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 $x_{1}l>m>n$, 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 $\frac{\sum_{1 \leq i0, b>0)$, 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 $\overline{A D}=\overline{B E}$. ![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-16.jpg?height=656&width=831&top_left_y=500&top_left_x=515) 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 $-12r$, 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,![](https://cdn.mathpix.com/cropped/2024_05_06_1f1bf0225c3b69484645g-23.jpg?height=422&width=507&top_left_y=92&top_left_x=469),$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? ![](https://cdn.mathpix.com/cropped/2024_05_06_535cfcd84333341737d0g-06.jpg?height=571&width=797&top_left_y=588&top_left_x=641)",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) $Sb$, 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 $i1$, 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] ![](https://cdn.mathpix.com/cropped/2024_05_06_a6149daf2f3a04ecb66bg-03.jpg?height=574&width=654&top_left_y=287&top_left_x=241)",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) $60 \%$ ![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-020.jpg?height=318&width=441&top_left_y=2237&top_left_x=1319)",(C),easy b9d5c26e3669,"Example 11 Let real numbers $a, b$ satisfy $0 1$ 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,"## Task $1 / 74$ 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 $10\}, \\ 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} $$","1, y=1$ and $x+y=0$.",medium 6452eeccd22e,"Example 3 Let $a \leqslant b0)$.,See reasoning trace,medium d0e3a260f9dd,,68 minutes,medium d20746b6b476,"Question 74, Given $a \geq b \geq c \geq d>0, a^{2}+b^{2}+c^{2}+d^{2}=\frac{(a+b+c+d)^{2}}{3}$, find the maximum value of $\frac{a+c}{b+d}$.",\frac{x}{3-x} \leq \frac{\frac{6+\sqrt{6}}{4}}{3-\frac{6+\sqrt{6}}{4}}=\frac{7+2 \sqrt{6}}{5}$. Equa,medium ae2eb33e08bf,"[ Auxiliary area. The area helps to solve the The sides of the triangle are 13, 14, and 15. Find the radius of the circle that has its center on the middle side and touches the other two sides.",6,medium d58916b8b5ff,"## Task Condition Find the derivative. $$ 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? #",## 7 light bulbs,easy 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 $150$ 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 $[a, b](a, b \in \mathbf{Z}$, $an)$, 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 $x<608 c$. 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 $01$ 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? ![](https://cdn.mathpix.com/cropped/2024_05_06_7d113eccc6d3ed32fb40g-06.jpg?height=571&width=797&top_left_y=588&top_left_x=641)",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 $\angle A O B$. ![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-04.jpg?height=471&width=517&top_left_y=638&top_left_x=655) 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 $\angle C A B - \angle E A B = 80^{\circ}$. ![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-04.jpg?height=434&width=699&top_left_y=1645&top_left_x=587)",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 $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}1$, $q>1$. 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: $a_{1}=0, a_{2}=2, a_{3}=3, a_{n}=\max _{11$ 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 ![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-053.jpg?height=263&width=358&top_left_y=1403&top_left_x=1339)",(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: $H-F-C, C-G-K$. ![](https://cdn.mathpix.com/cropped/2024_05_06_2fdcf97aa9799d0d4cd6g-09.jpg?height=359&width=601&top_left_y=886&top_left_x=420)","$B, G, H$",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 ![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-101.jpg?height=360&width=415&top_left_y=1224&top_left_x=1321)",(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 $$ \sum_{1 \leqslant i0$, it follows that $\sqrt[12]{x}=2$, and thus $x=2^{12}=4096$. T",easy fe8ccb7c2651,"2. (8 points) Shuaishuai finished memorizing English words in five days. It is known that in the first three days, he memorized $\frac{1}{2}$ of all the words, and in the last three days, he memorized $\frac{2}{3}$ of all the words, and he memorized 120 fewer words in the first three days than in the last three days. Therefore, Shuaishuai memorized $\qquad$ English words on the third day.",Shuai Shuai memorized 120 English words on the third day,easy c66931a3612c,"In triangle $A B C$, the lengths of two sides are given: $A B=6, B C=16$. Additionally, it is known that the center of the circle passing through vertex $B$ and the midpoints of sides $A B$ and $A C$ lies on the bisector of angle $C$. Find $A C$.",18,medium f90c0a29ba5e,"Let X and Y be the following sums of arithmetic sequences: \begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end{eqnarray*} What is the value of $Y - X?$ $\textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112$",92 \ \mathbf{(A),easy 71ae786563c9,"## Task 6 - 020716 In a house with 28 windows, some missing shutters need to be procured so that each window has 2 shutters. Some windows still have 2 shutters, the same number of windows are missing both, and the rest have one shutter each. How many new shutters are needed? Justify your answer!",See reasoning trace,easy 33f28b3cd136,"Write in ascending order the numbers $$ \sqrt{121}, \quad \sqrt[3]{729} \quad \text { and } \quad \sqrt[4]{38416} $$",See reasoning trace,easy 269f562f9147,"4. If for any $x \in(0,+\infty)$, the inequality $a \mathrm{e}^{a e x+a} \geqslant \ln (\mathrm{e} x+1)$ always holds, then the range of values for $a$ is $\qquad$","\mathrm{e} x+1$, then $a \mathrm{e}^{a u} \geqslant \ln u \Rightarrow a u \mathrm{e}^{a u} \geqslant",medium 3a8637077154,"2. 55 In decimal, find the smallest natural number: its square number starts with 19 and ends with 89","3$, we similarly examine the two possibilities, and so on. Among the $x$ values we obtained, the sma",medium 7a1143320e75,17. What is the value of the sum: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{1024}$ ?,$1 \frac{1023}{1024}$,easy 4b873aef485f,"4-4. A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ lying on the circle to the line $A B$ is 8. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 30 less than the other.",34,easy 16f5f2cb55fc,"Variant 9.2.3. On the sides $AB$ and $AD$ of rectangle $ABCD$, points $M$ and $N$ are marked, respectively. It is known that $AN=3$, $NC=39$, $AM=10$, $MB=5$. (a) (1 point) Find the area of rectangle $ABCD$. (b) (3 points) Find the area of triangle $MNC$. ![](https://cdn.mathpix.com/cropped/2024_05_06_1f4981519479532effc1g-10.jpg?height=425&width=614&top_left_y=389&top_left_x=420)",(a) 585,easy edbcc04cc190,"4. Let the edge length of a regular octahedron be 1, then the distance between its two parallel faces is $\qquad$","As shown in the figure, the regular octahedron $P-ABCD-Q$, it is easy to know that $PQ=\sqrt{2}$",easy 4061ea1b83b2,"### 3.34. Compute the integral $$ \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? ![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-072.jpg?height=663&width=916&top_left_y=1459&top_left_x=570)",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 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)$, $01$ 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) $\frac{\sqrt{3}}{2}$ ![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-034.jpg?height=360&width=374&top_left_y=262&top_left_x=1339)",(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