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 | 3. In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=$ $\frac{2-\sqrt{3}}{4}$. Then the maximum value of $\cos A \cos B$ is $\qquad$ . | \frac{2+\sqrt{3}}{4} | 56 | 12 |
math | 5. In triangle $A B C, \angle A=45^{\circ}$ and $M$ is the midpoint of $\overline{B C}$. $\overline{A M}$ intersects the circumcircle of $A B C$ for the second time at $D$, and $A M=2 M D$. Find $\cos \angle A O D$, where $O$ is the circumcenter of $A B C$. | -\frac{1}{8} | 92 | 7 |
math | 11.3. Find the roots of the equation: $\left(x^{3}-2\right)\left(2^{\sin x}-1\right)+\left(2^{x^{3}}-4\right) \sin x=0$. | \sqrt[3]{2},\pin(n\in{Z}) | 54 | 15 |
math | 17.1.7 * Find the integer solution to the equation $\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\cdots+\left[\frac{x}{10!}\right]=1001$. | 584 | 54 | 3 |
math | XXXVIII OM - II - Task 1
From an urn containing one ball marked with the number 1, two balls marked with the number 2, ..., $ n $ balls marked with the number $ n $, we draw two balls without replacement. We assume that drawing each ball from the urn is equally probable. Calculate the probability that both drawn balls... | \frac{4}{3(n+2)} | 78 | 10 |
math | Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus. | ABCD | 38 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{h \rightarrow 0} \frac{\sin (x+h)-\sin (x-h)}{h}$ | 2\cosx | 38 | 4 |
math | The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:
[i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\
Determine the smallest possible value of $N$. | 6 | 89 | 1 |
math | A tournament will take place with 100 competitors, all with different skill levels. The most skilled competitor always wins against the least skilled competitor. Each participant plays exactly twice, with two randomly drawn opponents (once against each). A competitor who wins two matches receives a medal. Determine the... | 1 | 70 | 1 |
math | Two squares of a $7 \times 7$ board are painted yellow and the rest is painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many non-equivalent color schemes can we obtain?
# | 300 | 58 | 3 |
math | 6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is 2, and
the later... | \frac{15\sqrt{2}}{4} | 91 | 13 |
math |
4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
| 2,3,6 | 40 | 5 |
math | 10.325. The base of the triangle is 20 cm, the medians of the lateral sides are 18 and 24 cm. Find the area of the triangle. | 288\mathrm{~}^{2} | 42 | 11 |
math | 3. The equation $x^{2}+a x+3=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}-\frac{99}{2 x_{2}^{2}}=x_{2}^{3}-\frac{99}{2 x_{1}^{2}}
$$
Find all possible values of $a$. | -6 | 92 | 2 |
math | 8th Irish 1995 Problem A2 Find all integers n for which x 2 + nxy + y 2 = 1 has infinitely many distinct integer solutions x, y. | all\n\except\-1,0,1 | 40 | 10 |
math | Example 4. Find $\lim _{y \rightarrow 0} \frac{e^{y}+\sin y-1}{\ln (1+y)}$. | 2 | 35 | 1 |
math | 237. $\left\{\begin{array}{l}x+y+\sqrt{x^{2}+y^{2}}=\frac{x y}{2} \\ x y=48 .\end{array}\right.$ | 8,6 | 48 | 3 |
math | 4. If the system of inequalities about $x$ $\left\{\begin{array}{l}x^{3}+3 x^{2}-x-3>0, \\ x^{2}-2 a x-1 \leqslant 0\end{array},(a>0)\right.$ has exactly one integer solution, then the range of values for $a$ is $\qquad$ | [\frac{3}{4},\frac{4}{3}) | 86 | 14 |
math | How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)? | 2,002,001 | 53 | 9 |
math | 2.204. $\sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$. | 1 | 71 | 1 |
math | 14. Let $a_{1}=2006$, and for $n \geq 2$,
$$
a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n} .
$$
What is the value of $2005 a_{2005}$ ? | 2 | 72 | 1 |
math | Let's find the values of $x$ for which the functions
$$
y=\frac{1}{\tan x}+\frac{1}{\cot x} \quad \text{and} \quad y=\frac{1}{\tan x}-\frac{1}{\cot x}
$$
will have their minimum values. | 45 | 71 | 2 |
math | Khachaturyan A.V.
Petr was born in the 19th century, and his brother Pavel - in the 20th century. Once, the brothers met to celebrate their shared birthday. Petr said: "My age is equal to the sum of the digits of the year of my birth." - "Mine too," replied Pavel. How much younger is Pavel than Petr? | 9 | 80 | 1 |
math | 1A. Determine the second term of the arithmetic progression if the sum of the first 10 terms is 300, and the first, second, and fifth terms, in that order, form a geometric progression. | 9 | 46 | 1 |
math | For some positive integer $n$, the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$, inclusive. Compute $n$.
[i]2020 CCA Math Bonanza Lightning Round #3.1[/i] | 21 | 73 | 2 |
math | 7. Let $\alpha, \beta$ be a pair of conjugate complex numbers. If $|\alpha-\beta|=2 \sqrt{3}$ and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha|=$ $\qquad$ . | 2 | 59 | 1 |
math | 10. If for all positive real numbers $a, b, c, d$, the inequality $\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geqslant \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}$ always holds, find all real numbers $x$ that satisfy the condition. | \frac{15}{8} | 98 | 8 |
math | 5. Given that vectors $\boldsymbol{\alpha}, \boldsymbol{\beta}$ are two unit vectors in a plane with an angle of $60^{\circ}$ between them, and $(2 \boldsymbol{\alpha}-\boldsymbol{\gamma}) \cdot(\boldsymbol{\beta}-\boldsymbol{\gamma})=0$, then the maximum value of $|\gamma|$ is $\qquad$ . | \frac{\sqrt{7}+\sqrt{3}}{2} | 85 | 15 |
math | A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i] | 91 | 113 | 2 |
math | Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$. | 7 | 41 | 3 |
math | 25. Between 1 and 8000 inclusive, find the number of integers which are divisible by neither 14 nor 21 but divisible by either 4 or 6 . | 2287 | 41 | 4 |
math | 1089. Given an infinite arithmetic progression $3 ; 16 ; 29 ; 42 ; \ldots$. Find:
a) the smallest term of the sequence that can be written using only sevens;
b) all such terms. | 777\ldots7,wheretheofsevensis6k+4(k=0,1,2,\ldots) | 54 | 30 |
math | Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that:
[b](i)[/b] $a_k < b_k,$
[b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb R,$
prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit. | 0 | 127 | 1 |
math | For what value of $\lambda$ does the following equation represent a pair of straight lines:
$$
\lambda x^{2}+4 x y+y^{2}-4 x-2 y-3=0
$$ | \lambda=4 | 45 | 4 |
math | 4. $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ 的最大值为 $\qquad$
The maximum value of $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ is $\qquad$ | \frac{8\sqrt{3}}{9} | 67 | 12 |
math | Example 5. Find $\lim _{x \rightarrow \infty} \frac{\ln x}{x^{\alpha}}$ for $\alpha>0, x>0$. | \lim_{xarrow\infty}\frac{\lnx}{x^{\alpha}}=0\text{for}\alpha>0 | 38 | 29 |
math | Example 8. Find the differential equation for which the function $y=C_{1} x+C_{2}$, depending on two arbitrary constants, is the general solution. | y^{\\}=0 | 35 | 5 |
math | 6. The reading on a factory's electricity meter is 52222 kilowatts. After several days, the meter reading (a five-digit number) again shows four identical digits. How much electricity, in kilowatts, did the factory use at least in these days?
untranslated portion: $\qquad$ | 333 | 66 | 3 |
math | 6. Variant 1. In the kindergarten, 5 children eat porridge every day, 7 children eat porridge every other day, and the rest never eat porridge. Yesterday, 9 children ate porridge. How many children will eat porridge today? | 8 | 55 | 1 |
math | Condition of the problem
Find the derivative.
$$
y=\cos (\ln 13)-\frac{1}{44} \cdot \frac{\cos ^{2} 22 x}{\sin 44 x}
$$ | \frac{1}{4\sin^{2}22x} | 51 | 15 |
math | 1. Does there exist a natural number for which the sum of the digits of its square is:
a) 80;
b) 81? | 111111111 | 32 | 9 |
math | 17. Mingming's mother found an interesting phenomenon while shopping. Every time she paid, the amount of money in her wallet was exactly 5 times the amount she paid. After settling the bill twice, she still had 320 yuan left in her wallet. How much money did she have in her wallet before shopping? | 500 | 67 | 3 |
math | Example 1. Select 4 people from 6 boys and 4 girls to participate in an extracurricular interest group. How many ways are there to select them?
(1) At least one boy and one girl participate;
(2) At most 3 boys participate. | 195 | 58 | 3 |
math | 8. Given $a b=1$, and $\frac{1}{1-2^{x} a}+\frac{1}{1-2^{y+1} b}=1$, then the value of $x+y$ is $\qquad$. | -1 | 53 | 2 |
math | $\left.\begin{array}{l}\text { [Inclusion-Exclusion Principle]} \\ {[\quad \text { Word Problems (Miscellaneous). }}\end{array}\right]$
In the garden, Anya and Vitya had 2006 rose bushes. Vitya watered half of all the bushes, and Anya watered half of all the bushes. It turned out that exactly three bushes, the most be... | 3 | 117 | 1 |
math | Problem 6.6. Several oranges (not necessarily of equal weight) were picked from a tree. When they were weighed, it turned out that the weight of any three oranges taken together is less than $5 \%$ of the total weight of the remaining oranges. What is the smallest number of oranges that could have been picked? | 64 | 67 | 2 |
math | Problem 3. Determine how many roots of the equation
$$
4 \sin 2 x + 3 \cos 2 x - 2 \sin x - 4 \cos x + 1 = 0
$$
are located on the interval $\left[10^{2014!} \pi ; 10^{2014!+2015} \pi\right]$. In your answer, write the sum of all digits of the found number. | 18135 | 105 | 5 |
math | 1. Let $\log _{2} x=m \in Z, m>0, \log _{6} y=n \in Z, n>0$.
Then $x=2^{m}, y=6^{n}$. As a result, we have
$$
\text { GCD }(x, y)=\text { GCD }\left(2^{m}, 6^{n}\right)=\text { GCD }\left(2^{m}, 2^{n} \cdot 3^{n}\right)=8=2^{3} .
$$
Case 1. $m \geq n$. Then $n=3, \quad y=6^{3}=216$,
GCD $\left(\log _{2} x, 3\right)... | 8^{k},k=1,2,\ldots;216\text | 225 | 18 |
math | Example 6.24. Five televisions have been put on subscription service. It is known that for a group of five televisions, the expected number of failures per year is one. If the televisions have the same probability of working without failure, what is the probability that at least one repair will be needed within a year? | 0.67 | 68 | 4 |
math | 16. (15 points) Let the function
$$
f(x)=x^{2}-\left(k^{2}-5 a k+3\right) x+7(a, k \in \mathbf{R}) \text {. }
$$
For any $k \in[0,2]$, if $x_{1}, x_{2}$ satisfy
$$
x_{1} \in[k, k+a], x_{2} \in[k+2 a, k+4 a] \text {, }
$$
then $f\left(x_{1}\right) \geqslant f\left(x_{2}\right)$. Find the maximum value of the positive ... | \frac{2\sqrt{6}-4}{5} | 153 | 13 |
math | Six, in all integers that start and end with 1 and alternate between 1 and 0 (i.e., $101$, $10101$, $1010101$, etc.), how many of them are prime numbers? | 101 | 55 | 3 |
math | 7.4. Given nine cards with the numbers $5,5,6,6,6,7,8,8,9$ written on them. From these cards, three three-digit numbers $A, B, C$ were formed, each with all three digits being different. What is the smallest value that the expression $A+B-C$ can have? | 149 | 75 | 3 |
math | # 3.1. Condition:
The number 4597 is displayed on the computer screen. In one move, it is allowed to swap any two adjacent digits, but after this, 100 is subtracted from the resulting number. What is the largest number that can be obtained by making no more than two moves? | 8357 | 69 | 4 |
math | ## Task 2 - 200732
Given are seven line segments with lengths $1 \mathrm{~cm}, 3 \mathrm{~cm}, 5 \mathrm{~cm}, 7 \mathrm{~cm}, 9 \mathrm{~cm}, 11 \mathrm{~cm}$, and $15 \mathrm{~cm}$.
a) Give the number of all different ways to select three of these seven line segments! Ways that differ only in the order of the selec... | 31.4 | 184 | 4 |
math | 2. Let $a, b$ be positive real numbers,
$$
A=\frac{a+b}{2}, B=\frac{2}{\frac{1}{a}+\frac{1}{b}} \text {. }
$$
If $A+B=a-b$, then $\frac{a}{b}=$ | 3+2 \sqrt{3} | 66 | 8 |
math | Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1\\
wxy^2 + wx^2z + w^2yz + xyz^2 &=2\\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*... | 24 | 136 | 2 |
math | Determine the $gcd$ of all numbers of the form $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ where $a, b, c, d$ range over the integers.
## Solutions | 12 | 48 | 2 |
math | Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows:
-for each $i = 1, . . . , n$, he computes the number $a_i$ of integers currently on the board that are at most $i$,
-he erases all integers on the board,
... | 2n | 264 | 4 |
math | 5-2. In a sports tournament, a team of 10 people participates. The regulations stipulate that 8 players from the team are always on the field, changing from time to time. The duration of the match is 45 minutes, and all 10 participants on the team must play an equal number of minutes. How many minutes will each player ... | 36 | 83 | 2 |
math | 3. Six points A, B, C, D, E, F are connected with segments length of $1$. Each segment is painted red or black probability of $\frac{1}{2}$ independence. When point A to Point E exist through segments painted red, let $X$ be. Let $X=0$ be non-exist it. Then, for $n=0,2,4$, find the probability of $X=n$. | P(X=0) = \frac{69}{128}, P(X=2) = \frac{7}{16}, P(X=4) = \frac{3}{128} | 97 | 46 |
math | 12. (18 points) Let $n (n \geqslant 11)$ be a positive integer. The set $A$ consists of the sums of 10 consecutive positive integers not greater than $n$, and the set $B$ consists of the sums of 11 consecutive positive integers not greater than $n$. If the number of elements in $A \cap B$ is 181, find the maximum and m... | 2011 \text{ and } 2001 | 100 | 14 |
math | 81. Let positive real numbers $x, y, z$ satisfy the condition $2 x y z=3 x^{2}+4 y^{2}+5 z^{2}$, find the minimum value of the expression $P=3 x+2 y+z$.
| 36 | 59 | 2 |
math | 2. Given $x, y \in \mathbf{R}, 2 x^{2}+3 y^{2} \leqslant 12$, then the maximum value of $|x+2 y|$ is $\qquad$ | \sqrt{22} | 53 | 6 |
math | Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where... | 7 | 118 | 1 |
math | 5. Given the inequality $\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta<3 a+6$, for $\theta \in\left[0, \frac{\pi}{2}\right]$ to always hold. Find the range of $\theta$. (1st China Southeast Mathematical Olympiad) | a>3 | 96 | 3 |
math | II. Find all integer values of $a$ for which the equation $(a+1) x^{2}-\left(a^{2}+1\right) x$ $+2 a^{3}-6=0$ has integer roots. | -1,0,1 | 51 | 6 |
math | Given a triangle with sides $A B=2, B C=3, A C=4$. A circle is inscribed in it, and the point $M$ where the circle touches side $B C$ is connected to point $A$. Circles are inscribed in triangles $A M B$ and $A M C$. Find the distance between the points where these circles touch the line $A M$. | 0 | 85 | 1 |
math | Shapovalov A.V.
There are three piles of 40 stones each. Petya and Vasya take turns, Petya starts. On a turn, one must combine two piles, then divide these stones into four piles. The player who cannot make a move loses. Which of the players (Petya or Vasya) can win, regardless of how the opponent plays? | Vasya | 83 | 3 |
math | Gardener Mr. Malina was selling strawberries. In the last nine crates, he had 28, 51, 135, 67, 123, 29, 56, 38, and 79 strawberry plants, respectively. He sold the crates whole, never removing any plants from the crates. The gardener wanted to sell the crates to three customers so that nothing was left and each of thes... | 202 | 121 | 3 |
math | Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\] | a = 2 + \sqrt{3} | 56 | 11 |
math |
4. Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
| 5412 | 47 | 4 |
math | 1. Let $x, y$ and $a$ be real numbers such that $x+y=a-1$ and $x y=a^{2}-7 a+12$. For which $a$ does the expression $x^{2}+y^{2}$ attain its maximum possible value? What are $x$ and $y$ then? | 6,2,3or3,2 | 73 | 9 |
math | ## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\ln t \\
y=\operatorname{arctg} t
\end{array}\right.
$$ | \frac{\cdot(1-^2)}{(1+^2)^2} | 65 | 18 |
math | 4B. Find all five-digit numbers $\overline{a b c d e}$ such that $\overline{a b}$, $\overline{b c}$, $\overline{c d}$, and $\overline{d e}$ are perfect squares. | 81649 | 56 | 5 |
math | 1. On the meadow, there are children and adults. The percentage of boys among all children is equal to the percentage of girls among all present people and also the number of all adults. How many boys, girls, and adults are on the meadow? | 32 | 53 | 2 |
math | $ABCD$ is a square and $AB = 1$. Equilateral triangles $AYB$ and $CXD$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $XY$? | \sqrt{3} - 1 | 51 | 8 |
math | ## Task 4 - 190614
Three pioneers from a school, Karin, Dieter, and Frank, were delegated to the district mathematics olympiad and won a first, a second, and a third prize (each of the three pioneers exactly one of these prizes). Later, Anette inquired about the performance of the three olympiad participants. She was ... | Karin:2,Dieter:3,Frank:1 | 162 | 13 |
math | 1. $x, y$ are real numbers, and $\left(x+\sqrt{x^{2}+1}\right)(y+$ $\left.\sqrt{y^{2}+1}\right)=1$. Then $x+y=$ $\qquad$ . | 0 | 54 | 1 |
math | Proizvoov V.v.
Ten consecutive natural numbers were written on the board. When one of them was erased, the sum of the nine remaining numbers turned out to be 2002.
What numbers remained on the board? | 218,219,220,221,222,224,225,226,227 | 48 | 35 |
math | 6th Mexico 1992 Problem A2 Given a prime number p, how many 4-tuples (a, b, c, d) of positive integers with 0 < a, b, c, d < p-1 satisfy ad = bc mod p? | (p-2)(p^2-5p+7) | 57 | 13 |
math | 27.15. How many digits does the number $2^{100}$ have?
## 27.3. Identities for logarithms | 31 | 33 | 2 |
math | 3. The sum of the digits of the number $X$ is $Y$, and the sum of the digits of the number $Y$ is $Z$. If $X+Y+Z=60$, determine the number $X$.
| 44,50,47 | 51 | 8 |
math | $$
a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b) \geqslant k a b c
$$
holds for all right triangles, and determine when equality occurs. | 2+3\sqrt{2} | 47 | 8 |
math | 6.005. $\frac{1}{x(x+2)}-\frac{1}{(x+1)^{2}}=\frac{1}{12}$. | x_{1,2}\in\varnothing,x_{3}=-3,x_{4}=1 | 38 | 21 |
math |
Problem 8'.3. Find all natural numbers $n$ such that there exists an integer number $x$ for which $499\left(1997^{n}+1\right)=x^{2}+x$.
| 1 | 53 | 1 |
math | 3. In the coordinate plane, the area enclosed by the curve ||$x|+| x-1||+|y|=2$ is equal to $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 | 61 | 1 |
math | Let $p$ be a prime number. Find all possible values of the remainder when $p^{2}-1$ is divided by 12 . | 3,8,0 | 31 | 5 |
math | Bootin D. ..
Several guard brigades of the same size slept for the same number of nights. Each guard slept more nights than there are guards in a brigade, but fewer than the number of brigades. How many guards are in a brigade if all the guards together slept for 1001 person-nights? | 7 | 66 | 1 |
math | 4. Problem: Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and
$$
2 x(x+5)=p^{n}+3(x-1) \text {. }
$$ | (2,5,2)(0,3,1) | 50 | 13 |
math | 13 If a positive integer $n$ can be written in the form $a^{b}$ (where $a, b \in \mathbf{N}, a \geqslant 2, b \geqslant 2$), then $n$ is called a "good number". Among the positive integers adjacent to the positive integer powers of 2, find all the "good numbers". | 9 | 85 | 1 |
math | 15. In the sequence $20,18,2,20,-18, \ldots$ the first two terms $a_{1}$ and $a_{2}$ are 20 and 18 respectively. The third term is found by subtracting the second from the first, $a_{3}=a_{1}-a_{2}$. The fourth is the sum of the two preceding elements, $a_{4}=a_{2}+a_{3}$. Then $a_{5}=a_{3}-a_{4}$, $a_{6}=a_{4}+a_{5}$,... | 38 | 155 | 2 |
math | 10.270. Find the ratio of the sum of the squares of all medians of a triangle to the sum of the squares of all its sides. | \frac{3}{4} | 34 | 7 |
math | 1. Given that $a$, $b$, $c$, and $d$ are prime numbers, and $a b c d$ is the sum of 77 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$ | 32 | 55 | 2 |
math | The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter | 9 | 39 | 1 |
math | ## 22. Math Puzzle $3 / 67$
Peter wants to use a balance scale, whose beam lengths $a$ and $b$ are no longer exactly equal. He first places a $5 \mathrm{~kg}$-"weight" on the left pan and weighs, and then the $5 \mathrm{~kg}$ piece on the right pan and weighs the rest of the apples.
Is the weighed amount heavier or l... | G>10 | 102 | 4 |
math | 6.156. $20\left(\frac{x-2}{x+1}\right)^{2}-5\left(\frac{x+2}{x-1}\right)^{2}+48 \frac{x^{2}-4}{x^{2}-1}=0$. | x_{1}=\frac{2}{3},x_{2}=3 | 63 | 16 |
math | [ Residue arithmetic (other).]
Solve the equation $x^{2}+y^{2}=z^{2}$ in natural numbers. | {x,y}={nk,\frac{1}{2}k(^{2}-n^{2})},\frac{1}{2}k(^{2}+n^{2}) | 30 | 39 |
math | There are 2023 cups numbered from 1 through 2023. Red, green, and blue balls are placed in the cups according to the following rules.
- If cups $m$ and $n$ both contain a red ball, then $m-n$ is a multiple of 2 .
- If cups $m$ and $n$ both contain a green ball, then $m-n$ is a multiple of 3 .
- If cups $m$ and $n$ both... | 538 | 131 | 3 |
math | 5. Each football is sewn from pieces of leather in the shape of regular pentagons and regular hexagons. The ball has a total of 32 leather pieces. Each piece of leather in the shape of a pentagon is connected along its sides only to pieces of leather in the shape of hexagons. Each piece of leather in the shape of a hex... | 20 | 120 | 2 |
math | In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively, and $G$ is the centroid of $\triangle A B C$. If
$$
a \overrightarrow{G A}+b \overrightarrow{G B}+\frac{\sqrt{3}}{3} c \overrightarrow{G C}=0 \text {, }
$$
then $\angle A=$ . $\qquad$ | 30^{\circ} | 104 | 6 |
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