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 | ## Task B-2.2.
A natural number is said to be a palindrome if it reads the same from left to right and from right to left in decimal notation. We can order palindromes by size. Determine the 2010th palindrome in this order. | 1011101 | 58 | 7 |
math |
4. If we divide the number 13 by the three numbers 5,7 , and 9 , then these divisions leave remainders: when dividing by 5 the remainder is 3 , when dividing by 7 the remainder is 6 , and when dividing by 9 the remainder is 4 . If we add these remainders, we obtain $3+6+4=13$, the original number.
(a) Let $n$ be a po... | 395 | 196 | 3 |
math | 1. Solve the inequality $\left(x^{2}-x+1\right)^{16 x^{3}-6 x} \leq\left(x^{2}-x+1\right)^{13 x^{2}+x^{3}}$. | x\in(-\infty;-\frac{1}{3}]\cup{0}\cup[1;\frac{6}{5}] | 55 | 30 |
math | In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$. | 84 | 60 | 2 |
math | [ Average values $\quad]$ [ Area of a circle, sector, and segment ]
At a familiar factory, metal disks with a diameter of 1 m are cut out. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius... | 4 | 116 | 1 |
math | 1. Let $a=\operatorname{tg} x, b=\operatorname{tg} \frac{y}{2}$. Then $a+b=\frac{4}{\sqrt{3}}$ and $\frac{1}{a}+\frac{1}{b}=\frac{4}{\sqrt{3}}$. From the second equation, it follows that $a b=1$. By Vieta's theorem, $a$ and $b$ satisfy the equation $t^{2}-\frac{4}{\sqrt{3}} t+1=0$, whose roots are $\sqrt{3}$ and $\frac... | \frac{\pi}{3}+\pin,\frac{\pi}{3}+2\pikor\frac{\pi}{6}+\pin,\frac{2\pi}{3}+2\pik | 262 | 45 |
math | ## Task 22/75
How many different triangles are there where the measure of the perimeter is 50 and the measures of the sides are natural numbers? | 52 | 35 | 2 |
math | Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q... | 71 | 140 | 2 |
math | Exercise 3. Find all real numbers $x, y$ and $z$ such that
$$
\left\{\begin{array}{r}
x^{2}-4 y+7=0 \\
y^{2}-6 z+14=0 \\
z^{2}-2 x-7=0
\end{array}\right.
$$ | (1,2,3) | 74 | 7 |
math | Find all natural numbers $n> 1$ for which the following applies:
The sum of the number $n$ and its second largest divisor is $2013$.
(R. Henner, Vienna) | n = 1342 | 44 | 8 |
math | Given points $A, B, C$, and $D$ such that segments $A C$ and $B D$ intersect at point $E$. Segment $A E$ is 1 cm shorter than segment $A B$, $A E = D C$, $A D = B E$,
$\angle A D C = \angle D E C$. Find the length of $E C$. | 1 | 82 | 1 |
math | Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 4)[/i] | f(x) = x | 93 | 6 |
math | 14. (15 points) Given positive real numbers $x, y$ satisfy
$$
\begin{array}{l}
a=x+y, \\
b=\sqrt{x^{2}+7 x y+y^{2}} .
\end{array}
$$
(1) When $y=1$, find the range of $\frac{b}{a}$;
(2) If $c^{2}=k x y$, and for any positive numbers $x, y$, the segments with lengths $a, b, c$ can always form a triangle, find the range ... | (1,25) | 124 | 6 |
math | 10. (20 points) Let $A$ and $B$ be two points on the hyperbola $x^{2}-\frac{y^{2}}{2}=1$.
$O$ is the origin, and it satisfies $\overrightarrow{O A} \cdot \overrightarrow{O B}=0, \overrightarrow{O P}=\alpha \overrightarrow{O A}+(1-\alpha) \overrightarrow{O B}$.
(1) When $\overrightarrow{O P} \cdot \overrightarrow{A B}=... | 2 \sqrt{2} | 149 | 6 |
math | There are positive integers $x$ and $y$ that satisfy the system of equations
\begin{align*} \log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) &= 60\\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*}
Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $... | 880 | 146 | 3 |
math | Example 5 Given $a>0, b>0, a+2b=1$, find the minimum value of $t=\frac{1}{a}+\frac{1}{b}$. | 3+2\sqrt{2} | 42 | 8 |
math | $100 \mathrm{~kg} 1.5 \mathrm{~mm}$ diameter copper wire is insulated with $0.5 \mathrm{~mm}$ thick rubber. How many $\mathrm{kg}$ of rubber is needed, if a $1 \mathrm{~m}$ long cord, whose cross-section is a square with a side length of $3 \mathrm{~mm}$, weighs 10 g when made from the same rubber. The specific weight ... | 22.07\, | 108 | 7 |
math | ## Task B-3.5.
Christmas ornaments are packed in two types of boxes, red and green. In red boxes, the ornaments are packed in five rows with four ornaments each, and in green boxes, in three rows with six ornaments each. In how many different ways can we choose the number of red and green boxes to pack 2018 Christmas ... | 12 | 93 | 2 |
math | Example 11 (16th "Hope Cup" Senior High School Competition Problem) In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{n}-a_{n-1}=a_{n+1}+3 a_{n} a_{n+1}$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Given that $y=f(x)$ is an even function, and for any $x$ there is $f(1+x)=f(1-x)$, w... | {x\lvert\,-1+2k<x<-\frac{1}{2}+2k.,k\in{Z}}\text{or}{x\lvert\,\frac{1}{2}+2k<x<1+2k.,k\in{Z}} | 190 | 62 |
math | 4. In $\triangle A B C$, $\sin A: \sin B: \sin C=2: 3: 4$, then $\angle A B C=$ $\qquad$ (the result should be expressed using inverse trigonometric function values). | \arccos\frac{11}{16} | 54 | 13 |
math | 25th Swedish 1985 Problem 2 Find the smallest positive integer n such that if the first digit is moved to become the last digit, then the new number is 7n/2. | 153846 | 43 | 6 |
math | 6. Given that $a$ is a constant, $x \in \mathbf{R}, f(x)=\frac{f(x-a)-1}{f(x-a)+1},$ the period of $f(x)$ is $\qquad$ . | 4a | 53 | 2 |
math | 8. For any subset $S \subseteq\{1,2, \ldots, 15\}$, a number $n$ is called an "anchor" for $S$ if $n$ and $n+|S|$ are both members of $S$, where $|S|$ denotes the number of members of $S$. Find the average number of anchors over all possible subsets $S \subseteq\{1,2, \ldots, 15\}$. | \frac{13}{8} | 102 | 8 |
math | 8.054. $\sin x+\sin 7 x-\cos 5 x+\cos (3 x-2 \pi)=0$. | x_{1}=\frac{\pik}{4},x_{2}=\frac{\pi}{8}(4n+3),k,n\inZ | 31 | 33 |
math | 719. Calculate the values of the partial derivatives of the given functions at the specified values of the arguments:
1) $f(\alpha, \beta)=\cos (m \alpha-n \beta) ; \alpha=\frac{\pi}{2 m}, \beta=0$;
2) $z=\ln \left(x^{2}-y^{2}\right) ; x=2, y=-1$. | f_{\alpha}^{\}(\frac{\pi}{2},0)=-;f_{\beta}^{\}(\frac{\pi}{2},0)=n;z_{x}^{\}(2;-1)=\frac{4}{3};z_{y}^{\}(2;-1)=\frac{2}{3} | 87 | 73 |
math | $5 \cdot 17$. The sum of two numbers is 667, and the quotient of their least common multiple divided by their greatest common divisor is 120. Find these two numbers.
(Kyiv Mathematical Olympiad, 1954) | x_{1}=232,y_{1}=435,orx_{2}=552,y_{2}=115 | 57 | 30 |
math | 9. Several rooks have beaten all the white cells of a $40 \times 40$ chessboard. What is the maximum number of black cells that could remain unbeaten? (A rook beats the cell it stands on.) | 400 | 50 | 3 |
math | 1. Solve the equation $1+2^{n}+3^{n}+5^{n}=2 k$ in integers. | n=0,k=2 | 28 | 6 |
math | 9.1. The square root of the number 49 can be extracted using this "formula": $\sqrt{49}=4+\sqrt{9}$. Are there other two-digit numbers whose square roots can be extracted in a similar manner and are integers? List all such two-digit numbers. | 6481 | 61 | 4 |
math | A convex polyhedron has $n$ vertices. What is the maximum number of edges the polyhedron can have? | 3n-6 | 25 | 4 |
math | 4. Let the $n$ roots of the equation $x^{n}=1$ (where $n$ is an odd number) be $1, x_{1}, x_{2}, \cdots, x_{n-1}$. Then $\sum_{i=1}^{n-1} \frac{1}{1+x_{i}}=$ $\qquad$ . | \frac{n-1}{2} | 80 | 8 |
math | 8. The numbers from 1 to 1000 are written in a circle. Starting from the first, every 15th number is crossed out (i.e., the numbers $1, 16, 31$, etc.), and during subsequent rounds, already crossed-out numbers are also taken into account. The crossing out continues until it turns out that all the numbers to be crossed ... | 800 | 100 | 3 |
math | \section*{Problem 5 - V01005}
When asked about his license plate number, a mathematician replies:
"It is called III Z ... You can calculate the number yourself. Of the four digits, the last 3 are the same. The cross sum is 22.
If you place the first digit at the end, the resulting number is 1998 less than the actual... | 7555 | 87 | 4 |
math | In a grid containing 50 points by 50 points, each point has been painted either blue or red. Directly adjacent points, whether horizontally or vertically, that are of the same color are connected by segments of the same color, while points of different colors are connected by black segments. Among the points, 1510 were... | 1976 | 111 | 4 |
math | 3.063. $1-\sin \left(\frac{\alpha}{2}-3 \pi\right)-\cos ^{2} \frac{\alpha}{4}+\sin ^{2} \frac{\alpha}{4}$. | 2\sqrt{2}\sin\frac{\alpha}{4}\sin(\frac{\alpha+\pi}{4}) | 52 | 24 |
math | 14. If the function $f(x)$ defined on the set $A$ satisfies: for any $x_{1}, x_{2} \in A$, we have
$$
f\left(\frac{x_{1}+x_{2}}{2}\right) \leqslant \frac{1}{2}\left[f\left(x_{1}\right)+f\left(x_{2}\right)\right],
$$
then the function $f(x)$ is called a concave function on $A$.
(1) Determine whether $f(x)=3 x^{2}+x$ is... | m \geqslant 0 | 182 | 8 |
math | 3B. We will say that a number is "fancy" if it is written with an equal number of even and odd digits. Determine the number of all four-digit "fancy" numbers written with different digits? | 2160 | 45 | 4 |
math | 2.236. $\frac{\sqrt{z^{2}-1}}{\sqrt{z^{2}-1}-z} ; \quad z=\frac{1}{2}\left(\sqrt{m}+\frac{1}{\sqrt{m}}\right)$ | \frac{-1}{2},if\in(0;1);\frac{1-}{2},if\in[1;\infty) | 58 | 31 |
math | 11. (10 points) The remainder when the three-digit number $\mathrm{abc}$ is divided by the sum of its digits is 1, and the remainder when the three-digit number $\mathrm{cba}$ is divided by the sum of its digits is also 1. If different letters represent different digits, and $a>c$, then $\overline{\mathrm{abc}}=$ $\qqu... | 452 | 85 | 3 |
math | Grandma told her grandchildren: "Today I am 60 years and 50 months and 40 weeks and 30 days old."
When was the last birthday Grandma had?
(L. Hozová)
Hint. How many whole years are 50 months? | 65 | 58 | 2 |
math | Solve the following equation for real values of $x$:
\[
2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
\] | x = 0 \text{ or } 1 | 53 | 11 |
math | 1. Nils has a goose farm. Nils calculated that if he sells 75 geese, the feed will last 20 days longer than if he doesn't sell any. If he buys an additional 100 geese, the feed will run out 15 days earlier than if he doesn't make such a purchase. How many geese does Nils have? | 300 | 80 | 3 |
math | 1. Given 2117 cards, on which natural numbers from 1 to 2117 are written (each card has exactly one number, and the numbers do not repeat). It is required to choose two cards such that the sum of the numbers written on them is divisible by 100. In how many ways can this be done? | 22386 | 74 | 5 |
math | 3. Among the natural numbers from 1 to 144, the number of ways to pick three numbers that form an increasing geometric progression with an integer common ratio is $\qquad$ . | 78 | 40 | 2 |
math | 5. Entrepreneurs Vasiliy Petrovich and Pyotr Gennadiyevich opened a clothing factory called "ViP". Vasiliy Petrovich invested 200 thousand rubles, and Pyotr Gennadiyevich invested 350 thousand rubles. The factory proved to be successful, and after a year, Anastasia Alekseevna approached them with an offer to buy part o... | 1,000,000 | 163 | 9 |
math | ## Task 24/72
Determine all solutions in the domain of natural numbers for the system of equations
$$
\begin{aligned}
\binom{y+1}{x+1}-\binom{y}{x+1} & =6 \\
\binom{x}{x-2} \cdot \frac{2}{x-1}+\binom{y}{y-1} \cdot \frac{1}{y} & =3
\end{aligned}
$$ | 2,4 | 107 | 3 |
math | [Thales' Theorem and the Proportional Segments Theorem]
On the median $A M$ of triangle $A B C$, a point $K$ is taken such that $A K: K M=1: 3$.
Find the ratio in which the line passing through point $K$ parallel to side $A C$ divides side $B C$. | 1:7 | 78 | 3 |
math | ## Zadatak B-1.1.
Odredite $x^{y}$ ako je $\frac{x}{4 \cdot 64^{7}}+\frac{1024^{-4} x}{16}=16$ i $y=\frac{3^{4+n}+5 \cdot 3^{n}}{3^{3+n}-25 \cdot 3^{n}}$.
| 2^{2021} | 88 | 7 |
math | 16. Let $n$ be a given positive integer, $S_{n} \subseteq\left\{\alpha \mid \alpha=\left(p_{1}, p_{2}, \cdots, p_{n}\right), p_{k} \in\{0,1\}, k=1,2, \cdots, n\right\}$. For any elements $\beta=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ and $\gamma=\left(y_{1}, y_{2}, \cdots, y_{n}\right)$ in the set $S_{n}$, the follow... | 5 | 259 | 1 |
math |
Problem 6. Find the number of non-empty sets of $S_{n}=\{1,2, \ldots, n\}$ such that there are no two consecutive numbers in one and the same set.
| f_{n}=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n+2}-(\frac{1-\sqrt{5}}{2})^{n+2})-1 | 46 | 52 |
math | Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ to $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.
| 880 | 84 | 3 |
math | $10 \cdot 3$ Try to find the six-digit integer $\overline{a b c d e f}, \overline{a b c d e f} \cdot 3=$ $\overline{e f a b c d}$. Here $a, b, c, d, e, f$ represent different digits, and $a, e \neq 0$.
(Wuhan, Hubei Province, China Math Summer Camp, 1987) | 153846,230769,307692 | 104 | 20 |
math | Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had t... | EDACB | 121 | 4 |
math | Task 1. Find the value of the expression $\sqrt[3]{7+5 \sqrt{2}}-\sqrt[3]{5 \sqrt{2}-7}$ | 2 | 36 | 1 |
math | Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for ever... | \frac{1 + \sqrt{5}}{2} | 131 | 13 |
math | Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$. Adam says, "$n$ leaves a remainder of $2$ when divided by $3$." Bendeguz says, "For some $k$, $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$. Then $2n - s = 20$." Dennis says,... | 210 | 141 | 3 |
math | 3. proposed by A. Shen
There is an infinite one-way strip of cells, numbered with natural numbers, and a bag with ten stones. Initially, there are no stones in the cells of the strip. The following actions are allowed:
- moving a stone from the bag to the first cell of the strip or back;
- if there is a stone in the ... | 1000 | 128 | 4 |
math | Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors)... | 3 | 107 | 1 |
math | ## Task B-1.5.
The PIN of Karlo's mobile phone is a four-digit natural number greater than 2021 and divisible by 5 and 6. If in this number we swap the first and third, and the second and fourth digits, we will get a number that is 3960 greater than it. Determine Karlo's mobile phone PIN. | 4080 | 80 | 4 |
math | 6. [40] Consider five-dimensional Cartesian space
$$
\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\},
$$
and consider the hyperplanes with the following equations:
- $x_{i}=x_{j}$ for every $1 \leq i<j \leq 5$;
- $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=-1$;
- $x_{1}+x_{2}+x... | 480 | 212 | 3 |
math | Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$ f(x+f(y))=f(x+y)+f(y) $$
for all $x, y \in \mathbb{R}^{+}$. (Symbol $\mathbb{R}^{+}$ denotes the set of all positive real numbers.) (Thailand)
Answer. $f(x)=2x$. | f(x)=2x | 97 | 5 |
math | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=1500$, if $f(0)=1$ and for any $x$ the equality $f(x+3)=f(x)+2 x+3$ holds. | 750001 | 61 | 6 |
math | Example 12 Given $f(\theta)=\sin ^{2} \theta+\sin ^{2}(\theta+\alpha)$ $+\sin ^{2}(\theta+\beta)$, where $\alpha, \beta$ are constants satisfying $0 \leqslant \alpha \leqslant \beta \leqslant \pi$. For what values of $\alpha, \beta$ is $f(\theta)$ a constant value? | \alpha=\frac{\pi}{3}, \beta=\frac{2 \pi}{3} | 96 | 20 |
math | ## Task A-2.3.
Determine all prime numbers $p$ for which $2^{p}+p^{2}$ is also a prime number. | 3 | 34 | 1 |
math | 10. $1991^{2000}$ divided by $10^{6}$, the remainder is | 880001 | 26 | 6 |
math | Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$? | 552 | 34 | 3 |
math | Example 1 Find the minimum value of the function
$$
f(x)=|x-1|+|x-2|+\cdots+|x-10|
$$
(2012, National High School Mathematics League Tianjin Preliminary Contest) | 25 | 56 | 2 |
math | 35th IMO 1994 shortlist Problem C1 Two players play alternately on a 5 x 5 board. The first player always enters a 1 into an empty square and the second player always enters a 0 into an empty square. When the board is full, the sum of the numbers in each of the nine 3 x 3 squares is calculated and the first player's sc... | 6 | 114 | 1 |
math | Solve the following equation:
$$
x^{2 \cdot \log _{2} x}=8
$$ | x_{1}=2^{\sqrt{3/2}}\approx2.337x_{2}=2^{-\sqrt{3/2}}\approx0.428 | 24 | 40 |
math | 1. For two consecutive days, Petya went to the store and each time bought buns for exactly 100 rubles. On the second day, the buns were one ruble more expensive than on the first day. The buns always cost an integer number of rubles.
How many buns did Petya buy over the two days? If there are multiple possible answers... | 150;45 | 110 | 6 |
math | 6. Given in a Cartesian coordinate system there are two moving points $P\left(\sec ^{2} \alpha, \operatorname{tg} \alpha\right), Q(\sin \beta, \cos \beta+5)$, where $\alpha, \beta$ are any real numbers. Then the shortest distance between $P$ and $Q$ is $\qquad$
Translate the above text into English, please retain the ... | 2 \sqrt{5}-1 | 105 | 7 |
math | 5. Given an ellipse $\frac{x^{2}}{2}+k y^{2}=1$ with its foci on the $x$-axis, points $A$ and $B$ are the two intersection points of a line passing through the origin with the ellipse. If the number $k$ allows for another point $C$ on the ellipse such that $\triangle A B C$ is an equilateral triangle, then for all such... | \frac{2 \sqrt{3}}{3} | 111 | 12 |
math | 101. Append three digits to 523 ... so that the resulting six-digit number is divisible by 7, 8, and 9. | 523152523656 | 33 | 12 |
math | 6. Two flagpoles of heights 5 meters and 3 meters are erected on a level ground. If the coordinates of the bottoms of the two flagpoles are determined as $A(-5,0)$ and $B(5,0)$, then the locus of points on the ground where the angles of elevation to the tops of the poles are equal is $\qquad$ . | (x-\frac{85}{8})^{2}+y^{2}=(\frac{75}{8})^{2} | 80 | 29 |
math | 13.060. A load with a mass of 60 kg presses on a support. If the mass of the load is reduced by 10 kg, and the area of the support is reduced by 5 dm², then the mass per square decimeter of the support will increase by 1 kg. Determine the area of the support. | 15 | 74 | 2 |
math | (3) Solve the inequality $m x-2 \geqslant 3 x-4 m$ for $x$, where the constant $m \in \mathbf{R}$. | x\geqslant\frac{2-4}{-3}when>3;x\leqslant\frac{2-4}{-3}when<3;xisanyrealwhen=3 | 41 | 45 |
math | Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three foots of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{... | 600 | 320 | 3 |
math | 99. How many diagonals does a convex decagon have? | 35 | 14 | 2 |
math | 7.4. Does there exist a six-digit natural number that, when multiplied by 9, is written with the same digits but in reverse order? | 109989 | 31 | 6 |
math | Three. (25 points) Select $k$ numbers from 1 to 2012, such that among the selected $k$ numbers, there are definitely three numbers that can form the lengths of the sides of a triangle (the lengths of the three sides of the triangle must be distinct). What is the minimum value of $k$ that satisfies the condition? | 17 | 76 | 2 |
math | B1. For the number $x$, it holds that: $x=\frac{1}{1+x}$. Calculate $x-\frac{1}{x}$. Simplify your answer as much as possible. | -1 | 44 | 2 |
math | 5. In a cube, the probability that 3 randomly chosen edges are pairwise skew is $\qquad$
In a cube, randomly select 3 edges, the probability that they are pairwise skew is $\qquad$ | \frac{2}{55} | 44 | 8 |
math | I3.2 Two bags $U_{1}$ and $U_{2}$ contain identical red and white balls. $U_{1}$ contains $A$ red balls and 2 white balls. $U_{2}$ contains 2 red balls and $B$ white balls. Take two balls out of each bag. If the probability of all four balls are red is $\frac{1}{60}$, find the value of $B$. | 3 | 92 | 1 |
math | One afternoon at the park there were twice as many dogs as there were people, and there were twice as many people as there were snakes. The sum of the number of eyes plus the number of legs on all of these dogs, people, and snakes was $510$. Find the number of dogs that were at the park. | 60 | 67 | 2 |
math | [ Law of Sines ]
In triangle $ABC$, it is known that $\angle A=\alpha, \angle C=\beta, AB=a$; $AD$ is the angle bisector. Find $BD$.
# | \cdot\sin\alpha/2/\sin(\alpha/2+\beta) | 46 | 17 |
math | Konagin S.
Find $x_{1000}$, if $x_{1}=4, x_{2}=6$, and for any natural $n \geq 3$, $x_{n}$ is the smallest composite number greater than $2 x_{n-1}-x_{n-2}$. | 501500 | 68 | 6 |
math | 1.4.3 * Given real numbers $a, x, y$ satisfy:
$$
x \sqrt{a(x-a)}+y \sqrt{a(y-a)}=\sqrt{|\lg (x-a)-\lg (a-y)|} .
$$
Find the value of the algebraic expression $\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$. | \frac{1}{3} | 91 | 7 |
math |
A5. Find the largest positive integer $n$ for which the inequality
$$
\frac{a+b+c}{a b c+1}+\sqrt[n]{a b c} \leq \frac{5}{2}
$$
holds for all $a, b, c \in[0,1]$. Here $\sqrt[1]{a b c}=a b c$.
| 3 | 84 | 1 |
math | 8、Two candidates, A and B, participate in an election, with A receiving $n$ votes and B receiving $m$ votes $(n>m)$. The probability that A's cumulative vote count always exceeds B's cumulative vote count during the vote counting process is $\qquad$ . | \frac{-n}{+n} | 59 | 8 |
math | 24. S2 (POL) $)^{1 \mathrm{MO}}$ The positive real numbers $x_{0}, x_{1}, \ldots, x_{1996}$ satisfy $x_{0}=$ $x_{1995}$ and
$$
x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}
$$
for $i=1,2, \ldots, 1995$. Find the maximum value that $x_{0}$ can have. | 2^{997} | 126 | 6 |
math | 11. A three-digit number in base seven, when expressed in base nine, has its digits in reverse order of the original number. Find this number. | 503 | 32 | 3 |
math | (P4 IMO 2019, untreated)
Find all pairs \((k, n)\) of strictly positive integers such that:
$$
k!=\left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \ldots\left(2^{n}-2^{n-1}\right)
$$ | (k,n)=(1,1)(k,n)=(3,2) | 82 | 14 |
math | 23. The line $l$ passing through point $E\left(-\frac{p}{2}, 0\right)$ intersects the parabola $C: y^{2}=2 p x(p>0)$ at points $A$ and $B$, and the inclination angle of line $l$ is $\alpha$. Then the range of $\alpha$ is $\qquad$; $F$ is the focus of the parabola, and the area of $\triangle A B F$ is $\qquad$ (expresse... | \left(0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \pi\right), \frac{p^{2} \sqrt{\cos 2 \alpha}}{\sin \alpha} | 123 | 53 |
math | Let $ABC$ be an isosceles obtuse-angled triangle, and $D$ be a point on its base $AB$ such that $AD$ equals to the circumradius of triangle $BCD$. Find the value of $\angle ACD$. | 30^\circ | 54 | 4 |
math | 4a.a * Let $x$ be a natural number. If a sequence of natural numbers $x_{0}=1, x_{1}, x_{2}, \cdots, x_{t-1}$, $x_{t}=x$ satisfies $x_{i-1} \mid x_{i}, x_{t-1}<x_{i}, i=1,2, \cdots, t$, then $\left\{x_{0}, x_{1}, \cdots, x_{t}\right\}$ is called a divisor chain of $x$, and $t$ is the length of this divisor chain. $T(x)... | T(x)=3n+k+,R(x)=\frac{(3n+k+)!}{(n!)^2\cdot!\cdot(k+n)!} | 221 | 32 |
math | N7 (10-6, UK) Let $[x]$ denote the greatest integer not exceeding $x$. For any positive integer $n$, compute the sum
$$
\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right] \text {. }
$$ | n | 73 | 1 |
math | 19.3.7 * Find positive integers $a, b$ satisfying $a \leqslant 2000 \leqslant b$, such that $2, a(a+1), b(b+1)$ form an arithmetic sequence. | =1477,b=2089 | 55 | 11 |
math | ## Task 2 - 020922
A car is traveling at a speed of $100 \frac{\mathrm{km}}{\mathrm{h}}$. It is being braked.
a) In what time will it come to a stop if, due to braking, its speed decreases by $5 \frac{\mathrm{m}}{\mathrm{s}}$ every second?
b) What braking distance does it cover in this time? | 5.56\mathrm{~},77.3\mathrm{~} | 94 | 18 |
math | How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$? | 6 | 29 | 1 |
math | 4. Let the two branches of the hyperbola $xy=1$ be $C_{1}$ and $C_{2}$, and let the three vertices of the equilateral triangle $PQR$ lie on this hyperbola.
(1) Prove that $P$, $Q$, and $R$ cannot all lie on the same branch of the hyperbola;
(2) Suppose $P(-1,-1)$ is on $C_{2}$, and $Q$, $R$ are on $C_{1}$. Find the coo... | (2+\sqrt{3}, 2-\sqrt{3}),(2-\sqrt{3}, 2+\sqrt{3}) | 138 | 28 |
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