| |
| (16 points)",25920,medium |
| 886b924775d8,"In this |
| |
| [i]2016 CCA Math Bonanza Lightning #5.2[/i]",90,medium |
| 56b2c4ff3710,"Let $ABC$ be an equilateral triangle with side length $2$, and let $M$ be the midpoint of $\overline{BC}$. Points $X$ and $Y$ are placed on $AB$ and $AC$ respectively such that $\triangle XMY$ is an isosceles right triangle with a right angle at $M$. What is the length of $\overline{XY}$?",3 - \sqrt{3,medium |
| b5032b7b6b85,"In rectangle $ABCD$, $AB=1, BC=m, O$ is its center, $EO \perp$ plane $ABCD, EO=n$, and there exists a unique point $F$ on side $BC$ such that $EF \perp FD$. What conditions must $m, n$ satisfy for the angle between plane $DEF$ and plane $ABCD$ to be $60^{\circ}$?",See reasoning trace,medium |
| 7294fe2e6bc1,"Suppose that for the positive integer $n$, $2^{n}+1$ is prime. What remainder can this prime give when divided by $240$?",See reasoning trace,medium |
| 28349e1220b8,"How many three-digit integers are exactly 17 more than a two-digit integer? |
| (A) 17 |
| (B) 16 |
| (C) 10 |
| (D) 18 |
| (E) 5",See reasoning trace,medium |
| 7ae375a6582e,6. $\frac{2 \cos 10^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}}$ The value is $\qquad$,See reasoning trace,easy |
| 0fb7e2b3cb3a,"8.5. 20 numbers: $1,2, \ldots, 20$ were divided into two groups. It turned out that the sum of the numbers in the first group is equal to the product of the numbers in the second group. a) What is the smallest and b) what is the largest number of numbers that can be in the second group?",. a) 3; b) 5,medium |
| b01025827769,"To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. |
| Which of the following is the best approximation of $\log_5 10$? |
| $\textbf{(A) }\frac{8}{7}\qquad \textbf{(B) }\frac{9}{7}\qquad \textbf{(C) }\frac{10}{7}\qquad \textbf{(D) }\frac{11}{7}\qquad \textbf{(E) }\frac{12}{7}$",\textbf{(C),easy |
| 5b6ef54c6b01,"5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\sqrt{1+a^{2}}+\sqrt{1+2 b}$ the integer part is",See reasoning trace,easy |
| 445e28dc8210,"11. Let $0<\alpha<\pi, \pi<\beta<2 \pi$. If for any $x \in \mathbf{R}$, we have |
| $$ |
| \cos (x+\alpha)+\sin (x+\beta)+\sqrt{2} \cos x=0 |
| $$ |
| |
| always holds, find the value of $\alpha \beta$.",\frac{7 \pi}{4}$.,medium |
| 7647c8036d09,"A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It turns out that each child receives the same amount of property. The father has $\qquad$ children.",See reasoning trace,medium |
| 15598a896a69,"2. (3 points) A student, while solving a calculation",of 180,easy |
| 8115341ba7b7,"6.5. 101 people bought 212 balloons of four colors, and each of them bought at least one balloon, but no one had two balloons of the same color. The number of people who bought 4 balloons is 13 more than the number of people who bought 2 balloons. How many people bought only one balloon? Provide all possible answers and prove that there are no others.",52 people,medium |
| f62594016b02,"3. A. If $a, b$ are given real numbers, and $1<a<b$, then the absolute value of the difference between the average and the median of the four numbers $1, a+1, 2a+b, a+b+1$ is ( ). |
| (A) 1 |
| (B) $\frac{2a-1}{4}$ |
| (C) $\frac{1}{2}$ |
| (D) $\frac{1}{4}$",See reasoning trace,easy |
| a60b356d6084,"Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\tfrac{p}{q}$ for relatively prime positive integers $p, q$. What is $p + q$?",11,medium |
| 0794c4087b2a,"4. Find all triples of real numbers $x, y, z$ for which |
| |
| $$ |
| \lfloor x\rfloor-y=2 \cdot\lfloor y\rfloor-z=3 \cdot\lfloor z\rfloor-x=\frac{2004}{2005} |
| $$ |
| |
| where $\lfloor a\rfloor$ denotes the greatest integer not exceeding the number $a$. |
| |
| The second round of category B takes place |
| |
| on Tuesday, March 22, 2005 |
| |
| so that it starts in the morning and the contestants have 4 hours of pure time to solve the",See reasoning trace,medium |
| f867d369c185,"On a $16 \times 16$ torus as shown all 512 edges are colored red or blue. A coloring is good if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the 4 edges of an arbitrary cell. What is the largest number of good colorings such that none of them can be converted to another by a sequence of moves? |
| |
| #",4,medium |
| 9ba393e35d44,"2. Three points $A, B$, and $C$ are placed in a coordinate system in the plane. Their abscissas are the numbers $2, -3, 0$, and their ordinates are the numbers $-3, 2, 5$, not necessarily in that order. Determine the coordinates of these points if it is known: |
| |
| - No point is in the III. quadrant. |
| - Point $A$ does not lie on a coordinate axis. |
| - The ordinate of point $B$ is equal to the sum of the ordinates of the other two points. |
| - The abscissa of point $C$ is equal to the ordinate of point $B$.",See reasoning trace,medium |
| bbfc09196c66,"9. In the sequence $\left\{a_{n}\right\}$, $a_{1}=-1, a_{2}=1, a_{3}=-2$. If for all $n \in \mathbf{N}_{+}$, $a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+$ $a_{n+2}+a_{n+3}$, and $a_{n+1} a_{n+2} a_{n+3} \neq 1$, then the sum of the first 4321 terms $S_{4321}$ of the sequence is",should be -4321,medium |
| 9954d5dc792e,"8.1. Five buses stand in a row behind each other in a traffic jam, and in any two of them, there is a different non-zero number of passengers. Let's call two different people sympathizers if they are either in the same bus or in adjacent ones. It turned out that each passenger has either exactly 20 or exactly 30 sympathizers. Provide an example of how this can be possible.",See reasoning trace,medium |
| d8fc20b3b3a4,"Find the complex numbers $ z$ for which the series
|
| \[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\]
|
| converges and find its sum.",S(z) = \frac{1,hard |
| 520481442955,"3. The sum of the first 1997 terms of the sequence $1,1,2,1,2,3,1,2,3,4,1,2 \cdots$ is $\qquad$ .",See reasoning trace,medium |
| baa59084690f,6. We will call a divisor $d$ of a natural number $n>1$ good if $d+1$ is also a divisor of $n$. Find all natural $n$ for which at least half of the divisors are good.,"\operatorname{HOK}(2,3)=6$ and $n=\operatorname{HOK}(2,3,4)=12$. Both of them work.",easy |
| e75770f5abd6,"6. $A$ is the foot of the mountain, $B$ is the peak, and $C$ is a point on the slope, with $A C=\frac{1}{3} A B$. Person A and Person B start from the foot of the mountain at the same time, reach the peak, and return to the foot, repeating this back-and-forth movement. The speed ratio of A to B is $6: 5$, and the downhill speed of both A and B is 1.5 times their respective uphill speeds. After some time, A first sees B climbing on the $A C$ segment from the peak; after some more time, A sees B climbing on the $A C$ segment from the peak for the second time. How many times has A reached the peak by the time A sees B climbing on the $A C$ segment for the second time (including this time)?",See reasoning trace,medium |
| 330f91b94ea1,"1. The price of one pencil is a whole number of eurocents. The total price of 9 pencils is greater than 11, but less than 12 euros, while the total price of 13 pencils is greater than 15, but less than 16 euros. How much does one pencil cost?",123$ eurocents.,easy |
| 42bad8a256f1,"I4.2 When the integers $1108+a$, $1453$, $1844+2a$ and 2281 are divided by some positive integer $n(>1)$, they all get the same remainder $b$. Determine the value of $b$.",s for $b: 0$ or 4,medium |
| c280ff16cdaa,"Mr. Kostkorád owned a rectangular garden, on which he gradually paved paths from one side to the other. The paths were of the same width, intersected at two points, and once a paved area was created, it was skipped during further paving. |
| |
| When Mr. Kostkorád paved a path parallel to the longer side, he used $228 \mathrm{~m}^{2}$ of paving stones. Then he paved a path parallel to the shorter side and used $117 \mathrm{~m}^{2}$ of paving stones. |
| |
| Finally, he paved another path parallel to the first path, this time using only $219 \mathrm{~m}^{2}$ of paving stones. |
| |
| Determine the dimensions of Mr. Kostkorád's garden. |
| |
| (M. Petrová) |
| |
| Hint. Why was less paving used for the third path than for the first one?",See reasoning trace,medium |
| 9d65941c92a5,"24. Find the number of integer solutions ( $x ; y ; z$ ) of the equation param1, satisfying the condition param2. |
| |
| | param1 | param2 | | |
| | :---: | :---: | :---: | |
| | $27^{a} \cdot 75^{b} \cdot 5^{c}=75$ | $\|a+b+c\| \leq 101$ | | |
| | $27^{a} \cdot 75^{b} \cdot 5^{c}=375$ | $\|a+b+c\|<98$ | | |
| | $27^{a} \cdot 75^{b} \cdot 5^{c}=1875$ | $\|a+b+c\| \leq 111$ | | |
| | $27^{a} \cdot 75^{b} \cdot 5^{c}=\frac{125}{3}$ | $\|a+b+c\| \leq 139$ | |",\frac{125}{3}$ | $\|a+b+c\| \leq 139$ | 69 |,easy |
| 131044a96fb4,"$29-35$ Let $a$ be a positive integer, $a<100$, and $a^{3}+23$ is divisible by 24. Then, the number of such $a$ is |
| (A) 4. |
| (B) 5. |
| (C) 9. |
| (D) 10. |
| (China High School Mathematics League, 1991)",$(B)$,medium |
| cfe537ab1896,"1. Detective Podberezyakov is pursuing Maksim Detochkin (each driving their own car). At the beginning, both were driving on the highway at a speed of 60 km/h, with Podberezyakov lagging behind Detochkin by 2 km. Upon entering the city, each of them reduced their speed to 40 km/h, and upon exiting the city, finding themselves on a good highway, each increased their speed to 70 km/h. When the highway ended, at the border with a dirt road, each had to reduce their speed to 30 km/h again. What was the distance between them on the dirt road?",1 km,easy |
| c3f0ca8a187e,"## Task 6 - 150736 |
| |
| If $z$ is a natural number, let $a$ be the cross sum of $z$, $b$ be the cross sum of $a$, and $c$ be the cross sum of $b$. |
| |
| Determine $c$ for every 1000000000-digit number $z$ that is divisible by 9!",See reasoning trace,medium |
| 784c23c60e5f,Russian,"BC 2 , k>=AC 2 and k<=AC.BC, with equality in the last case only if AC is perpendicular to BC. Hence",easy |
| 7630d91c2d2f,"8. In an acute-angled triangle $A B C$, a point $Q$ is chosen on side $A C$ such that $A Q: Q C=1: 2$. From point $Q$, perpendiculars $Q M$ and $Q K$ are dropped to sides $A B$ and $B C$ respectively. It is given that $B M: M A=4: 1, B K=K C$. Find $M K: A C$.",$M K: A C=\frac{2}{\sqrt{10}}$,medium |
| cb8286fe66aa,"3. Let $M$ be the midpoint of side $BC$ of $\triangle ABC$, $AB=4$, $AM=1$. Then the minimum value of $\angle BAC$ is $\qquad$ .",See reasoning trace,easy |
| b99b847cd105,"Let's divide 9246 crowns among 4 people in the following way: If $A$ gets $2 K$, then $B$ gets $3 K$, if $B$ gets $5 K$, then $C$ gets $6 K$, and if $C$ gets $3 K$, then $D$ gets $4 K$.",See reasoning trace,easy |
| 6d559edd1a5f,"What is the $21^{\varrho}$ term of the sequence |
| |
| $$ |
| 1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \ldots ? |
| $$",See reasoning trace,medium |
| 260749e6333b,"## Task 1 |
| |
| Solve the following equations. The same letters represent the same numbers. |
| |
| $$ |
| \begin{aligned} |
| 3280+a & =3330 \\ |
| a+b & =200 \\ |
| c: a & =4 \\ |
| a+b+c+d & =500 |
| \end{aligned} |
| $$",50 ; b=150 ; c=200 ; d=100$,easy |
| 86d283fecfb5,,"1$, from the division theorem, we get $\overline{a 1} = (a + 1) \cdot C + a$, from which $9 \cdot (a",medium |
| 6b3efc397413,"1. Let $\triangle A B C$ have interior angles $\angle A, \angle B, \angle C$ with opposite sides of lengths $a, b, c$ respectively. If $c=1$, and the area of $\triangle A B C$ is equal to $\frac{a^{2}+b^{2}-1}{4}$, then the maximum value of the area of $\triangle A B C$ is $\qquad$",b=\sqrt{\frac{2+\sqrt{2}}{2}}$.,easy |
| 34b59a0cdd8a,"33. Find the smallest number \( n > 1980 \) such that the number |
| |
| \[ |
| \frac{x_{1}+x_{2}+x_{3}+\ldots+x_{n}}{5} |
| \] |
| |
| is an integer for any assignment of integers \( x_{1}, x_{2}, x_{3}, \ldots, x_{n} \), none of which are divisible by 5.",See reasoning trace,easy |
| 1758fc59f0ec,"Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance between points $A$ and $C$ on the lateral surface of the frustum.",See reasoning trace,medium |
| 8aa629f75f83,"8. Determine the value of the sum |
| $$ |
| \frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2} \cdot 3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\cdots+\frac{29}{14^{2} \cdot 15^{2}} . |
| $$",See reasoning trace,easy |
| 2b732f25ce1f,"10.344. Calculate the area of the common part of two rhombuses, the lengths of the diagonals of the first of which are 4 and $6 \mathrm{~cm}$, and the second is obtained by rotating the first by $90^{\circ}$ around its center.",$9,medium |
| 2b5f6cfc011b,"5. Let $f(x)$ be a cubic polynomial. If $f(x)$ is divided by $2 x+3$, the remainder is 4 , while if it is divided by $3 x+4$, the remainder is 5 . What will be the remainder when $f(x)$ is divided by $6 x^{2}+17 x+12$ ?",See reasoning trace,easy |
| d14ae0e406f2,"2. Given rhombus $A B C D, \Gamma_{B}$ and $\Gamma_{C}$ are circles centered at $B$ and passing through $C$, and centered at $C$ and passing through $B$, respectively. $E$ is one of the intersection points of circles $\Gamma_{B}$ and $\Gamma_{C}$, and the line $E D$ intersects circle $\Gamma_{B}$ at a second point $F$. Find the size of $\angle A F B$.",See reasoning trace,medium |
| b7ca4acb708c,"Example 5 Find all integer arrays $(a, b, c, x, y, z)$, such that |
| $$\left\{\begin{array}{l} |
| a+b+c=x y z, \\ |
| x+y+z=a b c, |
| \end{array}\right.$$ |
| |
| where $a \geqslant b \geqslant c \geqslant 1, x \geqslant y \geqslant z \geqslant 1$.","(2,2,2,6,1,1),(5,2,1,8,1,1)$, $(3,3,1,7,1,1),(3,2,1,3,2,1),(6,1,1,2,2,2),(8,1,1,5,2,1)$ and $(7, 1,1",medium |
| dd7e71a8c79a,1. The range of the function $f(x)=\sqrt{x-5}-\sqrt{24-3 x}$ is $\qquad$ .,See reasoning trace,easy |
| ca7a10b751e0,"B3. Binnen een vierkant $A B C D$ ligt een punt $P . E$ is het midden van de zijde $C D$. |
| |
| Gegeven is : $A P=B P=E P=10$. |
| |
| Wat is de oppervlakte van vierkant $A B C D$ ?",See reasoning trace,easy |
| 80a144b9262b,"Question 1 Let $n$ be a positive integer, $D_{n}$ be the set of all positive divisors of $2^{n} 3^{n} 5^{n}$, $S \subseteq D_{n}$, and any number in $S$ cannot divide another number in $S$. Find the maximum value of $|S|$. ${ }^{[1]}$","to this question is $\left[\frac{3(n+1)^{2}+1}{4}\right]$, where $[x]$ represents the greatest integer not exceeding the real number $x$",easy |
| bf817a8ba84c,"168. Another house number puzzle. Brown lives on a street with more than 20 but fewer than 500 houses (all houses are numbered in sequence: $1,2,3$ and so on). Brown discovered that the sum of all numbers from the first to his own, inclusive, is half the sum of all numbers from the first to the last, inclusive. |
| |
| What is the number of his house",See reasoning trace,medium |
| 8f64e0b5cbe0,"5. If $x, y$ satisfy $|y| \leqslant 2-x$, and $x \geqslant-1$, then the minimum value of $2 x+y$ is |
| A. -7 |
| B. -5 |
| C. 1 |
| D. 4",$B$,easy |
| dc6b764aa566,"Determine all pairs $(x, y)$ of integers satisfying $x^{2}=y^{2}\left(x+y^{4}+2 y^{2}\right)$",0$ does not need to be treated again.,medium |
| 6d59429ba9b1,"3.1. Along the groove, there are 100 multi-colored balls arranged in a row with a periodic repetition of colors in the following order: red, yellow, green, blue, purple. What color is the ball at the $78-$th position? |
| |
| $$ |
| \text { (4-5 grades) } |
| $$",See reasoning trace,easy |
| d58d160d21f0,"11.006. The diagonal of a rectangular parallelepiped is 13 cm, and the diagonals of its lateral faces are $4 \sqrt{10}$ and $3 \sqrt{17}$ cm. Determine the volume of the parallelepiped.",$144 \mathrm{~cm}^{3}$,medium |
| 8c3e0f25e683,"$10 . B$ ship is at a position $45^{\circ}$ north of west from $A$ ship, the two ships are $10 \sqrt{2} \mathrm{~km}$ apart. If $A$ ship sails west, and $B$ ship sails south at the same time, and the speed of $B$ ship is twice that of $A$ ship, then the closest distance between $A$ and $B$ ships is $\qquad$ $\mathrm{km}$.","6$, $A_{1} B_{1}=2 \sqrt{5}$ is the minimum.",easy |
| e8bf87c18fb2,"3. Two numbers x and y satisfy the equation $280 x^{2}-61 x y+3 y^{2}-13=0$ and are the fourth and ninth terms, respectively, of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.",$d=-5$,medium |
| 04166c53a0b3,8. $\sum_{k=1}^{2006} \frac{1}{[\sqrt{k}]}-\sum_{k=1}^{44} \frac{1}{k}$ The value is $\qquad$,86+\frac{71}{44}+\sum_{k=1}^{43} \frac{1}{k}-\sum_{k=1}^{44} \frac{1}{k}=86+\frac{70}{44}=\frac{1927,medium |
| c59258353a05,"[b]p1.[/b] An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of $101$ ships. By the decision of the commander of the fleet, some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Destroyers can shoot rockets so that every rocket destroys one drone. During the attack each carrier will have enough time to provide each destroyer with one rocket but not more. How many destroyers and how many carriers should the commander assign to destroy the maximal number of drones and what is the maximal number of drones that the fleet can destroy? |
| |
| |
| [b]p2.[/b] Solve the inequality: $\sqrt{x^2-3x+2} \le \sqrt{x+7}$ |
| |
| |
| [b]p3.[/b] Find all positive real numbers $x$ and $y$ that satisfy the following system of equations: |
| $$x^y = y^{x-y}$$ |
| $$x^x = y^{12y}$$ |
| |
| |
| [b]p4.[/b] A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$. |
| |
| |
| [b]p5.[/b] Find all positive integer solutions $n$ and $k$ of the following equation: |
| $$\underbrace{11... 1}_{n} \underbrace{00... 0}_{2n+3} + \underbrace{77...7}_{n+1} \underbrace{00...0}_{n+1}+\underbrace{11...1}_{n+2} = 3k^3.$$ |
| |
| |
| [b]p6.[/b] The Royal Council of the planet Naboo consists of $12$ members. Some of these members mutually dislike each other. However, each member of the Council dislikes less than half of the members. The Council holds meetings around the round table. Queen Amidala knows about the relationship between the members so she tries to arrange their seats so that the members that dislike each other are not seated next to each other. But she does not know whether it is possible. Can you help the Queen in arranging the seats? Justify your answer. |
| |
| |
| PS. You should use hide for answers.","[-1, 1] \cup [2, 5]",medium |
| e367aa5cd4d1,"Three. (25 points) As shown in Figure 2, given points $A$ and $B$ are two distinct points outside circle $\odot O$, point $P$ is on $\odot O$, and $PA$, $PB$ intersect $\odot O$ at points $D$ and $C$ respectively, different from point $P$, and $AD \cdot AP = BC \cdot BP$. |
| (1) Prove: $\triangle OAB$ is an isosceles triangle; |
| (2) Let $p$ be a prime number, and $m$ be a positive integer. If $AD \cdot AP = p(2p + 1)$, $OA = m - 1$, and the radius of $\odot O$ is 3, find the length of $OA$.",8$.,medium |
| 7808aa5be5c2,Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.,k = 1,medium |
| 4086aef8b2a7,"II. (50 points $\}$ |
| $a, b, c \in \mathbf{R}$. Satisfy $|a|>1,|b|>1,|c|>1$ and $b=\frac{a^{2}}{2-a^{2}}, c=\frac{b^{2}}{2-b^{2}}, a=$ $\frac{c^{2}}{2-c^{2}}$. Find all possible values of $a+b+c$.",See reasoning trace,medium |
| 1d6f7389d531,"Let $S$ be the [set] of points whose [coordinates] $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The [probability] that the [midpoint] of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$",200,medium |
| 5b5fbedc9003,"10. Calculate: |
| $$ |
| 325 \times\left(\frac{1}{1 \times 2 \times 3 \times 4}+\frac{1}{2 \times 3 \times 4 \times 5}+\frac{2}{4 \times 5 \times 6 \times 7}+\frac{2}{5 \times 6 \times 7 \times 8}+\cdots+\frac{8}{22 \times 23 \times 24 \times 25}+\frac{8}{23 \times 24 \times 25 \times 26}\right) |
| $$",See reasoning trace,easy |
| d68b65d46ae1,"If $n$ is any integer, $n+3, n-9, n-4, n+6$, and $n-1$ are also integers. If $n+3, n-9, n-4, n+6$, and $n-1$ are arranged from smallest to largest, the integer in the middle is |
| (A) $n+3$ |
| (B) $n-9$ |
| (C) $n-4$ |
| (D) $n+6$ |
| (E) $n-1$","0$. Then the values of the 5 integers are 3 , $-9,-4,6$, and -1 . When we arrange these from smalles",easy |
| c0b121190a61,"For which real numbers $x$ is it true that |
| |
| $$ |
| \left\{\frac{1}{3}\left[\frac{1}{3}\left(\frac{1}{3} x-3\right)-3\right]-3\right\}=0 |
| $$ |
| |
| where $\{z\}$ is the fractional part of $z$ - see the 398th page of our November 1985 issue for the Gy. 2294 exercise -, $[z]$ is the integer part of $z$, i.e., $[z]=z-\{z\}$.",See reasoning trace,medium |
| 41770aa05ecf,110. Two equally skilled chess players are playing chess. What is more likely: to win two out of four games or three out of six (draws are not considered)?,See reasoning trace,medium |
| dc625ee1077a,"1. A commercial lock with 10 buttons, which can be opened by pressing the correct five numbers, regardless of the order. The figure below is an example using $\{1,2,3,6,9\}$ as its combination. If these locks are reprogrammed to allow combinations of one to nine digits, how many additional combinations (i.e., not using five digits) are possible?",See reasoning trace,easy |
| 8b5f4783461e,"Example. Calculate the triple integral |
| |
| $$ |
| \iiint_{\Omega} \frac{x^{2}}{x^{2}+y^{2}} d x d y d z |
| $$ |
| |
| where the region $\Omega$ is bounded by the surfaces |
| |
| $$ |
| z=\frac{9}{2} \sqrt{x^{2}+y^{2}}, \quad z=\frac{11}{2}-x^{2}-y^{2} |
| $$",. $\iiint_{\Omega} \frac{x^{2}}{x^{2}+y^{2}} d x d y d z=\pi$,medium |
| 0bb46aa9a3a3,"7) Every year, at the time of paying taxes, the user makes a declaration regarding the current year. If the declaration is true, they must pay the taxes; if it is false, they do not pay. A young mathematician, who considers the system unfair, finds a way to block it with one of the following statements: which one? |
| (A) ""Fish live in water"" |
| (B) ""I live in water"" |
| (C) ""Fish do not pay taxes"" |
| (D) ""I do not pay taxes"" |
| (E) ""I pay taxes"".",(D),easy |
| 1e6bcf741279,"Knowing that Luca paid 5 lei more than Vlad, and Adina paid 4 lei less than Vlad and Luca together, find out how much a pen costs, how much a notebook costs, and how much a box of colored pencils costs.",See reasoning trace,medium |
| b51066e5c763,"NT3 Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.",See reasoning trace,medium |
| f4f768fa1fcf,"2. For any real numbers $m$, $n$, $p$, it is required: |
| (1) to write the corresponding quadratic equation; |
| (2) to make the equation have $n^{2}-4 m p$ as the discriminant. |
| |
| Among the following answers: |
| $$ |
| \begin{array}{l} |
| m x^{2}+n x+p=0, \\ |
| p x^{2}+n x+m=0, \\ |
| x^{2}+n x+m p=0, \\ |
| \frac{1}{a} x^{2}+n x+a m p=0(a \neq 0), |
| \end{array} |
| $$ |
| |
| The equation(s) that fully meet the requirements ( ). |
| (A) exactly 1 |
| (B) exactly 2 |
| (C) exactly 3 |
| (D) all 4","0(a \neq 0)$ corresponds to a unique discriminant $\Delta=b^{2}-4 a c$, but conversely, a quadratic ",medium |
| d58fc8914ae3,4. Calculate $\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}$.,$\sqrt{2}$,easy |
| 5099564a2bc1,"202. Indicate the number of steps sufficient to assert with an error probability not exceeding 0.001 that the reduced speed of the chip is less than 0.01. |
| |
| Let us recall now that each movement of the chip is conditioned by the result of tossing a coin. If, in $n$ tosses of the coin, heads appear $l$ times and tails appear $n-l$ times, the chip will make $l$ steps to the right and $n-l$ steps to the left and will end up at the point |
| |
| $$ |
| l-(n-l)=2 l-n |
| $$ |
| |
| The reduced speed of the chip over $n$ steps will be expressed by the absolute value of the ratio |
| |
| $$ |
| \frac{2 l-n}{n}=2 \frac{l}{n}-1 |
| $$ |
| |
| The fraction $\frac{l}{n}$ characterizes the frequency of heads appearing. |
| |
| Suppose a certain allowable error probability is given. As we know, for large values of $n$, it can be asserted with practical certainty that the reduced speed is close to zero. From expression (8), it is clear that if the reduced speed is small, then $2 \frac{l}{n}$ is approximately equal to 1 and, consequently, the frequency $\frac{l}{n}$ is close to $\frac{1}{2}$. Thus: |
| |
| If a coin is tossed a large number of times, it is practically certain that the frequency of heads appearing will be close to $\frac{1}{2}$. |
| |
| Roughly speaking, it is practically certain that heads will appear in about half of the cases. A more precise formulation states: |
| |
| Choose an arbitrary allowable error probability $\varepsilon$ and specify any arbitrarily small number $\alpha$. If the number of coin tosses exceeds |
| |
| $$ |
| N=\frac{1}{\alpha^{2} \sqrt[3]{\varepsilon^{2}}} |
| $$ |
| |
| then with an error probability less than $\varepsilon$, it can be asserted that the frequency of heads appearing differs from $\frac{1}{2}$ by less than $\alpha$. |
| |
| The proof of this precise formulation easily follows from statement b) on page 151: if $n>\frac{1}{\alpha^{2} \sqrt[3]{\varepsilon^{2}}}$, then |
| |
| $$ |
| \frac{1}{\sqrt{n}}<\alpha \cdot \sqrt[3]{e} \text { and } \frac{\frac{2}{\sqrt[3]{\varepsilon}}}{\sqrt{n}}<2 \alpha |
| $$ |
| |
| Thus, with an error probability less than $\varepsilon$, the reduced speed of the chip is less in absolute value than $2 \alpha$. |
| |
| But the reduced speed in our case is the absolute value of $\frac{2 l-n}{n}=2 \frac{l}{n}-1$. Therefore, with an error probability less than $\varepsilon$, it can be asserted that $2 \frac{l}{n}$ differs from 1 by less than $2 \alpha$ or, in other words, $\frac{l}{n}$ differs from $\frac{1}{2}$ by less than $\alpha$.",See reasoning trace,medium |
| a1b494e7f2a9,27. Let $\xi$ and $\eta$ be independent random variables having exponential distributions with parameters $\lambda$ and $\mu$ respectively. Find the distribution functions of the variables $\frac{\xi}{\xi+\eta}$ and $\frac{\xi+\eta}{\xi}$.,See reasoning trace,medium |
| 0fd3c49c742f,7.099. $\lg (3-x)-\frac{1}{3} \lg \left(27-x^{3}\right)=0$.,0,easy |
| 0f853be16d61,"18 Given real numbers $x_{1}, x_{2}, \cdots, x_{10}$ satisfy |
| $$ |
| \sum_{i=1}^{10}\left|x_{i}-1\right| \leqslant 4, \sum_{i=1}^{10}\left|x_{i}-2\right| \leqslant 6, |
| $$ |
| |
| find the average $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$.",See reasoning trace,medium |
| ed17e79fb2e1,"Example 12. There are three urns with balls. The first contains 5 blue and 3 red balls, the second - 4 blue and 4 red, and the third - 8 blue. One of the urns is randomly chosen, and a ball is randomly drawn from it. What is the probability that it will be red (event $A$).","1,2, \ldots, n)$ of events $H_{1}, H_{2}, \ldots, H_{n}$ before the experiment are called prior prob",medium |
| 7fddcbcfdbd8,"5. Let the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, then $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$ .",4000,medium |
| 081ba544d832,"Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$.
|
|
|
| Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.","k \in \{0, \alpha_2, \ldots, \alpha_n\",medium |
| 3f8045df3a28,"Four, on a plane there are $n(n \geqslant 4)$ lines. For lines $a$ and $b$, among the remaining $n-2$ lines, if at least two lines intersect with both lines $a$ and $b$, then lines $a$ and $b$ are called a ""congruent line pair""; otherwise, they are called a ""separated line pair"". If the number of congruent line pairs among the $n$ lines is 2012 more than the number of separated line pairs, find the minimum possible value of $n$ (the order of the lines in a pair does not matter).",See reasoning trace,medium |
| 104499e4a966,"2. Find the real numbers $a, b, c>0$ if |
| |
| $$ |
| \lim _{n \rightarrow \infty}\left(\sqrt{a^{2} n^{2}+2014 n+1}-b n+c\right)=\sqrt{\frac{2 c}{a}} \cdot \sqrt{2014} \text { and } a+c=72 \text {. } |
| $$ |
| |
| Prof.Voiculeț Septimius, Videle","53, b=53, c=19),(a=19, b=19, c=53) \ldots \ldots . .1 p$",medium |
| 295794b237e8,"For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. |
| |
| Note : Please solve the |
| |
| (1) Find $f(\sqrt{3})$ |
| |
| (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ |
| |
| (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.",\frac{\pi,hard |
| 900ae209a369,"Find the surface generated by the solutions of \[ \frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy}, \] which intersects the circle $y^2+ z^2 = 1, x = 0.$",y^2 + z^2 = 1 + 2x^2,medium |
| b26df81f32da,"Two $5\times1$ rectangles have 2 vertices in common as on the picture.
|
| (a) Determine the area of overlap
|
| (b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$.
|
|
|
| [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=290[/img]",\frac{\sqrt{26,medium |
| cc2e2abddd7d,"Find all seven-digit numbers that contain each of the digits 0 to 6 exactly once and for which the first and last two-digit numbers are divisible by 2, the first and last three-digit numbers are divisible by 3, the first and last four-digit numbers are divisible by 4, the first and last five-digit numbers are divisible by 5, and the first and last six-digit numbers are divisible by 6. |
| |
| (M. Mach)",See reasoning trace,medium |
| 27b9f5ca843a,"In the diagram, square $P Q R S$ has side length 2. Points $M$ and $N$ are the midpoints of $S R$ and $R Q$, respectively. The value of $\cos (\angle M P N)$ is |
| (A) $\frac{4}{5}$ |
| (B) $\frac{\sqrt{2}}{2}$ |
| (C) $\frac{\sqrt{5}}{3}$ |
| (D) $\frac{1}{3}$ |
| (E) $\frac{\sqrt{3}}{2}$ |
| |
| ",(A),medium |
| 1fe7d49c5453,"Example 1. A pile of toothpicks 1000 in number, two people take turns to pick any number from it, but the number of toothpicks taken each time must not exceed 7. The one who gets the last toothpick loses. How many toothpicks should the first player take on the first turn to ensure victory? (New York Math Competition)","125 \times 8$, we know that one should first dare to take 7 moves, so that the latter can achieve a ",easy |
| 3ab5f37b14de,"$17 \cdot 150$ The perimeter of a right triangle is $2+\sqrt{6}$, and the median to the hypotenuse is 1. Then the area of this triangle is |
| (A) 1 . |
| (B) 2 . |
| (C) 4 . |
| (D) $\frac{1}{2}$. |
| (China Jilin Province Seven Cities and Prefectures Junior High School Mathematics Competition, 1987)",$(D)$,easy |
| b4b2123156ee,"Example 11 There are 20 points distributed on a circle, and now we connect them using 10 chords that have no common endpoints and do not intersect each other. How many different ways can this be done?","2, a_{3}=5, a_{4}=14, a_{5}=42, \cdots, a_{10}=16796$.",medium |
| 052399165269,"1.5.2 * Let real numbers $a, x, y$ satisfy the following conditions |
| $$ |
| \left\{\begin{array}{l} |
| x+y=2 a-1, \\ |
| x^{2}+y^{2}=a^{2}+2 a-3 . |
| \end{array}\right. |
| $$ |
| |
| Find the minimum value that the real number $xy$ can take.",\frac{3}{2}(a-1)^{2}+\frac{1}{2} \geqslant \frac{3}{2}\left(2-\frac{\sqrt{2}}{2}-1\right)^{2}+\frac{,medium |
| 729dd1320da1,"4. Given arithmetic sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, the sums of the first $n$ terms are $S_{n}, T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=\frac{3 n+2}{2 n+1}$. Then $\frac{a_{7}}{b_{5}}=$ $\qquad$",\frac{41 k}{19 k}=\frac{41}{19}$.,easy |
| 4480cdd20315,"4. Given two circles $C_{1}: x^{2}+y^{2}=1$ and $C_{2}$ : $(x-2)^{2}+y^{2}=16$. Then the locus of the center of the circle that is externally tangent to $C_{1}$ and internally tangent to $C_{2}$ is $\qquad$ |
| $\qquad$",1$,easy |
| 303b1b78fc7d,"A1. What is the maximum number of odd sums among $x+y, x+z, x+w, y+z, y+w$ and $z+w$, if $x, y, z$ and $w$ are natural numbers? |
| (A) 2 |
| (B) 3 |
| (C) 4 |
| (D) 5 |
| (E) 6",See reasoning trace,easy |
| 36105ae479d8,"6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. For the sake of secrecy, he will not visit the same village twice, and he will not visit villages in a row that are directly connected by a road. How many villages can he guarantee to fly over? He can start from any village.",See reasoning trace,medium |
| e33fb3ebf519,"7.2. There are 11 kg of cereal. How can you measure out 1 kg of cereal using two weighings on a balance scale, if you have one 3 kg weight?","7$ kg (grain) (since $3+x=11-x=>x=4$). Second weighing: from the obtained 4 kg of grain, pour out 3 ",easy |
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