task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$.
Note: $N = \{0,1,2,...\}$ | (f(n), g(n)) = (an^2 + 2bn + c, an + b) | 69 | 24 |
math | 5.4. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 83.25 | 134 | 5 |
math | 13. The function $f(x)$ defined on $\mathbf{R}$ satisfies: $f(x+2)=2-f(x), f(x+3) \geqslant f(x)$, try to find $f(x)$. | f(x)=1 | 51 | 4 |
math | Ha $9^{-1 \frac{1}{6}}=0.0770401$, what is $9^{-\frac{2}{3}}$? | 0.2311203 | 37 | 9 |
math | Example 5. Find the value of $\sum_{k=1}^{n} k^{2} C_{n}^{k}$. (3rd Putnam Mathematical Competition, USA) | n(n+1) \cdot 2^{n-2} | 40 | 14 |
math | Example 12. Let $M, x, y$ be positive integers, and $\sqrt{M-\sqrt{28}}=\sqrt{x}-\sqrt{y}$. Then the value of $x+y+M$ is ( ).
(1994, Hope Cup Mathematics Competition) | 16 | 62 | 2 |
math | 12.240. The ratio of the volume of a sphere inscribed in a cone to the volume of a circumscribed sphere is $k$. Find the angle between the slant height of the cone and the plane of its base and the permissible values of $k$. | \arccos\frac{1\\sqrt{1-2\sqrt[3]{k}}}{2},0<k\leq\frac{1}{8} | 58 | 36 |
math | 1. Let $a_{1}, a_{2}, \cdots, a_{2015}$ be a sequence of numbers taking values from $-1, 0, 1$, satisfying
$$
\sum_{i=1}^{2015} a_{i}=5 \text {, and } \sum_{i=1}^{2015}\left(a_{i}+1\right)^{2}=3040,
$$
where $\sum_{i=1}^{n} a_{i}$ denotes the sum of $a_{1}, a_{2}, \cdots, a_{n}$.
Then the number of 1's in this sequenc... | 510 | 154 | 3 |
math | Example 18. Solve the inequality
$$
5^{\frac{1}{4} \log _{5}^{2} x} \geqslant 5 x^{\frac{1}{5} \log _{5} x}
$$ | (0;5^{-2\sqrt{5}}]\cup[5^{2}\sqrt{5};+\infty) | 56 | 26 |
math | 3. On the board is written a three-digit number **8. A trio of students guessed the properties of this number.
Zoran: All its digits are even and it has an even number of different prime divisors.
Daniel: It is divisible by 9 and is the square of some natural number.
Nikola: It is less than 400 and 13 times the squa... | 108or468 | 112 | 7 |
math | 2. Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence defined by: $a_{1}=\frac{1}{2}$ and $a_{n+1}+a_{n}=\frac{2}{n^{2}+2 n}, \forall n \geq 1$.
a. Find the general term of the sequence.
b. Calculate the sum $S=\sum_{k=1}^{m}(2 k+1) a_{k}^{2}$ and show that $S<1$.
Prof. Traian Tămâian | 1-\frac{1}{(n+1)^2} | 129 | 13 |
math | [ Counting in two ways ] [ Different tasks on cutting ]
Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the 100 marked points and the vertices of the square, and for each triangle in the partition, each marked point either lies out... | 202 | 95 | 3 |
math | 1. Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$. | 4,6 | 56 | 3 |
math | 12.206. A section is made through the vertex of a regular triangular pyramid and the midpoints of two sides of the base. Find the area of the section and the volume of the pyramid, given the side $a$ of the base and the angle $\alpha$ between the section and the base. | \frac{^{2}\sqrt{3}}{48\cos\alpha};\frac{^{3}\operatorname{tg}\alpha}{48} | 65 | 34 |
math | 5. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1$, and for each $n \in \mathbf{N}^{*}, a_{n}, a_{n+1}$ are the roots of the equation $x^{2}+3 n x+b_{n}=0$, then $\sum_{k=1}^{20} b_{k}=$ $\qquad$ . | 6385 | 93 | 4 |
math | 1. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal, and among all sets with this property, the sum of $S$ is the largest. Find the sum of the set $S$.
(4th American Invitational Mathem... | 61 | 87 | 2 |
math | 2. There are 20 teams participating in the national football championship finals. To ensure that in any group of three teams, at least two teams have played against each other, what is the minimum number of matches that need to be played? | 90 | 49 | 2 |
math | In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.) | 2 | 69 | 1 |
math | Example 7. From the first machine, 200 parts were sent to assembly, of which 190 are standard; from the second - 300, of which 280 are standard. Find the probability of event $A$, which consists in a randomly taken part being standard, and the conditional probabilities of it relative to events $B$ and $\bar{B}$, if eve... | 0.94,0.95,\frac{14}{15}\approx0.93 | 99 | 23 |
math | 5】 Given $x_{i}$ are non-negative real numbers, $i=1,2,3,4 . x_{1}+x_{2}+x_{3}+x_{4}=1$. Let $S=1-$ $\sum_{i=1}^{4} x_{i}^{3}-6 \sum_{1 \leqslant i<j<k \leqslant 4} x_{i} x_{j} x_{k}$, find the range of $S$. | S \in \left[0, \frac{3}{4}\right] | 110 | 17 |
math | 7. Let $a, b>0$, satisfy: the equation $\sqrt{|x|}+\sqrt{|x+a|}=b$ has exactly three distinct real solutions $x_{1}, x_{2}, x_{3}$, and $x_{1}<x_{2}<x_{3}=b$, then the value of $a+b$ is $\qquad$ . | 144 | 79 | 3 |
math | Determine all pairs $(a, b)$ of strictly positive integers such that $a b-a-b=12$. | (2,14),(14,2) | 24 | 11 |
math | Let $l$ be a line passing the origin on the coordinate plane and has a positive slope. Consider circles $C_1,\ C_2$ determined by the condition (i), (ii), (iii) as below.
(i) The circles $C_1,\ C_2$ are contained in the domain determined by the inequality $x\geq 0,\ y\geq 0.$
(ii) The circles $C_1,\ C_2$ touch the li... | 7 | 207 | 1 |
math | The surface area of the circumscribed sphere of a cube $K_{1}$ is twice as large as the surface area of the inscribed sphere of a cube $K_{2}$. Let $V_{1}$ denote the volume of the inscribed sphere of cube $K_{1}$, and $V_{2}$ the volume of the circumscribed sphere of cube $K_{2}$. What is the ratio $\frac{V_{1}}{V_{2}... | \frac{2\sqrt{2}}{27} | 101 | 13 |
math | Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space.
Let $V$ be the cone obtained by rotating the triangle around the $x$-axis.
Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis. | \frac{8\pi}{3} | 93 | 9 |
math | Let $n \geqslant 3$ be integer. Given convex $n-$polygon $\mathcal{P}$. A $3-$coloring of the vertices of $\mathcal{P}$ is called [i]nice[/i] such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of differen... | 2^n - 3 - (-1)^n | 112 | 10 |
math | The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$. | \frac{289}{259} | 110 | 11 |
math | 1. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$ the following holds
$$
(x+1) f(x f(y))=x f(y(x+1))
$$ | f(x)=0f(x)=x | 59 | 8 |
math | 3. Let the function be
$$
f(x)=\sqrt{2 x^{2}+2 x+41}-\sqrt{2 x^{2}+4 x+4}(x \in \mathbf{R}) \text {. }
$$
Then the maximum value of $f(x)$ is $\qquad$ | 5 | 70 | 1 |
math | 15 Let vectors $\vec{i}$ and $\vec{j}$ be the unit vectors in the positive directions of the $x$-axis and $y$-axis, respectively, in a rectangular coordinate plane. If $\vec{a}=(x+2) \vec{i}+y \vec{j}$, $\vec{b}=(x-2) \vec{i}+y \vec{j}$, and $|\vec{a}|-|\vec{b}|=2$.
(1) Find the equation of the locus of point $P(x, y)$... | \lambda=2 | 180 | 4 |
math | 4. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: x^{2}-\frac{y^{2}}{24}=1$, respectively, and let $P$ be a point on the hyperbola $C$ in the first quadrant. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=\frac{4}{3}$, then the radius of the incircle of $\triangle P F_{1} F_{2}$ is . $\q... | 2 | 127 | 1 |
math | 14. How many four-digit numbers can be formed using the digits $1$, $9$, $8$, $8$ that leave a remainder of 8 when divided by 11? | 4 | 40 | 1 |
math | Example 3 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 value $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$. (2012, Zhejiang Province High School Mathematics Competition) | 1.4 | 124 | 3 |
math | 5. (3 points) Find all functions continuous on the entire number line that satisfy the identity $5 f(x+y)=f(x) f(y)$ and the condition $f(1)=10$.
Answer: $f(x)=5 \cdot 2^{x}$. | f(x)=5\cdot2^{x} | 58 | 10 |
math | Four siblings are sitting down to eat some mashed potatoes for lunch: Ethan has 1 ounce of mashed potatoes, Macey has 2 ounces, Liana has 4 ounces, and Samuel has 8 ounces. This is not fair. A blend consists of choosing any two children at random, combining their plates of mashed potatoes, and then giving each of those... | \frac{1}{54} | 109 | 8 |
math | 5.2. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere was outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its moveme... | 60.75 | 133 | 5 |
math | Example 2.3.5 Derangement Problem: Find the number of permutations $\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$ that satisfy $\forall i$, $x_{i} \neq i,(i=1,2, \cdots, n)$, denoted as $D_{n}$. | D_{n}=n!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^{n}\frac{1}{n!}) | 77 | 48 |
math | 53. Point $K$ lies on the base $AD$ of trapezoid $ABCD$, such that $|AK|=\lambda|AD|$. Find the ratio $|AM|:|AD|$, where $M$ is the point of intersection with $AD$ of the line passing through the points of intersection of the lines $AB$ and $CD$ and the lines $BK$ and $AC$.
Taking $\lambda=1 / n, n=1,2,3, \ldots$, obt... | \frac{|AM|}{|AD|}=\frac{\lambda}{\lambda+1} | 138 | 20 |
math | -、(Full marks 10 points) There are two decks of playing cards, each deck arranged in such a way that the first two cards are the Big Joker and the Small Joker, followed by the four suits of Spades, Hearts, Diamonds, and Clubs, with each suit arranged in the order of $1,2,3$, $\cdots, J, Q, K$. Someone stacks the two de... | 6 \text{ of Diamonds} | 140 | 7 |
math | 4- 123 Let $a, b, c$ be given positive real numbers, try to determine all positive real numbers $x, y, z$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
x+y+z=a+b+c, \\
4 x y z-\left(a^{2} x+b^{2} y+c^{2} z\right)=a b c .
\end{array}\right.
$$ | \frac{b+}{2},\frac{+}{2},\frac{+b}{2} | 97 | 23 |
math | 5. From the set of numbers $\{1,2,3, \ldots, 200\}$, one number is randomly selected. Calculate the probability that the following random event will occur $A=\{$ A number that is not divisible by 6 is selected \}. | \frac{167}{200} | 59 | 11 |
math | Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of... | n \equiv 1 \pmod{2} | 94 | 12 |
math | Find all integer solutions $(a, b)$ of the equation $3 a^{2} b^{2}+b^{2}=517+30 a^{2}$. | (2,7),(-2,7),(2,-7),(-2,-7) | 38 | 19 |
math | 3. Find the number of natural numbers $k$, not exceeding 267000, such that $k^{2}-1$ is divisible by 267. | 4000 | 38 | 4 |
math | Question 228, Set $S=\{1,2, \ldots, 10\}$ has several five-element subsets satisfying: any two elements in $S$ appear together in at most two five-element subsets. Ask: What is the maximum number of five-element subsets? | 8 | 60 | 1 |
math | 17. Let $p$ and $q$ be positive integers such that $\frac{72}{487}<\frac{p}{q}<\frac{18}{121}$. Find the smallest possible value of $q$.
(2 marks)
設 $p \vee q$ 為滿足 $\frac{72}{487}<\frac{p}{q}<\frac{18}{121}$ 的正整數。求 $q$ 的最小可能值。
(2 分) | 27 | 115 | 2 |
math | 10. For the curve $C: x^{4}+y^{2}=1$, consider the following statements:
(1) The curve $C$ is symmetric with respect to the origin;
(2) The curve $C$ is symmetric with respect to the line $y=x$;
(3) The area enclosed by the curve $C$ is less than $\pi$;
(4) The area enclosed by the curve $C$ is greater than $\pi$.
The... | (1), (4) | 117 | 6 |
math | Find the number of positive integers x satisfying the following two conditions:
1. $x<10^{2006}$
2. $x^{2}-x$ is divisible by $10^{2006}$ | 3 | 48 | 1 |
math | 5. [5 points] Around a hook with a worm, in the same plane as it, a carp and a minnow are swimming along two circles. In the specified plane, a rectangular coordinate system is introduced, in which the hook (the common center of the circles) is located at the point $(0 ; 0)$. At the initial moment of time, the carp and... | (\sqrt{2}-4;-4-\sqrt{2}),(-4-\sqrt{2};4-\sqrt{2}),(4-\sqrt{2};4+\sqrt{2}),(4+\sqrt{2};\sqrt{2}-4) | 163 | 53 |
math | 13.108. On the plots allocated by the agrolaboratory for experiments, $c$ from two plots, 14.7 tons of grain were collected. The next year, after the application of new agricultural techniques, the yield on the first plot increased by $80 \%$, and on the second - by $24 \%$, as a result of which from these same plots, ... | 10.26 | 114 | 5 |
math | 3. A swimming pool is in the shape of a circle with diameter $60 \mathrm{ft}$. The depth varies linearly along the east-west direction from $3 \mathrm{ft}$ at the shallow end in the east to $15 \mathrm{ft}$ at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at al... | 8100\pi | 107 | 6 |
math | 6.50. $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$. | 4 | 28 | 1 |
math | How many unordered triples $A,B,C$ of distinct lattice points in $0\leq x,y\leq4$ have the property that $2[ABC]$ is an integer divisible by $5$?
[i]2020 CCA Math Bonanza Tiebreaker Round #3[/i] | 300 | 63 | 3 |
math | 1. When one of two integers was increased 1996 times, and the other was reduced 96 times, their sum did not change. What can their quotient be? | 2016 | 38 | 4 |
math | ## 9. Distribution of Euros
Ana, Janko, and Tara have certain amounts of euros and want to redistribute them among themselves. First, Ana gives Janko and Tara a portion of her money so that after this, both Janko and Tara have twice as much money as they had before. Then, Janko gives Ana and Tara a portion of his mone... | 511 | 165 | 3 |
math | Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$. | 99 | 87 | 2 |
math | 1.4. Let initially each island is inhabited by one colony, and let one of the islands have $d$ neighboring islands. What can the maximum possible number of colonies that can settle on this island be equal to? | +1 | 45 | 2 |
math | Example 6. Given the equation $x^{2}+(a-6) x+a=0$ ( $a$ $\neq 0$ ) with both roots being integers. Try to find the integer $a$. (1989, Sichuan Province Junior High School Mathematics Competition) | 16 | 63 | 2 |
math | Task 4. Find the smallest natural $m$, for which there exists such a natural $n$ that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are the same, and $a<b$. | 671 | 76 | 3 |
math | 6. In triangle $A B C$, it is known that $A B=4, A C=6, \angle B A C=60^{\circ}$. The extension of the angle bisector $A A_{1}$ intersects the circumcircle of triangle $A B C$ at point $A_{2}$. Find the areas of triangles $O A_{2} C$ and $A_{1} A_{2} C$. ( $O$ - the center of the circumcircle of triangle $\left.A B C\r... | S_{OA_{2}C}=\frac{7}{\sqrt{3}};S_{A_{1}A_{2}C}=\frac{7\sqrt{3}}{5} | 115 | 43 |
math | 6. What are the acute angles of a right triangle if for its legs $a, b$ and hypotenuse (20) $c$ it holds that $4 a b=c^{2} \sqrt{3}$? | 30 | 48 | 2 |
math | 2. Let the complex numbers be
$$
\begin{array}{l}
z_{1}=(6-a)+(4-b) \mathrm{i}, \\
z_{2}=(3+2 a)+(2+3 b) \mathrm{i}, \\
z_{3}=(3-a)+(3-2 b) \mathrm{i},
\end{array}
$$
where, $a, b \in \mathbf{R}$.
When $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|$ reaches its minimum value, $3 a+4 b$ $=$ | 12 | 134 | 2 |
math | 4. In the division equation $26 \div$ $\square$
$\square$ $\square$.. .2, both the divisor and the quotient are single-digit numbers. Please write down all the division equations that meet the requirements: $\qquad$ . | 26\div3=8\ldots\ldots\ldots0.2;26\div4=6\ldots\ldots00.2;26\div6=4\ldots\ldots\ldots0.2;26\div8=3\ldots\ldots\ldots0.2 | 53 | 77 |
math | Egy, a tízes számrendszerben felírt valódi hatjegyú számról a következőket tudjuk:
a) egyik jegye 7,
b) osztható 9-cel,
c) ha sorra kivonjuk belőle az $i$-edik és $j$-edik jegyének felcserélésével keletkező számokat, a nullától különböző különbségek között van 2525-tel, 2168-cal, 4375-tel és 6875 -tel osztható.
Mel... | 924741 | 274 | 6 |
math | Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 101 | 58 | 3 |
math | Five. (20 points) Given that the side lengths of $\triangle A B C$ are $a, b, c$, and they satisfy
$$
a b c=2(a-1)(b-1)(c-1) .
$$
Does there exist a $\triangle A B C$ with all side lengths being integers? If so, find the three side lengths; if not, explain the reason. | 3,7,8or4,5,6 | 86 | 11 |
math | $\left[\begin{array}{l}\text { Arithmetic. Mental calculation, etc. } \\ {[\underline{\text { Rebus }}]}\end{array}\right.$
Authors: Galnierein G.A., Grieorenko D.:
2002 is a palindrome year, meaning it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum n... | 109 | 118 | 3 |
math | 5. If the equation $x^{2}=a \mathrm{e}^{x}$ has three distinct real roots, then the range of the real number $a$ is $\qquad$ | (0,4\mathrm{e}^{-2}) | 40 | 12 |
math | 7,8
What is the maximum number of rooks that can be placed on an 8x8 chessboard so that they do not attack each other
# | 8 | 34 | 1 |
math | 10. (20 points) How many zeros does the number $4^{5^{6}}+6^{5^{4}}$ end with in its decimal representation?
# | 5 | 37 | 1 |
math | 9.3. Find five different numbers if all possible sums of triples of these numbers are equal to $3,4,6$, $7,9,10,11,14,15$ and 17. The numbers do not have to be integers. | -3,2,4,5,8 | 58 | 10 |
math | ## Task 6/82
Given the inequality $|x|+|y| \leq n$ with $n \in N, x ; y \in G$ (where $G$ denotes the set of integers). Determine the number of ordered solution pairs $(x ; y)$. | n^{2}+(n+1)^{2} | 62 | 12 |
math | Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$. | 420 | 30 | 3 |
math | Find all polynomials $P$ with real coefficients such that $P\left(X^{2}\right)=P(X) P(X-1)$. | (1+X+X^{2})^{n} | 31 | 12 |
math | 7. In order to cater to customers' psychology, a mall bundles two items with different cost prices for sale. Both items are sold for 216 yuan, with one item at a loss of $20 \%$, but the overall profit is $20 \%$. The cost price of the other item should be $\qquad$ yuan. | 90 | 71 | 2 |
math | [ Tangent Circles ]
Circle $S$ with its center at the vertex of the right angle of a right triangle is tangent to the circle inscribed in this triangle. Find the radius of circle $S$, given that the legs of the triangle are 5 and 12.
# | 2(\sqrt{2}\1) | 59 | 8 |
math | 2. Solve the inequality $\frac{\sqrt{x^{2}-5}}{x}-\frac{x}{\sqrt{x^{2}-5}}<\frac{5}{6}$. (8 points) | x\in(-\infty;-3)\cup(\sqrt{5};+\infty) | 43 | 20 |
math | \section*{Problem \(3-340933=341032\)}
Calculate the number
\(123456785 \cdot 123456787 \cdot 123456788 \cdot 123456796 - 123456782 \cdot 123456790 \cdot 123456791 \cdot 123456793\)
without calculating the values of the two products individually! | 22222222020 | 132 | 11 |
math | Solve the following system of equations:
$$
\frac{4}{\sqrt{x+5}}-\frac{3}{\sqrt{y+2}}=1, \quad \frac{2}{\sqrt{x+5}}+\frac{9}{\sqrt{y+2}}=4 .
$$ | -1,7 | 65 | 4 |
math | (9) In the Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$. If the "rectangular distance" from $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real num... | 5(\sqrt{2}+1) | 184 | 9 |
math | ## Task 2.
Let $a \geqslant 2018$ be a real number. In each of 2018 jars, there is a finite number of balls, with the mass of each ball being of the form $a^{k}$, where $k \in \mathbb{Z}$. The total mass of the balls in each jar is the same. What is the minimum number of balls of the same mass among the balls in the j... | 2018 | 101 | 4 |
math | 3. Define the function on $\mathbf{R}$
$$
f(x)=\left\{\begin{array}{ll}
\log _{2}(1-x), & x \leqslant 0 ; \\
f(x-1)-f(x-2), & x>0 .
\end{array}\right.
$$
Then $f(2014)=$ $\qquad$ | 1 | 86 | 1 |
math | 2. (1 mark) A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to $1: 00 \mathrm{pm}$ of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.
(1 分) 一時鐘的時針長為 3 , 分針長為 4 。問在同一天上午一時至下午一時的一段時間内, 時針與分針的端點的距離為整數多少次？ | 132 | 124 | 3 |
math | 6. It is known that all positive integers are in $n$ sets, satisfying that when $|i-j|$ is a prime number, $i, j$ belong to two different sets. Then the minimum value of $n$ is $\qquad$ . | 4 | 54 | 1 |
math | 8,9 One of the angles formed by the intersecting lines $a$ and $b$ is $15^{\circ}$. Line $a_{1}$ is symmetric to line $a$ with respect to line $b$, and line $b_{1}$ is symmetric to line $b$ with respect to $a$. Find the angles formed by the lines $a_{1}$ and $b_{1}$. | 45,45,135,135 | 88 | 13 |
math | Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
[i]Proposed by Ray Li[/i] | 200 | 98 | 3 |
math | We inscribe a straight circular cone in a sphere such that the center of the sphere divides the height of the cone according to the golden ratio. What is the ratio of the volumes of the two bodies to each other?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4:1 | 66 | 3 |
math | Task 2. (10 points) Find the greatest value of the parameter $b$ for which the inequality $b \sqrt{b}\left(x^{2}-10 x+25\right)+\frac{\sqrt{b}}{\left(x^{2}-10 x+25\right)} \leq \frac{1}{5} \cdot \sqrt[4]{b^{3}} \cdot\left|\sin \frac{\pi x}{10}\right|$ has at least one solution. | 0.0001 | 110 | 6 |
math | Task A-2.4. (8 points)
Given are complex numbers $z=7-i, w=-3+4 i$. Determine $\left|\frac{z^{20}}{\bar{w}^{10}}\right|$. | 10^{10} | 52 | 6 |
math | 2. The function
$$
f(x)=\sqrt{3} \sin ^{2} x+\sin x \cdot \cos x-\frac{\sqrt{3}}{2}\left(x \in\left[\frac{\pi}{12}, \frac{\pi}{2}\right]\right)
$$
has the range . $\qquad$ | \left[-\frac{1}{2}, 1\right] | 75 | 15 |
math | 16th USAMO 1987 Problem 2 The feet of the angle bisectors of the triangle ABC form a right-angled triangle. If the right-angle is at X, where AX is the bisector of angle A, find all possible values for angle A. Solution | 120 | 59 | 3 |
math | 1. A warehouse has coffee packed in bags of 15 kg and 8 kg. How many bags of coffee in total does the warehouseman need to prepare to weigh out 1998 kg of coffee, with the number of 8 kg bags being the smallest possible? | 136 | 58 | 3 |
math | 5. We know: $9=3 \times 3, 16=4 \times 4$, here, $9, 16$ are called "perfect squares". Among the first 300 natural numbers, if we remove all the "perfect squares", what is the sum of the remaining natural numbers? | 43365 | 68 | 5 |
math | 8. Let $\left(1+x+x^{2}\right)^{150}=\sum_{k=0}^{300} c_{k} x^{k}$, where $c_{0}$, $c_{1}, \cdots, c_{300}$ are constants. Then $\sum_{k=0}^{100} c_{3 k}=$ $\qquad$ . | 3^{149} | 89 | 6 |
math | Three. (50 points) On a line $l$, there are $n+1$ points numbered $1, 2, \cdots, n+1$ arranged from left to right. In the plane above $l$, connect these points with $n$ continuous curves, satisfying:
(1) Any curve connects only two different points, and at most one curve connects any two points;
(2) If point $i$ is con... | A_{n}=\frac{1}{n+1} \mathrm{C}_{2 n}^{n} | 206 | 24 |
math | 3. In $\triangle A B C$, the lengths of the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. Point $G$ satisfies
$$
\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=0, \overrightarrow{G A} \cdot \overrightarrow{G B}=0 \text {. }
$$
If $(\tan A+\tan B) \tan C=m... | \frac{1}{2} | 127 | 7 |
math | 9.5 What is the largest number of non-overlapping groups into which all integers from 1 to 20 can be divided so that the sum of the numbers in each group is a perfect square? | 11 | 42 | 2 |
math | 2. Determine the cardinality of the set $A=\left\{x_{n} \in \mathbb{Q} \left\lvert\, x_{n}=\frac{n^{2}+2}{n^{2}-n+2}\right., n \in \mathbb{N}, 1 \leq n \leq 2015\right\}$. | 2014 | 85 | 4 |
math | 13. $[\mathbf{9}]$ Find the smallest positive integer $n$ for which
$$
1!2!\cdots(n-1)!>n!^{2} .
$$ | 8 | 42 | 1 |
math | Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible? | 144 | 76 | 3 |
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