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 | 7. Find all pairs of integers ( $x, y$ ) for which the equation $x^{2}+y^{2}=x+y+2$ holds.
ANSWER: $(-1,0) ;(-1,1) ;(0,-1) ;(0,2) ;(1,-1),(1,2),(2,0) ;(2,1)$ | (-1,0);(-1,1);(0,-1);(0,2);(1,-1),(1,2),(2,0);(2,1) | 82 | 38 |
math | $4 \cdot 6$ Find a natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$ itself, it has exactly 10 divisors.
(Kyiv Mathematical Olympiad, 1982) | 5\cdot7^4 | 59 | 6 |
math | $8 \cdot 38$ Find the largest integer $x$ such that $4^{27}+4^{1000}+4^{x}$ is a perfect square.
(6th All-Soviet Union Mathematical Olympiad, 1972) | 1972 | 59 | 4 |
math | ## Task 4 - 080924
Four people $A, B, C$, and $D$ make three statements each about the same number $x$. By agreement, each person must have at least one true statement and at least one false statement.
$A$ says:
(1) The reciprocal of $x$ is not less than 1.
(2) $x$ does not contain the digit 6 in its decimal repres... | 25 | 250 | 2 |
math | Example 3 (APMO) Find all nonempty finite sets $S$ of positive integers such that if $m, n \in$ $S$, then $\frac{m+n}{(m, n)} \in \mathbf{S}, (m, n$ do not have to be distinct). | {2} | 64 | 3 |
math | 2.1. Find the smallest value of $a$, for which the sum of the squares of the roots of the equation $x^{2}-3 a x+a^{2}=0$ is $0.28$. | -0.2 | 46 | 4 |
math | 2. Let $a>b>0$, then the minimum value of $a^{2}+\frac{1}{b(a-b)}$ is
保留了源文本的换行和格式。 | 4 | 41 | 1 |
math | 7.210. $5^{x-1}+5 \cdot 0.2^{x-2}=26$. | 1;3 | 29 | 3 |
math | Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$. | 505 | 51 | 3 |
math | 1. Given $a_{n}=\frac{1}{n \sqrt{n+1}+(n+1) \sqrt{n}}$. Then $a_{1}+a_{2}+\cdots+a_{99}=$ $\qquad$
$(2008$, Shanghai Jiao Tong University Winter Camp) | \frac{9}{10} | 68 | 8 |
math | 9.5. On a circle, $n>1$ points, called positions, are marked, dividing it into equal arcs. The positions are numbered clockwise from 0 to $n-1$. Vasya places a chip in one of them. Then the following actions, called moves, are repeated an unlimited number of times: Petya names some natural number, and Vasya moves the c... | 2^{k} | 151 | 4 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(7 ; 0 ;-15)$
$C(2 ; 10 ;-12)$ | A(0;0;-4\frac{1}{3}) | 62 | 14 |
math | 4. Given $H$ is the orthocenter of $\triangle A B C$, $\angle A=75^{\circ}, B C=2$, the area of the circumcircle of $\triangle A B H$ is . $\qquad$ | 4\pi(2-\sqrt{3}) | 52 | 10 |
math | Task 1 - 091241
At an international camp, a group of 30 friends are participating, some of whom speak German, some Russian, and some French. Some friends speak only one language, some two languages, and some even all three languages.
The number of friends who speak exactly two languages is more than twice but less tha... | =2,r=1,f=4,s_{3}=7 | 193 | 13 |
math | What is the difference between the median and the mean of the following data set: $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$? | \frac{38}{13} | 68 | 9 |
math | 1. Given the set $S=\{1,2, \cdots, 3 n\}, n$ is a positive integer, $T$ is a subset of $S$, satisfying: for any $x, y, z \in T$ (where $x, y, z$ can be the same), we have $x+y+z \notin T$. Find the maximum number of elements in all such sets $T$. | 2n | 90 | 2 |
math | 8. In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{2}=2$, and $a_{n+2}=7 a_{n-1}-12 a_{n}$, then $a_{n}=$ | a_{n}=2\cdot3^{n-1}-4^{n-1}(n\in{N}^{*}) | 59 | 28 |
math | ## problem statement
Find the angle between the planes:
$x+2 y-1=0$
$x+y+6=0$ | \arccos\frac{3}{\sqrt{10}}\approx1826^{\}6^{\\} | 27 | 28 |
math | 1. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Set
$$
\begin{array}{l}
A=\{y \mid y=[x]+[2 x]+[4 x], x \in \mathbf{R}\}, \\
B=\{1,2, \cdots, 2019\} .
\end{array}
$$
Then the number of elements in $A \cap B$ is $\qquad$ | 1154 | 102 | 4 |
math | For $101 n$ as an integer not less than 2, determine the largest constant $C(n)$ such that
$$
\sum_{1 \leqslant i<j \leqslant n}\left(x_{j}-x_{i}\right)^{2} \geqslant C(n) \cdot \min \left(x_{i+1}-x_{i}\right)^{2}
$$
holds for all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}<x_{2}<\cdots<x_{n}$. | \frac{1}{12} n^{2}\left(n^{2}-1\right) | 132 | 21 |
math | 7 Find the coefficient of $x^{2}$ in the expansion of $(1+x)(1+2 x)(1+4 x) \cdots\left(1+2^{n-1} \cdot x\right)$. | \frac{1}{3}(2^{n}-1)(2^{n}-2) | 49 | 19 |
math | # Problem 5. (3 points)
Circles $O_{1}, O_{2}$, and $O_{3}$ are located inside circle $O_{4}$ with radius 6, touching it internally, and touching each other externally. Moreover, circles $O_{1}$ and $O_{2}$ pass through the center of circle $O_{4}$. Find the radius of circle $O_{3}$.
# | 2 | 89 | 1 |
math | The quadratic polynomial $f(x)$ has the expansion $2x^2 - 3x + r$. What is the largest real value of $r$ for which the ranges of the functions $f(x)$ and $f(f(x))$ are the same set? | \frac{15}{8} | 55 | 9 |
math | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(0 ; 2 ; -4), B(8 ; 2 ; 2), C(6 ; 2 ; 4)$ | 0.96 | 59 | 4 |
math | $1.16 \frac{x^{2}+4 x-5+(x-5) \sqrt{x^{2}-1}}{x^{2}-4 x-5+(x+5) \sqrt{x^{2}-1}}, x>1$.
$\mathbf{1 . 1 7}$
$$
9 b^{\frac{4}{3}}-\frac{a^{\frac{3}{2}}}{b^{2}}
$$ | \sqrt{\frac{x-1}{x+1}} | 98 | 12 |
math | 7. Given $\alpha, \beta \in\left(\frac{3 \pi}{4}, \pi\right), \sin (\alpha+\beta)=-\frac{3}{5}, \sin \left(\beta-\frac{\pi}{4}\right)=\frac{12}{13}$, then the value of $\cos \left(\alpha+\frac{\pi}{4}\right)$ is $\qquad$ . | -\frac{56}{65} | 91 | 9 |
math | 3. Arrange $1,2, \cdots, k$ in a row so that each number is strictly greater than all the numbers preceding it, or strictly less than all the numbers preceding it. Let the number of different arrangements be $a_{k}(k=1,2, \cdots)$. Then $a_{n}=$ $\qquad$ | 2^{n-1} | 75 | 6 |
math | Example 5 Find all positive integers that are coprime with all terms of the sequence $\left\{a_{n}=2^{n}+3^{n}+6^{n}-1, n \geqslant\right.$ $1\}$. | 1 | 56 | 1 |
math | 1. Let the function $f(x)=x^{2}(x \in D, D$ be the domain) have the range $\left\{1^{2}, 2^{2}, \cdots, 2012^{2}\right\}$. Then the number of functions $f(x)$ that satisfy this condition is $\qquad$ . | 3^{2012} | 75 | 7 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-7 ; 7]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 210 | 54 | 3 |
math | 4. (Average) Simplify $\sqrt[3]{5 \sqrt{2}+7}-\sqrt[3]{5 \sqrt{2}-7}$ into a rational number. | 2 | 39 | 1 |
math | (1) Given the set $M=\{2,0,11\}$, if $A \varsubsetneqq M$, and $A$ contains at least one even number. Then the number of sets $A$ that satisfy the condition is $\qquad$ . | 5 | 58 | 1 |
math | ## 9. By car and by bicycle
Jean and Jules set off simultaneously from city $A$ to city $B$, one by car and the other by bicycle.
After some time, it turned out that if Jean had traveled three times more, he would have had to travel twice less than he currently has left, and that if Jules had traveled half as much, h... | Jean | 100 | 1 |
math | Example 1 Given $a^{2}+b^{2}=6 a b$, and $a>b>0$. Then $\frac{a+b}{a-b}=$ $\qquad$
$(2001$, Beijing Middle School Mathematics Competition (Grade 8)) | \sqrt{2} | 56 | 5 |
math | Problem 2. The number $x$ satisfies the condition $\frac{\sin 3 x}{\sin x}=\frac{5}{3}$. Find the value of the expression $\frac{\cos 5 x}{\cos x}$ for such $x$ | -\frac{11}{9} | 55 | 8 |
math | 4. Find all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ such that for all non-zero numbers $x, y$ the following holds:
$$
x \cdot f(x y)+f(-y)=x \cdot f(x) .
$$ | f(x)=(1+\frac{1}{x}) | 67 | 11 |
math | Let's determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for any $x, y$, $f(x+y) + f(x) f(y) = x^{2} y^{2} + 2 x y$. | f(x)=x^{2}-1 | 58 | 8 |
math | 35. (ROM 1) Find all numbers $N=\overline{a_{1} a_{2} \ldots a_{n}}$ for which $9 \times \overline{a_{1} a_{2} \ldots a_{n}}=$ $\overline{a_{n} \ldots a_{2} a_{1}}$ such that at most one of the digits $a_{1}, a_{2}, \ldots, a_{n}$ is zero. | 0 \text{ and } N_{k}=10 \underbrace{99 \ldots 9}_{k} 89, \text{ where } k=0,1,2, \ldots | 107 | 46 |
math | 4. Solve the equation $6(x-1)(x+2)-4(x-3)(x+4)=2(x-5)(x-6)$. | 1 | 34 | 1 |
math | 5. The number $N$ is written as the product of consecutive natural numbers from 2019 to 4036: $N=2019 \cdot 2020 \cdot 2021 \cdot \ldots \cdot 4034 \cdot 4035 \cdot 4036$. Determine the power of two in the prime factorization of the number $N$.
(points) | 2018 | 96 | 4 |
math | 8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=73
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 40 | 102 | 2 |
math | ## Task 4 - 260914
Determine whether there exist natural numbers $n$ such that the solution $x$ of the equation $17 x + n = 6 x + 185$ is also a natural number! If this is the case, find the smallest such number $n$ and the corresponding solution $x$ of the given equation! | 9,16 | 81 | 4 |
math | 13.003. In two barrels, there are 70 liters of milk. If 12.5% of the milk from the first barrel is poured into the second barrel, then both barrels will have the same amount. How many liters of milk are in each barrel? | 40 | 60 | 2 |
math | $4 \cdot 94$ Solve the equation $\operatorname{arctg} x+\operatorname{arctg}(1-x)=2 \operatorname{arctg} \sqrt{x-x^{2}}$.
(Kyiv Mathematical Olympiad, 1936) | \frac{1}{2} | 62 | 7 |
math | Solve the following system of equations over the real numbers:
$2x_1 = x_5 ^2 - 23$
$4x_2 = x_1 ^2 + 7$
$6x_3 = x_2 ^2 + 14$
$8x_4 = x_3 ^2 + 23$
$10x_5 = x_4 ^2 + 34$ | (x_1, x_2, x_3, x_4, x_5) = (1, 2, 3, 4, 5) | 92 | 38 |
math | (4) In $\triangle A B C$, it is known that $A B=2, B C=4, \angle B$'s bisector $B D=\sqrt{6}$, then the median $B E$ on side $A C$ is $\qquad$. | \frac{\sqrt{31}}{2} | 60 | 11 |
math | 3. Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3. | 672 | 40 | 3 |
math | 1. The front tire of a motorcycle wears out after $25000 \mathrm{~km}$, while the rear tire wears out after $15000 \mathrm{~km}$. After how many kilometers should the tires be swapped so that they wear out simultaneously? After how many kilometers must the motorcyclist put on new tires? | 9375 | 74 | 4 |
math | [ Radii of the inscribed, circumscribed, and exscribed circles (other) [ Area of a triangle (through the semiperimeter and the radius of the inscribed or exscribed circle).
Through the center $O$ of the inscribed circle $\omega$ of triangle $A B C$, a line parallel to side $B C$ is drawn, intersecting sides $A B$ and ... | 8 | 143 | 1 |
math | Nils is playing a game with a bag originally containing $n$ red and one black marble.
He begins with a fortune equal to $1$. In each move he picks a real number $x$ with $0 \le x \le y$, where his present fortune is $y$. Then he draws a marble from the bag. If the marble is red, his fortune increases by $x$, but if i... | \frac{2^n}{n+1} | 149 | 11 |
math | Find four consecutive numbers whose product is 1680.
# | 5,6,7,8or-8,-7,-6,-5 | 14 | 16 |
math | Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$.
The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With tha... | 12 | 186 | 2 |
math | 10. Non-negative real numbers $a_{i}(i=1,2, \cdots, n)$, satisfy: $a_{1}+a_{2}+a_{3}+\cdots+a_{n}=1$, find the minimum value of $\frac{a_{1}}{1+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}}+\cdots+\frac{a_{n}}{1+a_{1}+a_{2}+\cdots+a_{n-1}}$. | \frac{n}{2n-1} | 132 | 9 |
math | ## SUBJECT I
Solve in $\mathbf{Z}$ the equation: $15 x y-35 x-6 y=3$. | -3,2 | 31 | 4 |
math | 6. The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$. Let $[x]$ denote the greatest integer not exceeding $x$, then $\left[\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2009}}\right]$ is $\qquad$ . | 1 | 109 | 1 |
math | 1. [4] Let $A B C$ be a triangle with area 1. Let points $D$ and $E$ lie on $A B$ and $A C$, respectively, such that $D E$ is parallel to $B C$ and $D E / B C=1 / 3$. If $F$ is the reflection of $A$ across $D E$, find the area of triangle $F B C$. | \frac{1}{3} | 93 | 7 |
math | In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column? | 720 | 38 | 3 |
math | 12. At a party, 9 celebrities performed $n$ "trio dance" programs. If in these programs, any two people have collaborated exactly once, then $n=$ $\qquad$ | 12 | 42 | 2 |
math | 6. Express $M=\frac{4 x^{2}+2 x+6}{x^{4}+x^{2}+1}$ as partial fractions. | M=\frac{x+2}{x^{2}+x+1}+\frac{-x+4}{x^{2}-x+1} | 35 | 31 |
math | Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$.
[i]2020 CCA Math Bonanza Lightning Round #4.3[/i] | 169 | 102 | 3 |
math | Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area? | 5 | 52 | 1 |
math | 1. Vasya's dad is good at math, but on the way to the garage, he forgot the code for the digital lock on the garage. In his memory, he recalls that all the digits of the code are different and their sum is 28. How many different codes does dad need to try to definitely open the garage, if the opening mechanism of the l... | 48 | 90 | 2 |
math | Determine all the positive real numbers $x_1, x_2, x_3, \dots, x_{2021}$ such that
$x_{i+1}=\frac{x_i^3+2}{3x_i^2}$
for every $i=1, 2, 3, \dots, 2020$ and $x_{2021}=x_1$ | x_1 = x_2 = \dots = x_{2021} = 1 | 90 | 22 |
math | Solve the following system of equations:
$$
\frac{1}{2-x+2 y}-\frac{1}{x+2 y-1}=2,
$$
$$
\frac{1}{2-x+2 y}-\frac{1}{1-x-2 y}=4 .
$$ | \frac{11}{6},\frac{1}{12} | 64 | 16 |
math | 17. (12 points) Ellipse $C$:
$A x^{2}+B y^{2}=1$ intersects with the line $l: x+2 y=7$ at points $P$ and $Q$. Point $R$ has coordinates $(2,5)$. If $\triangle P Q R$ is an isosceles right triangle, $\angle P R Q=90^{\circ}$, find the values of $A$ and $B$. | A=\frac{3}{35}, B=\frac{2}{35} | 103 | 18 |
math | 5. Find all positive integers $x, y$ such that $\frac{x^{3}+y^{3}-x^{2} y^{2}}{(x+y)^{2}}$ is a non-negative integer. | 2 | 46 | 1 |
math | 6.6. (New York, 73). Find all values of $x \in [0, \pi / 2]$, satisfying the equation $\cos ^{8} x + \sin ^{8} x = 97 / 128$. | \pi/12,5\pi/12 | 58 | 12 |
math | 5. Let $\angle A, \angle B, \angle C$ be the three interior angles of $\triangle ABC$. If $\sin A=a, \cos B=b$, where $a>0, b>0$, and $a^{2}+b^{2} \leqslant 1$, then $\tan C=$ $\qquad$ | \frac{a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}}{a \sqrt{1-b^{2}}-b \sqrt{1-a^{2}}} | 74 | 41 |
math | 9. (8th "Hope Cup" Invitational Competition Question) If $a+b+c=1$, what is the maximum value of $\sqrt{3 a+1}+\sqrt{3 b+1}+\sqrt{3 c+1}$? | 3\sqrt{2} | 53 | 6 |
math | 2. Find all positive integers $n$ such that $\phi(n)$ has the value
a) 1
d) 6
b) 2
e) 14
c) 3
f) 24 . | a) 1,2 \quad b) 3,4,6 \quad c) \text{no solution} \quad d) 7,9,14,18 \quad e) \text{no solution} \quad f) 35,39,45,52,56,70,72,78,84,90 | 51 | 84 |
math | 11. The sequence $\left\{a_{\mathrm{n}}\right\}$ has 1001 terms, $a_{1}=0, a_{1001}=2020$, and $a_{\mathrm{k}+1}-a_{\mathrm{k}}=1$ or $3, \mathrm{k}=1,2, \ldots, 1000$. Then the number of different sequences that satisfy this condition is
(use a combination number as the answer); | C_{1000}^{490} | 108 | 12 |
math | 4.3 For Eeyore's Birthday, Winnie-the-Pooh, Owl, and Piglet decided to give balloons. Winnie-the-Pooh prepared twice as many balloons as Piglet, and Owl prepared four times as many balloons as Piglet. When Piglet was carrying his balloons, he was in a great hurry, stumbled, and some of the balloons burst. Eeyore receiv... | 2 | 103 | 1 |
math | 18.14 In a Japanese chess round-robin tournament, each of the 14 participants plays against the other 13. There are no ties in the matches. Find the maximum number of "triangular ties" (here, a "triangular tie" refers to a situation where each of the 3 participants has one win and one loss against the other two). | 112 | 78 | 3 |
math | ## Task A-4.2.
Determine all pairs of integers $(m, n)$ such that $3 \cdot 2^{m}+1=n^{2}$. | (,n)\in{(0,2),(0,-2),(4,7),(4,-7),(3,5),(3,-5)} | 37 | 30 |
math | 2. Determine all values of the real parameter $m$ for which the equation
$$
(m+1) x^{2}-3 m x+4 m=0
$$
has two distinct real solutions that are both greater than -1. | \in(-\frac{16}{7},-1)\cup(-\frac{1}{8},0) | 51 | 25 |
math | 8. (26th Russian Mathematical Olympiad) Find the sum $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right]$. | \frac{1}{3}(2^{1001}-2)-500 | 81 | 19 |
math | $5 \cdot 34$ Polynomial
$$
(1-z)^{b_{1}}\left(1-z^{2}\right)^{b_{2}}\left(1-z^{3}\right)^{b_{3}} \cdots\left(1-z^{32}\right)^{b_{32}}
$$
In this polynomial, $b_{i}$ are positive integers $(i=1,2, \cdots, 32)$, and the polynomial has the following remarkable property: when expanded, and terms with powers of $z$ higher ... | 2^{27}-2^{11} | 161 | 10 |
math | 6. (10 points) Person A and Person B work together on a project, which can be completed in several days. If Person A completes half of the project alone, it will take 10 days less than if Person A and Person B work together to complete the entire project; if Person B completes half of the project alone, it will take 15... | 60 | 117 | 2 |
math | Example 6 From the 100 positive integers $1,2,3, \cdots, 100$, if $n$ numbers are taken, among these $n$ numbers, there are always 4 numbers that are pairwise coprime. Find the minimum value of $n$.
| 75 | 64 | 2 |
math | 14. Given sets $A, B$ are both sets composed of positive integers, and $|A|=20,|B|=16$, set $A$ satisfies the following condition: if $a, b, m, n \in A$, and $a+b=m+n$, then it must be that $\{a, b\}=\{m, n\}$. Define $A+B=\{a+b \mid a \in A, b \in B\}$, try to determine the minimum value of $|A+B|$. | 200 | 116 | 3 |
math | (7) From $\{1,2,3, \cdots, 100\}$, if 5 numbers (repetition allowed) are randomly selected, then the expected number of composite numbers is $\qquad$ . | \frac{37}{10} | 49 | 9 |
math | Example 2. Find the integral curve of the equation $y^{\prime \prime}=x+1$, passing through the point $M_{0}(1,1)$ and tangent to the line $y=\frac{1}{2} x+\frac{1}{2}$ at this point. | \frac{x^{3}}{6}+\frac{x^{2}}{2}-x+\frac{4}{3} | 62 | 26 |
math | How many three-digit numbers exist in which the digits 1, 2, 3 appear exactly once each?
# | 6 | 24 | 1 |
math | A company of $n$ soldiers is such that
(i) $n$ is a palindrome number (read equally in both directions);
(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.
Find the smallest $n$ satisfying these conditions and prove that there are infinitely... | 515 | 88 | 5 |
math | An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proo... | 27 | 79 | 2 |
math | Variant 134. From 10 to 20 minutes (including 10 and 20).
Grading criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; $\mathbf{1 0}$ points - the solution is reduced to correct inequalities with respect to $t$, but further errors are made; 5 points - the equation and two ineq... | 1.6 | 249 | 3 |
math | Example 9. Solve the inequality
$$
\frac{\sqrt{2-x}+4 x-3}{x} \geqslant 2
$$
## 148 CH. 3. INEQUALITIES WITH ONE UNKNOWN | (-\infty;0)\cup[1;2] | 52 | 13 |
math | 7. Let a set of three real numbers be represented both as $\left\{a, \frac{b}{a}, 1\right\}$ and as $\left\{a^{2}, a+b, 0\right\}$, then the value of $a^{2002}+b^{2003}$ is $\qquad$. | 1 | 77 | 1 |
math | Auxiliary similar triangles [ Angle between tangent and chord Circle inscribed in an angle
[ Theorem on the lengths of tangent and secant; product of the entire secant and its external part
A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$. Ray $O X$ intersects this circle at poin... | \frac{4}{3} | 125 | 7 |
math | 1. If the arithmetic mean of two positive numbers is $2 \sqrt{3}$, and the geometric mean is $\sqrt{3}$, what is the difference between these two numbers? | 6 | 39 | 1 |
math | Three. (Full marks 23 points) Given that $M$ is a point on the moving chord $AB$ of the parabola $y^{2}=$ $2 p x$, $O$ is the origin, $O A$ $\perp O B, O M \perp A B$. Find the equation of the locus of point $M$.
| (x-p)^{2}+y^{2}=p^{2} .(x \neq 0) | 78 | 24 |
math | [ Tangent Circles ]
Circle $S$ with its center at the vertex of the right angle of a right triangle is tangent to the circle inscribed in this triangle. Find the radius of circle $S$, given that the legs of the triangle are 5 and 12.
# | 2(\sqrt{2}\1) | 59 | 8 |
math | 1. In triangle $A B C, A B=6, B C=10$, and $C A=14$. If $D, E$, and $F$ are the midpoints of sides $B C$, $C A$, and $A B$, respectively, find $A D^{2}+B E^{2}+C F^{2}$. | 249 | 79 | 3 |
math |
Problem 10.1. Find all values of the real parameter $a$ for which the equation $x^{3}-3 x^{2}+\left(a^{2}+2\right) x-a^{2}=0$ has three distinct roots $x_{1}$, $x_{2}$ and $x_{3}$ such that $\sin \left(\frac{2 \pi}{3} x_{1}\right), \sin \left(\frac{2 \pi}{3} x_{2}\right)$ and $\sin \left(\frac{2 \pi}{3} x_{3}\right)$ ... | 0 | 143 | 1 |
math |
Opgave 3. Vind alle functies $f: \mathbb{Z} \rightarrow \mathbb{Z}$ die voldoen aan
$$
f(-f(x)-f(y))=1-x-y
$$
voor alle $x, y \in \mathbb{Z}$.
| f(x)=x-1 | 67 | 6 |
math | [ Rectangles and squares. Properties and characteristics ] [ Equilateral (equiangular) triangle ]
On the side $A B$ of the square $A B C D$, an equilateral triangle $A B M$ is constructed. Find the angle $D M C$.
# | 30 | 59 | 2 |
math | Example 1. Find $\int e^{x^{2}} \cdot x d x$. | \frac{1}{2}e^{x^{2}}+C | 19 | 15 |
math | Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\t... | \frac{13}{29} | 116 | 9 |
math | 11. B. Given the parabola $y=x^{2}$ and the moving line $y=(2 t-1) x-c$ have common points $\left(x_{1}, y_{1}\right) 、\left(x_{2}, y_{2}\right)$, and $x_{1}^{2}+x_{2}^{2}=t^{2}+2 t-3$.
(1) Find the range of real number $t$;
(2) For what value of $t$ does $c$ attain its minimum value, and find the minimum value of $c$. | \frac{11-6 \sqrt{2}}{4} | 130 | 15 |
math | 6. How many numbers of the form $\overline{2 a b c d 3}$ are there such that all digits are distinct, and the number $\overline{a b c d}$ is a four-digit multiple of 5? | 390 | 50 | 3 |
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