task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | A3. In a class, $55 \%$ of students scored at least $55 \%$ on a test. $65 \%$ of students scored at most $65 \%$ on the same test. What percentage of students scored between $55 \%$ and $65 \%$ (inclusive) on the test? | 20 | 70 | 2 |
math | 3 Let $F(x)=|f(x) \cdot g(x)|$, where $f(x)=a x^{2}+b x+c, x \in[-1,1]$; $g(x)=c x^{2}+b x+a, x \in[-1,1]$, and for any parameters $a, b, c$, it always holds that $|f(x)| \leqslant 1$. Find the maximum value of $F(x)$. | 2 | 101 | 1 |
math | 19. In quadrilateral $A B C D, \angle A B C=\angle A D C=90^{\circ}, A D=D C$ and $A B+B C=20 \mathrm{~cm}$.
What is the area in $\mathrm{cm}^{2}$ of quadrilateral $A B C D$ ? | 100 | 73 | 3 |
math | Exercise 2. Determine all pairs of positive integers $(n, p)$ where $p$ is a prime number and such that $n+p$ is a power of $n$.
A power of $n$ is of the form $n^{k}$ for $k$ a natural number. | (2,2) | 61 | 5 |
math | Circles $\omega$ and $\gamma$ are drawn such that $\omega$ is internally tangent to $\gamma$, the distance between their centers are $5$, and the area inside of $\gamma$ but outside of $\omega$ is $100\pi$. What is the sum of the radii of the circles?
[asy]
size(3cm);
real lw=0.4, dr=0.3;
real r1=14, r2=9;
pair A=(0,0... | 20 | 234 | 2 |
math | 4.4. Find the radius of the base and the height of a right circular cone of the smallest volume described around a sphere of radius $R$.
| r=R\sqrt{2},H=4R | 32 | 11 |
math | Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$.
(1) Find $a_n$.
(2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$.
Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$.
Last Edited | a_n = 1 - \frac{1}{n^2} | 139 | 16 |
math | ## Aufgabe 34/81
Man ermittle alle Paare $(a ; b)$ natürlicher Zahlen $a$ und $b$, für die gilt $a^{2}-b^{2}=1981$.
| (991,990),(145,138) | 51 | 17 |
math | 1. (10 points) The creative competition at the institute consisted of four tasks. In total, there were 70 applicants. The first test was successfully passed by 35, the second by 48, the third by 64, and the fourth by 63 people, with no one failing all 4 tasks. Those who passed both the third and fourth tests were admit... | 57 | 91 | 2 |
math | Example 1. Calculate approximately the value of the function $f(x)=$ $=\sqrt{x^{2}+x+3}$ at $x=1.97$ | 2.975 | 37 | 5 |
math | B1. Alice has five real numbers $a<b<c<d<e$. She adds each pair of these numbers and records the ten results. The three smallest results turn out to be 32, 36, and 37, while the two largest results are 48 and 51. Determine $e$. | \frac{55}{2} | 68 | 8 |
math | 7. In the expansion of $\left(x+\frac{4}{x}-4\right)^{5}$, the coefficient of $x^{3}$ is $\qquad$ (answer with a specific number). | 180 | 44 | 3 |
math | 8. Let $a_{n}$ be the coefficient of the $x$ term in the expansion of $(3-\sqrt{x})^{n}$ $(n=2,3,4, \cdots)$. Then
$$
\lim _{n \rightarrow \infty}\left(\frac{3^{2}}{a_{2}}+\frac{3^{3}}{a_{3}}+\cdots+\frac{3^{n}}{a_{n}}\right)=
$$
$\qquad$ | 18 | 109 | 2 |
math | Example 2. Given the function
$$
f(x, y)=x \sin \frac{1}{y}+y \sin \frac{1}{x} \quad\left(x^{2}+y^{2} \neq 0\right), \quad f(0, y)=0, \quad f(x, 0)=0
$$
find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} f(x, y)$. | 0 | 104 | 1 |
math | Solve the following equation:
$$
8^{x}+27^{x}+64^{x}+125^{x}=24^{x}+30^{x}+40^{x}+60^{x} .
$$ | 0 | 56 | 1 |
math | 1B. Determine all pairs of natural numbers $(a, b)$ such that $a \leq b$ and
$$
\left(a+\frac{6}{b}\right)\left(b+\frac{6}{a}\right)=25
$$ | (1,4),(2,2),(1,9),(3,3) | 54 | 17 |
math | Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming... | 314 | 136 | 3 |
math | For each real number $p > 1$, find the minimum possible value of the sum $x+y$, where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$. | \frac{p - 1}{\sqrt{p}} | 57 | 13 |
math | A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________. | \frac{2^{n+2} - n - 3}{3} | 70 | 17 |
math | 38 If $a, b, c>0, a \cos ^{2} \theta+b \sin ^{2} \theta<c$, then the size relationship between $\sqrt{a} \cos ^{2} \theta+\sqrt{b} \sin ^{2} \theta$ and $\sqrt{c}$ is
$\qquad$ | \sqrt{}\cos^{2}\theta+\sqrt{b}\sin^{2}\theta<\sqrt{} | 76 | 22 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{1+\cos 3 x}{\sin ^{2} 7 x}$ | \frac{9}{98} | 41 | 8 |
math | 7. Let $f(x)$ be a function such that $f(1)=1, f(2)=2$ and $f(x+2)=f(x+1)-f(x)$. Find $f(2016)$. | -1 | 51 | 2 |
math | 6. (9th "Hope Cup" Invitational Competition Question) Let $\alpha, \beta$ be the roots of the equations $\log _{2} x+x+2=0$ and $2^{x}+x+2=0$ respectively, then the value of $\alpha+\beta$ is equal to
保留源文本的换行和格式，翻译结果如下：
6. (9th "Hope Cup" Invitational Competition Question) Let $\alpha, \beta$ be the roots of the equ... | -2 | 148 | 2 |
math | 【Question 2】 Arrange $\frac{2}{3}, \frac{5}{8}, \frac{15}{23}, \frac{10}{17}$ in ascending order | \frac{10}{17}<\frac{5}{8}<\frac{15}{23}<\frac{2}{3} | 41 | 32 |
math | In the operation below, the letters $a, b$, and $c$ are distinct digits and different from 1.
$$
\begin{array}{r}
a b b \\
\times \quad c \\
\hline b c b 1
\end{array}
$$
Determine the values of $a, b$, and $c$. | =5,b=3,=7 | 74 | 8 |
math | 11.3. Find all solutions of the system of equations in real numbers:
$$
\left\{\begin{array}{l}
x^{5}=y^{3}+2 z \\
y^{5}=z^{3}+2 x \\
z^{5}=x^{3}+2 y
\end{array}\right.
$$ | (0,0,0),\(\sqrt{2},\sqrt{2},\sqrt{2}) | 73 | 23 |
math | 3. Find all possible values of the expression
$$
\frac{x+y}{x^{2}+y^{2}}
$$
where $x$ and $y$ are any real numbers satisfying the condition $x+y \geq 1$. | (0,2] | 52 | 5 |
math | Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$? | 3 | 26 | 1 |
math | Problem 3. (6 points) Is it possible to apply the functions sin, cos, tan, cot, arcsin, arccos, arctan, arccot to the number 1 in some order to obtain the number 2010? (Each of the listed functions can be used any number of times.)
Answer. It is possible. | 2010 | 75 | 4 |
math | For a natural number $n$, let $S_{n}$ be
$$
\sum_{k=1}^{n} \sqrt{(2 k-1)^{2}+a_{k}^{2}}
$$
the minimum value, where $a_{1}, a_{2}, \cdots, a_{n}$ are positive real numbers, and their sum is 17. If there exists a unique $n$ such that $S_{n}$ is also an integer, find $n .{ }^{(4]}$ | 12 | 114 | 2 |
math | Problem 4. Two lines intersect at one point and form four angles (two acute and two obtuse). The sum of the opposite acute angles is equal to half of one of the opposite obtuse angles. Determine the sizes of the angles? | 36,144 | 49 | 6 |
math | 30th Swedish 1990 Problem 5 Find all monotonic positive functions f(x) defined on the positive reals such that f(xy) f( f(y)/x) = 1 for all x, y. | f(x)=1/xorf(x)=1 | 48 | 9 |
math | 39. How far is it to school?
All the time while Alice and the Griffin were solving the previous problem, the Tortoise Quasi was weeping inconsolably.
- Can you tell me what is sad about this problem? - the Griffin barked at him angrily. - After all, the passenger caught up with the train. Or did I misunderstand somet... | 5 | 361 | 1 |
math | Problem 6. A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. Now, sev... | 98 | 110 | 2 |
math | 8. Let $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are coprime to $n$, and $g(n)$ satisfies that for any $n \in \mathbf{Z}_{+}$, $\sum_{d \| n} g(d)=\varphi(n)$, where $\sum_{d \| n}$ indicates that $d$ takes all positive divisors of $n$. Then $g(50)=$ $\qquad$ | 0 | 100 | 1 |
math | 10. A number lies between 2013 and 2156, and when divided by $5$, $11$, and $13$, the remainders are the same. The largest possible remainder is $\qquad$ _. | 4 | 52 | 1 |
math | 11.5. Solve the equation $2021 x^{2021}-2021+x=\sqrt[2021]{2022-2021 x}$. (7 points)
# | 1 | 50 | 1 |
math | 9. (16 points) In $\triangle A B C$, $B C=a, C A=b$, $A B=c$. If $b$ is the geometric mean of $a$ and $c$, and $\sin A$ is the arithmetic mean of $\sin (B-A)$ and $\sin C$, find the value of $\cos B$.
---
The translation retains the original text's formatting and structure. | \frac{\sqrt{5}-1}{2} | 86 | 11 |
math | Let $a_{1}, a_{2}, \ldots$ be a sequence of integers that has infinitely many positive and infinitely many negative elements, and for every $n$, it holds that $a_{1}, a_{2}, \ldots, a_{n}$ leave pairwise distinct remainders when divided by $n$. How many times does $2008$ appear in the sequence? | 1 | 82 | 1 |
math | We know the following about the digits of a four-digit number:
I. The sum of the first and second (thousands and hundreds place) digits is equal to the sum of the last two digits.
II. The sum of the second and fourth digits is equal to twice the sum of the first and third digits.
III. Adding the first and fourth dig... | 1854 | 112 | 4 |
math | 1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{3}+20 x+63$. | 20 | 50 | 2 |
math | 6.- Determine the function $f: N \rightarrow N$ (where $N=\{1,2,3, \ldots\}$ is the set of natural numbers) that satisfies, for any $s, n \in N$, the following two conditions:
a) $f(1)=1, f\left(2^{s}\right)=1$.
b) If $n<2^{\text {s }}$, then $f\left(2^{\mathrm{s}}+n\right)=f(n)+1$.
Calculate the maximum value of $f... | 2^{2001}-1 | 155 | 8 |
math | 7. Given a positive real number $k$, for any positive real numbers $a, b$, let $m=\min \left\{a, \frac{b}{k a^{2}+b^{2}}\right\}$, then the maximum value of $m$ is $\qquad$ .
(Note: $\min \{x, y\}$ denotes the smaller of the real numbers $x, y$) | \frac{\sqrt{2}}{2}k^{-\frac{1}{4}} | 89 | 19 |
math | 8. If the three sides of a triangle form a geometric sequence, then the range of the common ratio $q$ is $\qquad$ . | \frac{\sqrt{5}-1}{2} < q < \frac{\sqrt{5}+1}{2} | 30 | 26 |
math | $A, B, C$ and $D$ need to divide a certain amount of money in such a way that the ratio of their shares is $3: 4: 5: 6$. However, the distribution occurred in the ratio of $3: 4: 6: 7$, as a result, $C$ and $D$ together received 1400 crowns more. What was the total amount to be divided? | 36000 | 94 | 5 |
math | 9.204. $\log _{3}\left(\log _{2}\left(2-\log _{4} x\right)-1\right)<1$. | x\in(2^{-28};1) | 38 | 11 |
math | 57. The number $123456789(10)(11)(12)(13)(14)$ is written in the base-15 numeral system, i.e., this number is equal to
(14) $+(13) \cdot 15+(12) \cdot 15^{2}+(11) \cdot 15^{3}+\ldots+2 \cdot 15^{12}+15^{13}$. What remainder does it give when divided by 7? | 0 | 122 | 1 |
math | Task 3. Vlad wants to write one number on each side of several cubes so that he can get any 30-digit number by arranging some of the cubes side by side. What is the minimum number of cubes he needs? | 50 | 47 | 2 |
math | 2・104 Let $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n}$ be real numbers, and
$$
\sum_{i=1}^{n} x_{i}^{2}+\sum_{i=1}^{n-1} x_{i} x_{i+1}=1 .
$$
For each fixed $k(k \in N, 1 \leqslant k \leqslant n)$, find the maximum value of $\left|x_{k}\right|$. | \max|x_{k}|=\sqrt{\frac{2k(n-k+1)}{n+1}} | 123 | 23 |
math | Shnol D.e.
In the class, there are 27 students, but not all of them came to the physical education lesson. The teacher divided the students who came into two equal teams to play pioneerball. In the first team, there was half of all the boys who came and a third of all the girls who came, while in the second team, ther... | 2or4 | 117 | 3 |
math | Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$.
Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$. | \frac{1}{3} | 137 | 7 |
math | 3. A student wrote a program for recoloring a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 21, green - 45, blue - 75, dark blue - 87, purple - 91, black -... | 75 | 221 | 2 |
math | 8. The range of the function $f(x)=2|\sin x|+3|\cos x|$ is $\qquad$ . | [2, \sqrt{13}] | 28 | 9 |
math | 7.170. $\log _{2}(2-x)-\log _{2}(2-\sqrt{x})=\log _{2} \sqrt{2-x}-0.5$. | 0;\frac{16}{9} | 42 | 9 |
math | Example 2 Given that $x$ and $y$ are real numbers, and satisfy $x y + x + y = 17, x^{2} y + x y^{2} = 66$.
Find the value of $x^{4} + x^{3} y + x^{2} y^{2} + x y^{3} + y^{4}$. (2000, Shandong Province Junior High School Mathematics Competition) | 12499 | 99 | 5 |
math | 3. The sum of the first 1997 terms of the sequence $1,1,2,1,2,3,1,2,3,4,1,2 \cdots$ is $\qquad$ . | 42654 | 50 | 5 |
math | 87 Given $\sin \theta+\cos \theta=\frac{1}{5}$, and $\theta \in(0, \pi)$. Then $\tan \theta=$ $\qquad$ | -\frac{4}{3} | 41 | 7 |
math | # Task 4.
Masha tightly packed 165 identical balls into the shape of a regular triangular pyramid. How many balls are at the base? | 45 | 32 | 2 |
math | 12. Nick and Tim are playing a game, in which there are no ties in each round. In a round of the game, the losing side must take at least one candy, and the winning side takes more candies than the losing side. They also agreed in advance on the number of candies (two given positive integers) that the losing and winnin... | 8 | 138 | 1 |
math | 2. If positive integers $a$, $b$, and $c$ satisfy $a b + b c = 518$, $a b - a c = 360$, then the maximum value of $a b c$ is $\qquad$ . | 1008 | 56 | 4 |
math | 49. A waterworks' reservoir contains 400 tons of water. The waterworks can inject 60 tons of water into the reservoir per hour, while the reservoir continuously supplies water to residential areas, with the total water supply in $t$ hours being $120 \sqrt{6 t}$ tons $(0 \leqslant t \leqslant 24)$.
(1) From the start of... | 8 | 157 | 1 |
math | Example 3 Given $p^{3}+q^{3}=2$, where $p, q$ are real numbers. Then the maximum value of $p+q$ is $\qquad$
(1987, Jiangsu Province Junior High School Mathematics Competition) | 2 | 56 | 1 |
math | The roots of the equation
\[ x^3-3ax^2+bx+18c=0 \]
form a non-constant arithmetic progression and the roots of the equation
\[ x^3+bx^2+x-c^3=0 \]
form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$. | (a, b) = (2, 9) | 86 | 13 |
math | ## Task A-2.1.
Determine all ordered triples $(x, y, z)$ of real numbers for which
$$
x^{2}+y^{2}=5, \quad x z+y=7, \quad y z-x=1
$$ | (2,1,3)(-\frac{11}{5},\frac{2}{5},-3) | 56 | 25 |
math | 10. In the methods for finding the zeros of a function, there is one called the Newton's method. The process of this method is: for the function $f(x)$, given $x_{1}$, and the intersection of the tangent line of $f(x)$ passing through the point $Q_{n}\left(x_{n}, f\left(x_{n}\right)\right)$ with the $x$-axis is denoted... | \sqrt{2} | 181 | 5 |
math | B4. If you try to divide the number 19 by 5, it doesn't come out exactly. The number 5 fits 3 times into 19 and you have 4 left over as the remainder. There are two positive integers $n$ with the property: if you divide $n^{2}$ by $2 n+1$, there is a remainder of 1000.
Which two numbers are these? | 6661999 | 91 | 7 |
math | 2.050. $\frac{2\left(x^{4}+4 x^{2}-12\right)+x^{4}+11 x^{2}+30}{x^{2}+6}$. | 1+3x^{2} | 50 | 7 |
math | G3.2 Let $n$ be the integral part of $\frac{1}{\frac{1}{1980}+\frac{1}{1981}+\cdots+\frac{1}{2009}}$. Find the value of $n$. | 66 | 59 | 2 |
math | What is the smallest eight-digit positive integer that has exactly four digits which are 4 ? | 10004444 | 18 | 8 |
math | Problem 5. The probability of at least one hit with two shots is 0.96. Find the probability of four hits with five shots. | 0.4096 | 31 | 6 |
math | 9. How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement?
(1 mark)
有多少種不同的方法可把「BRILLIANT 」一字中的字母重新排列, 使得排列後沒有兩個相鄰的字母相同？ | 55440 | 71 | 5 |
math | Four (This sub-question is worth 50 points)
$n$ chess players participate in a chess tournament, with each pair of players competing in one match. The winner gets 1 point, the loser gets 0 points, and in the case of a draw, both get 0.5 points. If it is found after the tournament that among any $m$ players, there is on... | 2m-3 | 191 | 4 |
math | 1. Let $\{x\}$ denote the fractional part of the real number $x$. Given $a=(5 \sqrt{2}+7)^{2017}$. Then $a\{a\}=$ $\qquad$ | 1 | 52 | 1 |
math | Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $ | 1 | 64 | 1 |
math | GS. 2 If $R$ is the remainder of $1^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6}$ divided by 7 , find the value of $R$. | 6 | 54 | 1 |
math | 4. One of the lateral sides of the trapezoid is perpendicular to the bases and equals $2 R$. A circle is constructed on this side as a diameter, which divides the other lateral side into three segments. The ratio of the lengths of these segments is 7:21:27 (counting from the upper base). Find the area of the trapezoid.... | \frac{100R^{2}}{11\sqrt{21}} | 104 | 19 |
math | Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the ... | n^2 | 110 | 3 |
math | 9. (16 points) Given that $f(x)$ is a function defined on the set of real numbers $\mathbf{R}$, $f(0)=2$, and for any $x \in \mathbf{R}$, we have
$$
\begin{array}{l}
f(5+2 x)=f(-5-4 x), \\
f(3 x-2)=f(5-6 x) .
\end{array}
$$
Find the value of $f(2012)$. | 2 | 113 | 1 |
math | 5. Given $m>0$. If the function
$$
f(x)=x+\sqrt{100-m x}
$$
has a maximum value of $g(m)$, find the minimum value of $g(m)$.
(2011, National High School Mathematics League Sichuan Province Preliminary Contest) | 10 | 68 | 2 |
math | Example 7. At two automatic machines, identical parts are manufactured. It is known that the productivity of the first machine is twice that of the second, and that the probability of producing a part of the highest quality on the first machine is 0.9, and on the second - 0.81. Parts manufactured during the shift on bo... | 0.87 | 97 | 4 |
math | Problem 5. Vasya and Petya live in the mountains and love to visit each other. When climbing up the mountain, they walk at a speed of 3 km/h, and when descending, they walk at a speed of 6 km/h (there are no horizontal road sections). Vasya calculated that it takes him 2 hours and 30 minutes to get to Petya's, and 3 ho... | 12 | 113 | 2 |
math | 23. (CZS 4) Find all complex numbers $m$ such that polynomial
$$
x^{3}+y^{3}+z^{3}+\text { mxyz }
$$
can be represented as the product of three linear trinomials. | -3,-3\omega,-3\omega^2 | 59 | 12 |
math | 4. In a decimal positive integer, if it contains an even number (including zero) of the digit 8, it is called a "good number"; otherwise, it is called a "non-good number". Then, the number of all "good numbers" with length (number of digits) not exceeding $n$ ($n$ is a positive integer) is $\qquad$ | \frac{1}{2}(8^{n}+10^{n})-1 | 78 | 19 |
math | 5. Let $a$ and $d$ be two positive integers. Construct a "triangle" similar to Pascal's triangle as follows:
$$
\begin{array}{l}
a \\
a+d \quad 2a \quad a+d \\
a+2d \quad 3a+d \quad 3a+d \quad a+2d \\
a+3d \quad 4a+3d \quad 6a+2d \quad 4a+3d \quad a+3d \\
\text{...... } \\
a+(n-1)d \\
\text{...... } \\
a+(n-1)d \\
\end... | n=6,=2,=30 | 334 | 10 |
math |
1. Alice and Ben play the game on a board with 72 cells around a circle. First, Ben chooses some cells and places one chip on each of them. Each round, Alice first chooses one empty cell and then Ben moves a chip from one of the adjacent cell onto the chosen one. If Ben fails to do so, the game ends; otherwise, anothe... | 36 | 114 | 2 |
math | 12.67 Find all integer solutions to the equation $2 x^{4}+1=y^{2}$.
(China National Training Team Selection Test, 1993) | (x,y)=(0,1),(0,-1) | 39 | 11 |
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 | 3. (physics) In a vessel, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He at a pressure $p$. The absolute temperature of the gas is doubled, during which all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of the gases are $\mu_{\... | \frac{9}{4}p | 104 | 8 |
math | 8.4. Given a right triangle, the height dropped to the hypotenuse is 4 times smaller than the hypotenuse. Find the acute angles of this triangle. | 15 | 36 | 2 |
math | Example 3. Given the sequence $\left\{a_{\mathbf{n}}\right\}$, where $a_{1}=1$, and it satisfies $a_{n+1}=1+2 a_{n}$. Find the general term $a_{n}$. | 2^{n}-1 | 59 | 5 |
math | 1. If a positive integer cannot be expressed as the difference of squares of two positive integers, then this positive integer is called a "non-wise number". If these non-wise numbers are arranged in ascending order, then the 2009th non-wise number is $\qquad$ | 8026 | 58 | 4 |
math | Question 4 Let $a, b, c \in \mathbf{C}$, satisfying
$$
\left\{\begin{array}{l}
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=9, \\
\frac{a^{2}}{b+c}+\frac{b^{2}}{c+a}+\frac{c^{2}}{a+b}=32, \\
\frac{a^{3}}{b+c}+\frac{b^{3}}{c+a}+\frac{c^{3}}{a+b}=122 .
\end{array}\right.
$$
Find the value of $a b c$.
[4]
(2017, Carnegie Me... | 13 | 171 | 2 |
math | Suppose you have three children and 40 pieces of candy. How many ways are there to distribute the candy such that each child gets more than one but fewer than 20 pieces? | 171 | 39 | 3 |
math | 9.1 Karlson eats three jars of jam and one jar of honey in 25 minutes, while Little Man takes 55 minutes. One jar of jam and three jars of honey Karlson eats in 35 minutes, while Little Man takes 1 hour 25 minutes. How long will it take them to eat six jars of jam together? | 20 | 74 | 2 |
math | 7. (6 points) In the grain store, there are 6 bags of flour, weighing $15, 16, 18, 19, 20, 31$ kilograms respectively. The canteen bought 5 of these bags in two purchases. It is known that the weight of the first purchase is twice that of the second purchase. The weight of the remaining bag is $\qquad$ kilograms. | 20 | 92 | 2 |
math | 26. (GBR 2) ${ }^{\mathrm{IMO} 3} \mathrm{~A}$ function $f$ defined on the positive integers (and taking positive integer values) is given by $$ \begin{aligned} f(1) & =1, \quad f(3)=3 \\ f(2 n) & =f(n) \\ f(4 n+1) & =2 f(2 n+1)-f(n) \\ f(4 n+3) & =3 f(2 n+1)-2 f(n) \end{aligned} $$ for all positive integers $n$. Det... | 92 | 161 | 2 |
math | B2. Let $x$ be such a real number for which $\cos \left(60^{\circ}-x\right) \neq 0$ and $\sin \left(120^{\circ}-x\right) \neq 0$. Without using a pocket calculator, calculate the exact value of the expression: $\frac{\sqrt{3}+4 \sin x \cos x}{\cos \left(60^{\circ}-x\right) \cdot \sin \left(120^{\circ}-x\right)}$. | 4 | 122 | 1 |
math | 42nd Swedish 2002 Problem 4 For which integers n ≥8 is n 1/(n-7) an integer? | 8,9 | 30 | 3 |
math | 5. Use $0,1, \cdots, 9$ to form a ten-digit number without repeated digits that is divisible by 11. Find the largest and smallest numbers among them. | 9876524130, 1024375869 | 41 | 22 |
math | 2. The teacher wrote a positive number $x$ on the board and asked Kolya, Petya, and Vasya to raise this number to the 4th, 5th, and 20th power, respectively. It turned out that Kolya's number has at least 8 digits before the decimal point, and Petya's number has no more than 9 digits before the decimal point. How many ... | 36 | 105 | 2 |
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