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 | 2. Let $\mathrm{i}=\sqrt{-1}$ be the imaginary unit, then $\mathrm{i}+2 \mathrm{i}^{2}+3 \mathrm{i}^{3}+\cdots+2013 \mathrm{i}^{2013}=$ | 1006+1007\mathrm{i} | 59 | 13 |
math | [ Arithmetic. Mental calculation, etc.]
From the ten-digit number 2946835107, five digits were erased. What is the largest number that could result from this?
# | 98517 | 42 | 5 |
math | Exercise 12. In how many ways can the integers from 1 to 2021 be colored such that each integer is colored either blue, green, or red, and such that two consecutive integers are never the same color? | 3\times2^{2020} | 49 | 10 |
math | Example 1 If the equations of the two asymptotes of a hyperbola are $y= \pm \frac{2}{3} x$, and it passes through the point $M\left(\frac{9}{2},-1\right)$, try to find its equation. | \frac{x^{2}}{18}-\frac{y^{2}}{8}=1 | 60 | 21 |
math | Example 5 Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy for any real numbers $x$, $y$,
$$
f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x) \text {. }
$$
Find the analytical expression of the function $f(x)$. [2]
(56th IMO) | f(x)=xorf(x)=2-x | 81 | 9 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{2 x}-e^{x}}{\sin 2 x-\sin x}$ | 1 | 42 | 1 |
math | ## Task A-1.2.
Determine all pairs $(n, k)$ of natural numbers for which
$$
n^{2}+n=k^{2}+2 k-9
$$ | (n,k)=(9,9)(n,k)=(2,3) | 42 | 14 |
math | 9. If $(2 x-1)^{8}=a_{8} x^{8}+a_{7} x^{7}+\cdots+a_{1} x$ $+a_{0}$, then $a_{8}+a_{6}+a_{4}+a_{2}=$ $\qquad$ | 3280 | 71 | 4 |
math | 83. Given a parallelogram $A B C D$. Points $M$ and $N$ are the midpoints of sides $B C$ and $C D$ respectively. Find the ratio of the area of quadrilateral AM ND to the area of the parallelogram. | \frac{5}{8} | 59 | 7 |
math | 18. Let $f(x)$ be the polynomial $\left(x-a_{1}\right)\left(x-a_{2}\right)\left(x-a_{3}\right)\left(x-a_{4}\right)\left(x-a_{5}\right)$ where $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ are distinct integers. Given that $f(104)=2012$, evaluate $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}$. | 17 | 121 | 2 |
math | [ | [ Investigation of a quadratic trinomial ] |
| :---: | :---: |
For what values of the parameter $a$ are both roots of the equation $(1+a) x^{2}-3 a x+4 a=0$ greater than 1? | -\frac{16}{7}<-1 | 58 | 10 |
math | 6. Let set $S$ be an $m(m \geqslant 3)$-element subset of $\{0,1, \cdots, 98\}$, satisfying that for any $x, y \in S$, there exists $z \in S$ such that $x+y \equiv 2 z(\bmod 99)$. Find all possible values of $m$.
(Wang Xinmao provided the problem) | 3,9,11,33,99 | 96 | 12 |
math | 7.253. $\log _{a} \sqrt{4+x}+3 \log _{a^{2}}(4-x)-\log _{a^{4}}\left(16-x^{2}\right)^{2}=2$. For which values of $a$ does the equation have a solution? | 4-^{2},where\in(0;1)\cup(1;2\sqrt{2}) | 70 | 23 |
math | Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$ | 16 | 58 | 2 |
math | Problem 3. A row of 101 numbers is written (the numbers are not necessarily integers). The arithmetic mean of all the numbers without the first one is 2022, the arithmetic mean of all the numbers without the last one is 2023, and the arithmetic mean of the first and last numbers is 51. What is the sum of all the writte... | 202301 | 83 | 6 |
math | 3. In the Cartesian coordinate system $x O y$, it is known that there are three points $A(a, 1), B(2, b), C(3,4)$.
If the projections of $\overrightarrow{O A}$ and $\overrightarrow{O B}$ in the direction of $\overrightarrow{O C}$ are the same, then $3 a-4 b=$ $\qquad$ | 2 | 86 | 1 |
math | Example 2 If the quadratic function $f(x)=a x^{2}+b x+c$ has values whose absolute values do not exceed 1 on $[0,1]$, what is the maximum value of $|a|+$ $|b|+|c|$? | 17 | 59 | 2 |
math | Example 9 Let the sequence $\left\{x_{n}\right\}$ satisfy: $x_{1}=2, x_{2}=3$, and
$$
\left\{\begin{array}{l}
x_{2 m+1}=x_{2 m}+x_{2 m-1}, m \geqslant 1, \\
x_{2 m}=x_{2 m-1}+2 x_{2 m-2}, m \geqslant 2 .
\end{array}\right.
$$
Find the general term formula for the sequence $\left\{x_{n}\right\}$.
(1983 Australian Olymp... | \begin{pmatrix}当n=2-1,时,x_{n}=\frac{1}{4}(3-\sqrt{2})(2+\sqrt{2})^{}+\frac{1}{4}(3+\sqrt{2})(2-\sqrt{2})^{};\\当n=2,时,x_{n}=\ | 145 | 73 |
math | 3. Determine the largest natural number $n$ for which there exist distinct sets $S_{1}, S_{2}, \ldots, S_{n}$ such that:
$1^{\circ}\left|S_{i} \cup S_{j}\right| \leqslant 2004$ for any two integers $1 \leqslant i, j \leqslant n$, and
$2^{\circ} S_{i} \cup S_{j} \cup S_{k}=\{1,2, \ldots, 2008\}$ for any three integers... | 32 | 196 | 2 |
math | ## 16. How many of you were there, children?
If you had asked me such a question, I would have answered you only that my mother dreamed of having no fewer than 19 children, but she did not manage to fulfill her dream; however, I had three times as many sisters as cousins, and brothers - half as many as sisters. How ma... | 10 | 82 | 2 |
math | 3. The sum of the upper base radius $r$ and the lower base radius $R (R > r)$ of a frustum is 6 times the slant height $l$, and the areas of the upper base, lateral surface, and lower base form a geometric sequence. The height of this frustum is $20 - 3r^2$, then the maximum volume of the frustum is $\qquad$ | \frac{500}{9} \pi(2+\sqrt{3}) | 88 | 18 |
math | Task B-3.3. Determine the real numbers $a, b$ and $c$ for which the function
$$
f(x)=a x^{2}+b x-c \sin x \cos x
$$
is odd. | 0,\quadb\in\mathbb{R},\quad\in\mathbb{R} | 51 | 22 |
math | Example 5 For integer pairs $(a, b)(0<a<b<1000)$, a set $S \subseteq\{1,2, \cdots, 2003\}$ is called a "jump set" of the pair $(a, b)$: if for any element pair $\left(s_{1}, s_{2}\right)$, $s_{1}, s_{2} \in S,\left|s_{1}-s_{2}\right| \notin\{a, b\}$.
Let $f(a, b)$ be the maximum number of elements in a jump set of the... | 668 | 153 | 3 |
math | ## Task 2 - 010612
In the Soviet rocket tests conducted in October 1961, at a target distance of about $12500 \mathrm{~km}$, all hits were within a circle whose radius was less than $1 \mathrm{~km}$. How large would the radius of the hit circle be for a student who, with the same accuracy, throws a softball at a targe... | 2\mathrm{~} | 103 | 6 |
math | # Problem 5. (3 points)
Positive numbers $x, y$, and $z$ are such that $x+y+z=5$. What is the smallest value that the quantity $x^{2}+y^{2}+2 z^{2}-x^{2} y^{2} z$ can take? | -6 | 67 | 2 |
math | I2.4 Determine the unit digit, $\delta$, of $\left(\left(\gamma^{\gamma}\right)^{\gamma}\right)^{\gamma}$. | 7 | 34 | 1 |
math | Let's explain the following facts. The sequence of digits "1221" is first considered as a number in the base-13, then in the base-12, followed by the base-11, and finally in the base-10 number system. In each case, we convert our number to the number system with the base one less. In the first three cases, the result o... | 1221_{b}=1596_{b-1} | 127 | 16 |
math |
N7. Find all perfect squares $n$ such that if the positive integer $a \geqslant 15$ is some divisor of $n$ then $a+15$ is a prime power.
| 1,4,9,16,49,64,196 | 47 | 18 |
math | 3. Solve the system of equations $\left\{\begin{array}{l}x^{2} y-x y^{2}-5 x+5 y+3=0, \\ x^{3} y-x y^{3}-5 x^{2}+5 y^{2}+15=0 .\end{array}\right.$ | (4;1) | 72 | 5 |
math | 8. If for all $\theta \in \mathbf{R}$, the complex number
$$
z=(a+\cos \theta)+(2 a-\sin \theta) \mathrm{i}
$$
has a modulus not exceeding 2, then the range of the real number $a$ is
$\qquad$. | \left[-\frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5}\right] | 67 | 25 |
math |
4. The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:
for any two different digits from 1, 2, 3, 4, 5, 6, 7, 8 there exists a number in $X$ which contains both of them.
Determine the smallest possible value of $N$.
| 6 | 95 | 1 |
math | Example 9. Find $\int \frac{\sqrt[6]{5-3 \sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x$. | -\frac{6}{7}\sqrt[6]{(5-3\sqrt[3]{x})^{7}}+C | 37 | 27 |
math | Example. Find the sum of the series
$$
\sum_{n=1}^{\infty} \frac{\sin ^{n} x}{n}
$$
and specify the domain of convergence of the series to this sum. | S(x)=-\ln(1-\sinx),\quadx\neq\pi/2+2\pik | 50 | 27 |
math | 4. While working on fruit harvesting, class 5A collected 560 kg of apples, 5B - 595 kg, and 5V - 735 kg. All the collected apples were packed into boxes, with each box containing the same maximum possible number of kilograms. How many such boxes were required for each class? | 16,17,21 | 73 | 8 |
math | 8. Let the general term formula of the sequence $\left\{a_{n}\right\}$ be $a_{n}=n^{3}-n$ $\left(n \in \mathbf{Z}_{+}\right)$, and the terms in this sequence whose unit digit is 0, arranged in ascending order, form the sequence $\left\{b_{n}\right\}$. Then the remainder when $b_{2018}$ is divided by 7 is $\qquad$ | 4 | 104 | 1 |
math | Let $x_{k} \in[-2,2](k=1,2, \cdots, 2013)$,
and
$x_{1}+x_{2}+\cdots+x_{2013}=0$. Try to find
$$
M=x_{1}^{3}+x_{2}^{3}+\cdots+x_{2013}^{3}
$$
the maximum value.
(Liu Kangning) | 4026 | 99 | 4 |
math | 8. Problem: Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots.
$$
P(x)=a x^{4}-8 a x^{3}+b x^{2}-32 c x+16 c
$$ | (,24a,) | 65 | 6 |
math | 9. Given the sets $M=\{x, x y, \lg x y\}, N=\{0,|x|, y\}$, and $M=N$, then the value of $\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\right.$ $\left.\frac{1}{y^{3}}\right)+\cdots+\left(x^{2 n+1}+\frac{1}{y^{2 n+1}}\right)$ is $\qquad$. | -2 | 127 | 2 |
math | 1. Friends Vasya, Petya, and Kolya live in the same house. One day, Vasya and Petya went fishing at the lake on foot. Kolya stayed home, promising to meet his friends on a bicycle on their way back. Vasya was the first to head home, and at the same time, Kolya set out on a bicycle to meet him. Petya, moving at the same... | 3 | 225 | 1 |
math | 7. If 653 divides $\overline{a b 2347}$, then $a+b=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 11 | 52 | 2 |
math | find all functions from the reals to themselves. such that for every real $x,y$.
$$f(y-f(x))=f(x)-2x+f(f(y))$$ | f(x) = x | 37 | 6 |
math | $8 、$ In Class One of Grade Five, there are 40 students. In the math exam, the average score of the top 8 students is 3 points higher than the class average, and the average score of the other students is $\qquad$ points lower than the average score of the top 8 students. | 3.75 | 69 | 4 |
math | 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 | 57 | 2 |
math | 13.320. Impurities constitute $20 \%$ of the total volume of the solution. What is the smallest number of filters through which the solution must be passed so that the final impurity content does not exceed $0.01 \%$, if each filter absorbs $80 \%$ of the impurities? (It is known that $\lg 2 \approx 0.30$.) | 5 | 86 | 1 |
math | If $a, b, c, d$ are positive real numbers such that $\frac{5 a+b}{5 c+d}=\frac{6 a+b}{6 c+d}$ and $\frac{7 a+b}{7 c+d}=9$, calculate $\frac{9 a+b}{9 c+d}$. | 9 | 65 | 1 |
math | In the interior of the square $ABCD$, the point $P$ lies in such a way that $\angle DCP = \angle CAP=25^{\circ}$. Find all possible values of $\angle PBA$. | 25^\circ | 47 | 4 |
math | Solve the following system of equations:
$$
\begin{gathered}
\frac{10}{2 x+3 y-29}+\frac{9}{7 x-8 y+24}=8 \\
\frac{2 x+3 y-29}{2}=\frac{7 x-8 y}{3}+8
\end{gathered}
$$ | 5,\quad7 | 82 | 4 |
math | ## Task B-2.4.
Solve the equation in the set of integers
$$
x^{8}+y^{2016}=32 x^{4}-256
$$ | (2,0)(-2,0) | 43 | 10 |
math | 15. (15 points) Two shopping malls, A and B, are offering New Year promotions. Mall A stipulates: "For every 200 yuan spent, 101 yuan is deducted." Mall B stipulates: "For every 101 yuan spent, 50 yuan is deducted." Xiao Ming's father is interested in a pair of sports shoes priced at 699 yuan and a wool sweater priced ... | 801 | 131 | 3 |
math | Example 5 Express 2006 as the sum of 5 positive integers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, and let $S=$ $\sum_{1 \leqslant i<j \leq 5} x_{i} x_{j}$. Questions:
(1) For what values of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, does $S$ reach its maximum value;
(2) Furthermore, if for any $1 \leqslant i, j \leqslant 5$, $\l... | 402, 402, 402, 400, 400 | 209 | 23 |
math | Find the natural integers $n$ such that the fraction
$$
\frac{21 n+4}{14 n+1}
$$
is irreducible.
The second theorem is due to Carl Friedrich Gauss.
Theorem 2.2.18 (Gauss's Theorem).
Let $a, b$, and $c$ be three natural integers such that $a$ divides $b \times c$. If $a$ and $b$ are coprime, then $a$ divides $c$.
... | n\not\equiv1(\bmod5) | 232 | 11 |
math | (2) Find all prime pairs $(p, q)$ such that $p q \mid 5^{p}+5^{q}$. | (2,3),(3,2),(2,5),(5,2),(5,5),(5,313),(313,5) | 30 | 33 |
math | Example 7. Integrate the equation
$$
y^{\prime \prime}+4 y^{\prime}+4 y=2 \sin 2 x+3 \cos 2 x
$$ | (C_{1}+C_{2}x)e^{-2x}-\frac{1}{4}\cos2x+\frac{3}{8}\sin2x | 44 | 35 |
math | 10.5. In a horse race, three horses participated. A betting office accepts bets with the following odds: for the first horse to win 4:1 (i.e., if the first horse wins, the player gets back the amount they bet and four times more; otherwise, the player loses all the money they bet), for the second horse -3:1, and for th... | )11,13,26\\b) | 294 | 12 |
math | 4. Find all odd natural numbers $n$ for which
$$
n=2019 \underbrace{\varphi(\varphi(\ldots \varphi}_{10 \text { times }}(n) \ldots)) .
$$ | 2019 | 53 | 4 |
math | Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$. | 25 | 54 | 2 |
math | For what values of $a, b$ does the system of equations
$$
19 x^{2}+19 y^{2}+a x+b y+98=0 \quad \text { and } \quad 98 x^{2}+98 y^{2}+a x+b y+19=0
$$
have a common solution? | ^{2}+b^{2}\geq117^{2} | 81 | 16 |
math | 10.6. Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya, in turn, called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice and wrote down the obtained values on the board from left to right. It turne... | 50 | 108 | 2 |
math | Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$. | (0, 3, 1) | 42 | 10 |
math | Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$. | n = 4 | 48 | 5 |
math | Tokaeva I.
Let $F_{1}, F_{2}, F_{3}, \ldots$ be a sequence of convex quadrilaterals, where $F_{k+1}$ (for $k=1,2,3, \ldots$) is obtained by cutting $F_{k}$ along a diagonal, flipping one of the parts, and gluing it back along the cut line to the other part. What is the maximum number of different quadrilaterals that t... | 6 | 124 | 1 |
math | 4.20. Given the equation of a line lying in the XY plane, $4 x^{2}+p y^{2}=1$ and the coordinates of point $A(3 / 2,1,1)$. It is required: 1) to derive the equation of the surface formed by rotating this line around the $O X$ axis; 2) to select the value of the parameter $p$ so that point $A$ lies on the surface of rot... | 4x^{2}-4\cdot(y^{2}+z^{2})=1 | 102 | 19 |
math | Determine all positive integers $n$ for which the equation
$$
x^{n}+(2+x)^{n}+(2-x)^{n}=0
$$
has an integer as a solution.
Answer: $n=1$.
# | 1 | 53 | 1 |
math | How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.)
[i]Based on a proposal by Yannick Yao[/i] | 256 | 74 | 3 |
math | In triangle $A B C$, $A B=A C$ and $\angle B A C=20^{\circ}$. Let $E$ and $F$ be the points on sides $A B$ and $A C$ such that $\angle B C E=50^{\circ}$ and $\angle C B F=60^{\circ}$. What angle do the lines $B C$ and $E F$ form? (H) | 30 | 96 | 2 |
math | 7.1. How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference of two squares of integers | 250 | 30 | 3 |
math | Let $n$ be a positive integer not divisible by $3$. A triangular grid of length $n$ is obtained by dissecting a regular triangle with length $n$ into $n^2$ unit regular triangles. There is an orange at each vertex of the grid, which sums up to
\[\frac{(n+1)(n+2)}{2}\]
oranges. A triple of oranges $A,B,C$ is [i]good[/i... | \frac{(n+1)(n+2)}{2} - 3 | 156 | 17 |
math | ## 3. Benches
In the park, there is a group of students. If 7 students sit on each bench in the park, 8 students will be left without a seat. If the students were to sit 9 to a bench, four benches would remain empty. How many students are in the park?
Result: $\quad 162$ | 162 | 75 | 3 |
math | 4. Which is greater: $\sqrt{2016}+\sqrt{2015+\sqrt{2016}}$ or $\sqrt{2015}+\sqrt{2016+\sqrt{2015}}$? | \sqrt{2016}+\sqrt{2015+\sqrt{2016}}>\sqrt{2015}+\sqrt{2016+\sqrt{2015}} | 56 | 46 |
math | Example 4 Choose $k$ numbers from $1,2, \cdots, 2004$, such that among the chosen $k$ numbers, there are definitely three numbers that can form the side lengths of a triangle (the three numbers must be distinct). What is the minimum value of $k$ that satisfies this condition? | 17 | 70 | 2 |
math | 3.13. The volume of a convex polyhedron is $V$, the surface area is $S$; the length of the $i$-th edge is $l_{i}$, the dihedral angle at this edge is $\varphi_{i}$. Consider the body consisting of points that are at a distance not greater than $d$ from the polyhedron. Find the volume and surface area of this body. | V+S+\frac{1}{2}^{2}\sum_{i}(\pi-\varphi_{i})l_{i}+\frac{4}{3}\pi^{3} | 91 | 39 |
math | 16. In four-dimensional space, the distance between point $A\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ and point $B\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ is defined as
$$
A B=\sqrt{\sum_{i=1}^{4}\left(a_{i}-b_{i}\right)^{2}} .
$$
Consider the set of points
$I=\left\{P\left(c_{1}, c_{2}, c_{3}, c_{4}\right) \mid c_{i}=... | 9 | 275 | 1 |
math | 116. Find the minimum value of the fraction $\frac{x^{2}-3 x+3}{1-x}$, if $x<1$. | 3 | 32 | 1 |
math | Question 1: Let set $A=\{1,2, m\}$, where $m$ is a real number. Let $B=\left\{a^{2} \mid a \in A\right\}, C=A \cup B$. If the sum of all elements in $C$ is 6, then the product of all elements in $\mathrm{C}$ is $\qquad$ | -8 | 85 | 2 |
math | For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. ... | 40320 | 140 | 5 |
math | [ Combinations and Permutations ] [ Arithmetic progression $]
How many different ways can a natural number $n$ be decomposed into the sum of three natural addends? Two decompositions that differ only in the order of the addends are considered different.
# | \frac{1}{2}(n-1)(n-2) | 54 | 15 |
math | Problem 4. A rectangular piece of paper $20 \times 19$, divided into unit squares, is cut into several square pieces, the cuts being made along the sides of the unit squares. A square piece of this kind is called an odd square if the length of its side is an odd number.
a) What is the minimum possible number of odd sq... | 4 | 97 | 1 |
math | 11.4. 2011 Warehouses are connected by roads in such a way that from any warehouse you can drive to any other, possibly by driving along several roads. On the warehouses, there are $x_{1}, \ldots, x_{2011}$ kg of cement respectively. In one trip, you can transport an arbitrary amount of cement from any warehouse to ano... | 2010 | 209 | 4 |
math | 10. Let $f(n)$ be the integer closest to $\sqrt[4]{n}$. Then $\sum_{k=1}^{2018} \frac{1}{f(k)}=$ $\qquad$ | \frac{2823}{7} | 48 | 10 |
math | Four years ago, Daryl was three times as old as Joe was.
In five years, Daryl will be twice as old as Joe will be.
How old is Daryl now? | 31 | 38 | 2 |
math | 2.15. The dihedral angle at the base of a regular $n$-sided pyramid is $\alpha$. Find the dihedral angle between adjacent lateral faces. | \cos(\varphi/2)=\sin\alpha\sin(\pi/n) | 38 | 18 |
math | $4 \cdot 242$ Let $a, b$ be integers, what is the smallest positive integer value $c$ of $\Delta=\left|\begin{array}{ll}36 & b \\ 81 & a\end{array}\right|$? For the positive integer solutions $(a, b)$ that satisfy $\Delta=c$, to make $a+b$ the smallest, what are $a, b$ respectively? | =7,b=3,+b=10 | 92 | 10 |
math | Let $k$ be a positive integer. Each cell of a $3 k \times 3 k$ grid is colored either red or blue. A triple of cells $\left(C_{1}, C_{2}, C_{3}\right)$ is called a good cell group if $C_{1}$ and $C_{2}$ are in the same row, $C_{2}$ and $C_{3}$ are in the same column, and $C_{1}, C_{3}$ are colored blue, while $C_{2}$ i... | 12k^{4} | 125 | 6 |
math | Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression \[(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n\] takes a minimal value. | n^2 | 95 | 4 |
math | A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider? ("Opposite" means "symmetric with respect to the center of the cube".)
Answer: no. | \frac{0<<(3-\sqrt{7})}{2} | 68 | 14 |
math | 2. Let real numbers $a, b$ satisfy:
$$
a^{3}-3 a b^{2}=39, b^{3}-3 a^{2} b=26 \text {. }
$$
Then the value of $a^{2}+b^{2}$ is $\qquad$ | 13 | 65 | 2 |
math | Solve the following system of equations:
$$
\begin{gathered}
x^{2}-y z=-23 \\
y^{2}-z x=-4 \\
z^{2}-x y=34
\end{gathered}
$$ | \5,\6,\8 | 52 | 6 |
math | ## Task 1 - 090811
Investigate whether there are polygons with any of the following properties:
a) The number of diagonals is three times the number of vertices.
b) The number of vertices is three times the number of diagonals. | 9 | 56 | 1 |
math | 18. (1) When $x_{1}=\frac{1}{3}, x_{n+1}=x_{n}^{2}+x_{n}$, in which interval of two consecutive integers does the expression $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2001}}+$ $\frac{1}{1+x_{2002}}$ lie? (2002 2003 Finnish Mathematical Olympiad Problem) | 2 < 3 - \frac{1}{x_{2003}} < 3 | 117 | 20 |
math | ## Task A-2.2.
Solve the equation $\sqrt{9-5 x}-\sqrt{3-x}=\frac{6}{\sqrt{3-x}}$ in the set of real numbers. | -3 | 45 | 2 |
math | $8 \cdot 58$ Find the positive integers $m$, $n$, $k$ greater than 1 that satisfy $1!+2!+3!+\cdots+m!=n^{k}$.
(Canadian Mathematical Olympiad Training Problem, 1988) | =3,n=3,k=2 | 61 | 8 |
math | A square piece of paper was punctured at 1965 points. From the puncture points and the vertices of the square, no three lie on the same straight line. Then, several straight, non-intersecting cuts were made, each starting and ending only at the punctured points or the vertices of the square. It turned out that the squa... | 5896 | 99 | 4 |
math | (11 Let $\sec x+\tan x=\frac{22}{7}, \csc x+\cot x=\frac{m}{n}(m, n$ have no common divisor greater than 1), then $m+n=$ $\qquad$ | 44 | 54 | 2 |
math | A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated directi... | P(x,0) | 132 | 6 |
math | 17th Mexico 2003 Problem B1 The quadrilateral ABCD has AB parallel to CD. P is on the side AB and Q on the side CD such that AP/PB = DQ/CQ. M is the intersection of AQ and DP, and N is the intersection of PC and QB. Find MN in terms of AB and CD. | MN=\frac{AB\cdotCD}{AB+CD} | 74 | 13 |
math | Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$. Find the minimum area of the part bounded by the curve $ C$ and two lines $... | \sqrt{\frac{2}{3}} | 122 | 9 |
math | ## Problem Statement
Find the $n$-th order derivative.
$$
y=\frac{11+12 x}{6 x+5}
$$ | y^{(n)}=(-1)^{n}\cdotn!\cdot6^{n}\cdot(6x+5)^{-n-1} | 33 | 31 |
math | Example 3 Let $g(n)$ denote the greatest odd divisor of the natural number $n$, for example, $g(3)=3$, $g(14)=7$, etc. Find the value of $g(1)+g(2)+g(3)+\cdots+g\left(2^{n}\right)$. | \frac{1}{3}(4^{n}+2) | 72 | 14 |
math | One, (40 points) Find the smallest positive integer $m$ such that the equation
$$
2^{x}+3^{y}-5^{z}=2 m
$$
has no positive integer solutions for $x, y, z$. | 11 | 53 | 2 |
math | ## Task A-3.2.
A quadruple of natural numbers $(a, b, c, d)$ is called green if
$$
b=a^{2}+1, \quad c=b^{2}+1, \quad d=c^{2}+1
$$
and $D(a)+D(b)+D(c)+D(d)$ is odd, where $D(k)$ is the number of positive divisors of the natural number $k$.
How many green quadruples are there whose all members are less than 1000000? | 2 | 119 | 1 |
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