task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | 11 Find the largest real number $k$ such that for any positive real numbers $a, b, c, d$, the following inequality always holds: $(a+b+c)\left[3^{4}(a+b+c+d)^{5}+2^{4}(a+b+c+2 d)^{5}\right] \geqslant$ $k$ abcd ${ }^{3}$. | 174960 | 85 | 6 |
math | 113 Given the function $f(x)=\sqrt{x+2}+k$, and there exist $a, b(a<b)$ such that the range of $f(x)$ on $[a, b]$ is $[a, b]$, find the range of the real number $k$. | k\in(-\frac{9}{4},-2] | 63 | 14 |
math | There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal... | 24 | 87 | 2 |
math | 2. Several numbers form an arithmetic progression, where their sum is 63, and the first term is one and a half times the common difference of the progression. If all terms of the progression are decreased by the same amount so that the first term of the progression is equal to the common difference of the progression, ... | 21/8or2 | 97 | 6 |
math | ## Task 1 - 130811
Determine all possibilities to specify a four-digit odd (natural) number $z$ such that it has the following properties:
(1) The number $z$ has four different digits.
(2) The product of the second and the third digit of $z$ is 21 times as large as the product of the first and the fourth digit.
(3)... | 1973 | 112 | 4 |
math | 10. Given that $f(x)$ is a function defined on $\mathbf{R}$. $f(1)=1$ and for any $x \in \mathbf{R}$, we have $f(x+5) \geqslant f(x)+5, \quad f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$ $\qquad$ | 1 | 102 | 1 |
math | $ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule:
1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students;
2) Each student received the maximum possible points in each problem or got $... | 60 | 137 | 2 |
math | 1. 205 Starting from any three-digit number $n$, we get a new number $f(n)$, which is the sum of the three digits of $n$, the pairwise products of the three digits, and the product of the three digits.
(1) When $n=625$, find $\frac{n}{f(n)}$ (take the integer value).
(2) Find all three-digit numbers $n$ such that $\fra... | 199,299,399,499,599,699,799,899,999 | 102 | 35 |
math | 4. In an arithmetic sequence with real number terms, the common difference is 4, and the square of the first term plus the sum of the remaining terms does not exceed 100. Such a sequence can have at most terms. | 8 | 50 | 1 |
math | 4. A test paper has 4 multiple-choice questions, each with three options (A), (B), (C). Several students take the exam, and after grading, it is found that: any 3 students have 1 question where their answers are all different. How many students can take the exam at most? | 9 | 65 | 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 | ## 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 | 3. (15 points) The vertices of the broken line $A B C D E F G$ have coordinates $A(-1 ; -7), B(2 ; 5), C(3 ; -8), D(-3 ; 4), E(5 ; -1), F(-4 ; -2), G(6 ; 4)$.
Find the sum of the angles with vertices at points $B, E, C, F, D$. | 135 | 96 | 3 |
math | 2.4. (SRP, 81). Solve the equation
$$
x^{6}+3 x^{3}+1=y^{4}
$$
in integers. | 0,\1 | 39 | 3 |
math | Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$. | \sqrt{5} | 48 | 5 |
math | 4. The numbers $x$ and $y$ are such that the equalities $\cos y + \cos x = \sin 3x$ and $\sin 2y - \sin 2x = \cos 4x - \cos 2x$ hold. What is the smallest value that the sum $\sin y + \sin x$ can take? | -1-\frac{\sqrt{2+\sqrt{2}}}{2} | 77 | 16 |
math | 714. Find the limits:
1) $\lim _{\substack{x \rightarrow 3 \\ y \rightarrow 0}} \frac{\tan(x y)}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x}{x+y}$. | 3 | 67 | 1 |
math | ## Task 3/70
It is $x^{0}=1$ and $0^{x}=0$ for $x \neq 0$. What is the value of $\lim _{x \rightarrow 0} x^{x}$ ? | 1 | 54 | 1 |
math | ## Exercise 11.
1) Alice wants to color the integers between 2 and 8 (inclusive) using $k$ colors. She wishes that if $m$ and $n$ are integers between 2 and 8 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ are of different colors. Determine the smallest integer $k$ for which Alice can color the in... | 4 | 223 | 1 |
math | 1. [2] Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. How many (potentially empty) subsets $T$ of $S$ are there such that, for all $x$, if $x$ is in $T$ and $2 x$ is in $S$ then $2 x$ is also in $T$ ? | 180 | 89 | 3 |
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 | Problem 9.2. During the first half of the year, lazy Pasha forced himself to solve math problems. Every day he solved no more than 10 problems, and if on any day he solved more than 7 problems, then for the next two days he solved no more than 5 problems per day. What is the maximum number of problems Pasha could solve... | 52 | 83 | 2 |
math | Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point?
(Alexey Panasenko) | 2.4 \, \text{cm} | 88 | 10 |
math | 119. What is the shortest distance on the surface of the given right parallelepiped (edges: $a>b>c$) between one of the vertices and the opposite vertex (the opposite vertex to $M$ is considered to be the vertex $N$, where the three faces parallel to the faces meeting at vertex $M$ meet)? | \sqrt{^{2}+(b+)^{2}} | 70 | 13 |
math | 1. Given $f(x) \in\left[\frac{3}{8}, \frac{4}{9}\right]$, then the range of $y=f(x)+\sqrt{1-2 f(x)}$ is $\qquad$ | [\frac{7}{9},\frac{7}{8}] | 52 | 14 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(n+1)^{3}+(n-1)^{3}}{n^{3}-3 n}
$$ | 2 | 52 | 1 |
math | Let the medians of the triangle $ABC$ meet at $G$. Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$, and let $P$ and $Q$ be points on the segments $BD$ and $BE$, respectively, such that $2BP=PD$ and $2BQ=QE$. Determine $\angle PGQ$. | 90^\circ | 86 | 4 |
math | 3. Find the smallest positive integer $n$, such that there exist rational coefficient polynomials $f_{1}, f_{2}, \cdots, f_{n}$, satisfying
$$
x^{2}+7=f_{1}^{2}(x)+f_{2}^{2}(x)+\cdots+f_{n}^{2}(x) .
$$ | 5 | 78 | 1 |
math | 8. Let $p_{k}=1+\frac{1}{k}-\frac{1}{k^{2}}-\frac{1}{k^{3}}$, where $k$ is a positive integer. Find the least positive integer $n$ such that the product $p_{2} p_{3} \ldots p_{n}$ exceeds 2010 . | 8038 | 79 | 4 |
math | Let $a$ and $b$ be integer solutions to $17a+6b=13$. What is the smallest possible positive value for $a-b$? | 17 | 36 | 2 |
math | 2. Let $n$ be a fixed integer, $n \geqslant 2$.
a) Determine the smallest constant $c$ such that the inequality $\sum_{1 \leqslant i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{1 \leqslant i \leqslant n} x_{i}\right)^{4}$ holds for all non-negative real numbers $x_{1}, \cdots, x_{n} \geqslant ... | \frac{1}{8} | 155 | 7 |
math | In how many ways, can we draw $n-3$ diagonals of a $n$-gon with equal sides and equal angles such that:
$i)$ none of them intersect each other in the polygonal.
$ii)$ each of the produced triangles has at least one common side with the polygonal. | \frac{1}{n-1} \binom{2n-4}{n-2} | 63 | 22 |
math | As $96 \div 8=12$, we have $8 \times 12=96$.
Notice that the solution is equivalent to solving the equation $8 x=96$, whose root is $x=\frac{96}{8}=12$. | 12 | 59 | 2 |
math | 1. The number n is the product of three (not necessarily distinct) prime numbers. If we increase each of them by 1, the product of the increased numbers will be 963 more than the original product. Determine the original number $n$. | 2013 | 53 | 4 |
math | Svatoovsky M.
On a plane, there is a grasshopper Kolya and 2020 of his friends. Kolya is going to jump over each of the other grasshoppers (in any order) such that the starting and ending points of each jump are symmetric relative to the grasshopper being jumped over. We will call a point a finishing point if Kolya ca... | C_{2020}^{1010} | 130 | 13 |
math | 3. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2$, $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$, and $a_{n+1} a_{n+2} \neq$ 1. Then $a_{1}+a_{2}+\cdots+a_{2004}=$ $\qquad$ | 4008 | 114 | 4 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{-9 ; 5 ; 5\}$
$p=\{4 ; 1 ; 1\}$
$q=\{2 ; 0 ;-3\}$
$r=\{-1 ; 2 ; 1\}$ | -p-q+3r | 74 | 5 |
math | 10. For any positive integer $n$, let $N_{n}$ be the set of integers from 1 to $n$, i.e., $N_{n}=$ $\{1,2,3, \cdots, n\}$. Now assume that $n \geq 10$. Determine the maximum value of $n$ such that the following inequality
$$
\mathrm{i}_{\substack{a, b \in A \\ a \neq b}}|a-b| \leq 10
$$
holds for each $A \subseteq N_{n... | 99 | 138 | 2 |
math | Given a triangle with side lengths $a,\ b,\ c$. Let $a,\ b,\ c$ vary, find the range of $\frac{a^2+b^2+c^2}{ab+bc+ca}$.
[i]2010 Tokyo Institute of Technology Admission Office entrance exam, Problem I-2/Science[/i] | 1 \leq \frac{a^2 + b^2 + c^2}{ab + bc + ca} < 2 | 71 | 28 |
math | 6. For what value of $x$ does the expression $x^{2}-600 x+369$ take its minimum value? | 300 | 31 | 3 |
math | # 2. CONDITION
Find all pairs $(a ; b)$ of real numbers $a$ and $b$ such that the equations $x^{2}+a x+b^{2}=0$ and $x^{2}+b x+a^{2}=0$ have at least one common root. | (0;0) | 64 | 5 |
math | Determine all pairs of real numbers $(x, y)$ for which $\left(4 x^{2}-y^{2}\right)^{2}+(7 x+3 y-39)^{2}=0$. | (x,y)=(3,6),(39,-78) | 46 | 13 |
math | Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i] | p(x) = \lambda x^2 + \mu x | 113 | 14 |
math | 13. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=\frac{1}{4}$, and $a_{n+1}=\frac{(n-1) a_{n}}{n-a_{n}}(n=2,3,4, \cdots)$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Prove that for all $n \in \mathbf{N}_{+}$,
$\sum_{k=1}^{n} a_{k}^{2}<\frac{7}{6}$. | \sum_{k=1}^{n}a_{k}^{2}<\frac{7}{6} | 146 | 24 |
math | Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true:
\begin{align*}
f(a)&=f(b), <br /> \\
f(a+b)&=\min\{f(a),f(b)\}.
\end{align*}
[i]Remarks:[/i] $\mathbb{N}$ denotes the set of all positive integers. A function $f:X\... | f(n) = \nu_2(n) + 1 | 141 | 14 |
math | Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\equiv 1\pmod n$. Find the maximum possible value of $n$. | 504 | 47 | 3 |
math | 1. Find the smallest natural number that is greater than the sum of its digits by 1755 (the year of the founding of Moscow University). | 1770 | 32 | 4 |
math | 74. Baxter's Dog. Here is an interesting puzzle, complementing the previous one. Anderson left the hotel in San Remo at 9 o'clock and was on the road for a whole hour when Baxter set out after him along the same route. Baxter's dog ran out at the same time as its owner and kept running back and forth between him and An... | 10 | 183 | 2 |
math | 3. Given the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ with its right focus at $F$, the upper vertex at $A$, and $P$ as a moving point on the ellipse in the first quadrant. Then the maximum value of the area of $\triangle A P F$ is $\qquad$ . | \sqrt{6}-1 | 77 | 6 |
math | 7.1. Two wheels of radii $r_{1}$ and $r_{2}$ roll along a straight line $l$. Find the set of points of intersection $M$ of their common internal tangents. | \frac{2r_{1}r_{2}}{r_{1}+r_{2}} | 45 | 22 |
math | 1. Find the smallest four-digit number $\overline{a b c d}$ such that the difference $(\overline{a b})^{2}-(\overline{c d})^{2}$ is a three-digit number written with three identical digits. | 2017 | 54 | 4 |
math | A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws. | \frac{55}{64} | 28 | 9 |
math | One, (15 points) Find the maximum value of the function
$$
y=\left[\sin \left(\frac{\pi}{4}+x\right)-\sin \left(\frac{\pi}{4}-x\right)\right] \sin \left(\frac{\pi}{3}+x\right)
$$
and the set of $x$ values at which the maximum value is attained. | \frac{3 \sqrt{2}}{4} | 87 | 12 |
math | 34. Given the equation $x^{3}-a x^{2}+b x-c=0$ has three positive roots (which can be equal), find the minimum value of $\frac{1+a+b+c}{3+2 a+b}-\frac{c}{b}$. (2008 Turkey Training Team Problem) | \frac{1}{3} | 70 | 7 |
math | 1. Due to the price reduction of $20 \%$, with 2400 denars, one can buy one more book than with 2700 denars before the price reduction. What was the price of one book before the price reduction? | 300 | 54 | 3 |
math | 11. Let $a, b$ be real numbers. Then the minimum value of $a^{2}+a b+b^{2}-$ $a-2 b$ is $\qquad$. | -1 | 42 | 2 |
math | ## Zadatak A-3.4. (10 bodova)
Izračunaj sumu
$$
\sum_{n=1}^{2012} \operatorname{tg} n \operatorname{tg}(n+1)
$$
| \frac{\operatorname{tg}2013}{\operatorname{tg}1}-2013 | 59 | 25 |
math | Exercise 7. Let $p \geqslant 3$ be a prime number. For $k \in \mathbb{N}$ satisfying $1 \leqslant k \leqslant p-1$, the number of divisors of $\mathrm{kp}+1$ that are strictly between $k$ and $p$ is denoted by $a_{k}$.
What is the value of $a_{1}+a_{2}+\ldots+a_{p-1}$? | p-2 | 109 | 3 |
math | 1. Does the number 1... (1000 ones) have a ten-digit divisor, all digits of which are different? | No | 28 | 1 |
math | On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square? | 20 | 35 | 2 |
math | 3. The number of zeros of the function $f(x)=x^{2} \ln x+x^{2}-2$ is . $\qquad$ | 1 | 32 | 1 |
math | Example 6 Solve the equation
$$
\sqrt{2-x}+\sqrt{x+3 y-5}+\sqrt{y+2}=\sqrt{12 y-3} .
$$ | x=-7, y=7 | 42 | 7 |
math | 5. Find all values of the parameter $a$ for which the equation $\left(2 \sin x+a^{2}+a\right)^{3}-(\cos 2 x+3 a \sin x+11)^{3}=12-2 \sin ^{2} x+(3 a-2) \sin x-a^{2}-a$ has two distinct solutions on the interval $\left[-\frac{\pi}{6} ; \frac{3 \pi}{2}\right]$. Specify these solutions for each found $a$.
Solution: $\lef... | \in[2.5,4),x_{1}=\arcsin(-3),x_{2}=\pi-\arcsin(-3);\in[-5,-2),x_{1}=\arcsin(0.5+2),x_{2}=\pi-\arcsin(0.5+2) | 918 | 71 |
math | A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive. | 1330 | 76 | 4 |
math | ## Problem 3.
Let the sequence of real numbers $\left(x_{n}\right)_{n \geq 0}$ be defined by
$$
\begin{aligned}
& x_{0}=0 \\
& x_{n}=1+\sin \left(x_{n-1}-1\right),(\forall) n \geq 1
\end{aligned}
$$
Study the convergence of the sequence and then determine its limit. | \lim_{narrow\infty}x_{n}=1 | 94 | 14 |
math | ## Task 2 - 150712
Two vessels, called $A$ and $B$, together have a capacity of exactly 8 liters. A certain amount of water $W$ is distributed between the two vessels such that $A$ is half full and $B$ is completely full. If water is poured from $B$ into $A$ until $A$ is completely full, then $B$ is still one-sixth fu... | )A=5 | 142 | 4 |
math | 7. The value of $\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}$ is | \frac{1}{128} | 55 | 9 |
math | 2. (10 points) Cars A and B start from points $A$ and $B$ respectively at the same time, heading towards each other. At the start, the speed ratio of cars A and B is 5:4. Shortly after departure, car A gets a flat tire, stops to change the tire, and then continues, increasing its speed by $20 \%$. As a result, 3 hours ... | 52 | 150 | 2 |
math | [Pythagorean Theorem (direct and inverse).]
One of the legs of a right triangle is 10 more than the other and 10 less than the hypotenuse. Find the hypotenuse of this triangle. | 50 | 48 | 2 |
math | A quantity of grey paint has a mass of $12 \mathrm{~kg}$. The grey paint is a mixture of black paint and white paint, of which $80 \%$ by mass is white paint. More white paint is added until the mixture is $90 \%$ white paint by mass. What is the resulting total mass of the paint, in $\mathrm{kg}$ ? | 24 | 81 | 2 |
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 | In a group of $n$ people, there are $k$ individuals who each have exactly two acquaintances among the present. Among the remaining $(n-k)$ members of the group, no two know each other.
What is the maximum number of handshakes that can occur if the strangers introduce themselves to each other? (We consider the acquaint... | S_{\max}=\frac{1}{2}n(n-1)-k | 87 | 18 |
math | 11.8. Find the real values of $u$ and $v$ that satisfy the equation
$$
\left(u^{2020}-u^{2019}\right)+\left(v^{2020}-v^{2019}\right)=u \ln u+v \ln v .
$$ | u=v=1 | 70 | 4 |
math | Example 5 A scientist stored the design blueprint of his time machine in a computer, setting the password to open the file as a permutation of $\{1,2, \cdots, 64\}$. He also designed a program that, when eight positive integers between $1 \sim 64$ are input each time, the computer will indicate the order (from left to ... | 45 | 112 | 2 |
math | 4.
$$
\frac{S_{M N C}}{S_{A B C}}=\frac{(p-c)^{2}}{a b}=\frac{(a+b-c)^{2}}{4 a b}=\frac{4(a+b)^{2}}{81 a b}
$$
Since $a+b \geqslant 2 \sqrt{a b}$ (equality is achieved only when $a=b$), then $\frac{16(a+b)^{2}}{81 a b} \geqslant \frac{4 \cdot 4 a b}{81 a b}=\frac{16}{81}$.
Rewrite the ratio of the areas in the foll... | [16/81;2/7) | 453 | 11 |
math | 6.003. $\frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$.
6.003. $\frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$. | x_{1}=-5,x_{2}=1,x_{3,4}=-1\\sqrt{6} | 79 | 24 |
math | 2. Determine all triples $(p, q, r)$ of prime numbers for which
$$
(p+1)(q+2)(r+3)=4 p q r
$$
(Jaromír Šimša) | (2,3,5),(5,3,3),(7,5,2) | 50 | 19 |
math | Example 4 (2003 China National Training Team) In $\triangle ABC$, $AC > AB$, $P$ is the intersection of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle A$. Draw $PX \perp AB$, intersecting the extension of $AB$ at point $X$, and $PY \perp AC$ intersecting $AC$ at point $Y$, $Z$ is the inte... | 1 | 116 | 1 |
math | 10. (12 points) 11 gardeners go to plant trees, 2 of them go to plant trees on Street $A$, and the remaining 9 go to plant trees on Street $B$. In the evening, after work, they return to their dormitory.
Gardener A says: "Although we are fewer, we took the same amount of time as you."
Gardener B says: "Although we ... | 44 | 167 | 2 |
math | ## Task B-3.1.
Let $x$ and $y$ be real numbers such that
$$
\sin x - \sin y = \sqrt{\frac{1008}{1009}}, \quad \cos x - \cos y = 1
$$
What is the value of $\cos (x-y)$? | \frac{1}{2018} | 74 | 10 |
math | 12. (10 points) $1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}+7^{3}+8^{3}+9^{3}=$ | 2025 | 54 | 4 |
math | 1. Given non-zero vectors $\boldsymbol{a}, \boldsymbol{b}$ with an angle of $120^{\circ}$ between them. If vector $a-b$ is perpendicular to $a+2 b$, then $\left|\frac{2 a-b}{2 a+b}\right|=$ $\qquad$ | \frac{\sqrt{10+\sqrt{33}}}{3} | 69 | 16 |
math | 8. (6 points) There is a three-digit number, the hundreds digit is the smallest prime number, the tens digit is the first digit after the decimal point in the result of the expression $(0.3+\pi \times 13)$, and the units digit is the units digit of the smallest three-digit number that can be divided by 17. What is this... | 212 | 111 | 3 |
math | Example 4 Let $X_{n}=\{1,2, \cdots, n\}$, for any subset $A$ of $X_{n}$, denote $t(A)$ as the smallest element in $A$. Find $t=$ $\sum_{A \subseteq X_{n}} t(A)$. | 2^{n+1}-2-n | 67 | 8 |
math | Touching Circles
Inside a right-angled triangle, two circles of the same radius are placed, each touching one of the legs, the hypotenuse, and the other circle. Find the radii of these circles if the legs of the triangle are equal to $a$ and $b$.
# | \frac{\sqrt{^{2}+b^{2}}}{(+b)(+b+\sqrt{^{2}+b^{2}})} | 63 | 31 |
math | Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now? | 45 | 59 | 2 |
math | \section*{Problem \(2-341042=340943\)}
On the side \(AB\) of the square \(ABCD\), a point \(X \neq A\) is chosen. The square is then divided into four regions by the segments \(AC\) and \(X D\).
Determine all possibilities for choosing \(X\) such that there exist natural numbers \(p, q\), and \(r\) for which the area... | |AX|=\frac{1}{r}\cdot|AB| | 118 | 14 |
math | 20. Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$ be positive integers such that
$$
a_{1}^{2}+\left(2 a_{2}\right)^{2}+\left(3 a_{3}\right)^{2}+\left(4 a_{4}\right)^{2}+\left(5 a_{5}\right)^{2}+\left(6 a_{6}\right)^{2}+\left(7 a_{7}\right)^{2}+\left(8 a_{8}\right)^{2}=204 \text {. }
$$
Find the value of... | 8 | 197 | 1 |
math | 5. (15 points) The density of a body $\rho$ is defined as the ratio of the body's mass $m$ to its volume $V$. A unit of mass used in jewelry is the carat (1 carat equals 0.2 grams). A unit of length used in many countries is the inch (1 inch equals 2.54 centimeters). It is known that the density of diamond is $\rho=3.5... | 287 | 119 | 3 |
math | In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$. | \sqrt{3} | 58 | 5 |
math | 1. Let $f(x)=\frac{x}{\sqrt{1+x^{2}}}, f_{1}(x)=f(x), f_{2}(x)=f\left(f_{1}(x)\right), \cdots, f_{n}(x)=f\left(f_{n-1}(x)\right)$, then $f_{n}(x)=$ | f_{n}(x)=\frac{x}{\sqrt{1+nx^{2}}} | 79 | 19 |
math | 10. If a convex $n$-sided polygon has exactly 4 obtuse interior angles, then the maximum number of sides $n$ of this polygon is $\qquad$ . | 7 | 40 | 1 |
math | ## Problem 1
Determine the smallest sum of the digits of the number $3 n^{2}+n+1$, where $n \in \mathbb{N}, n \geq 2$. | 3 | 45 | 1 |
math | ## Task 2 - 310732
A person answers the question about their birthday:
"In the year 1989, I was $a$ years old. I was born on the $t$-th day of the $m$-th month of the year $(1900+j)$. The numbers $a, j, m, t$ are natural numbers; for them, $a \cdot j \cdot m \cdot t=105792.$"
Determine whether the numbers $a, j, m, ... | 57,i=32,=2,=29 | 138 | 13 |
math | 6. (8 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=75 \\
y^{2}+y z+z^{2}=64 \\
z^{2}+x z+x^{2}=139
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 80 | 102 | 2 |
math | 3. How many of the integers from $2^{10}$ to $2^{18}$ inclusive are divisible by $2^{9}$ ? | 511 | 31 | 3 |
math | 15. Let $x>1, y>1, S=\min \left\{\log _{x} 2, \log _{2} y\right.$ , $\left.\log _{y}\left(8 x^{2}\right)\right\}$. Then the maximum value of $S$ is $\qquad$ . | 2 | 74 | 1 |
math | If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest
possible value of $p$. | 53 | 31 | 2 |
math | Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the ... | 486 | 142 | 3 |
math | 8. If a four-digit number $n$ contains at most two different digits among its four digits, then $n$ is called a "simple four-digit number" (such as 5555 and 3313). Then, the number of simple four-digit numbers is | 576 | 59 | 3 |
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