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 | Let $c > 1$ be a real number. A function $f: [0 ,1 ] \to R$ is called c-friendly if $f(0) = 0, f(1) = 1$ and $|f(x) -f(y)| \le c|x - y|$ for all the numbers $x ,y \in [0,1]$. Find the maximum of the expression $|f(x) - f(y)|$ for all [i]c-friendly[/i] functions $f$ and for all the numbers $x,y \in [0,1]$. | \frac{c + 1}{2} | 129 | 10 |
math | 1. What is the maximum number of sides a convex polygon can have if all its diagonals are of equal length? | 5 | 24 | 1 |
math | 9.1. What is the largest number of different natural numbers that can be chosen so that the sum of any three of them is a prime number? | 4 | 31 | 1 |
math | Example 1. Find the volume of the body bounded by the surfaces
$$
x=17 \sqrt{2 y}, \quad x=2 \sqrt{2 y}, \quad z=1 / 2-y, \quad z=0
$$ | 1 | 55 | 1 |
math | $g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$g(x)+g(y)=g(x+y)-xy.$$ If $g(23)=0$, what is the sum of all possible values of $g(35)$? | 210 | 59 | 3 |
math | 17. $N$ pieces of candy are made and packed into boxes, with each box containing 45 pieces. If $N$ is a non-zero perfect cube and 45 is one of its factors, what is the least possible number of boxes that can be packed? | 75 | 58 | 2 |
math | Kornél found two rings of the same size, and the side of each ring is divided into 36 equal parts. On both rings, 18 parts are painted yellow, and 18 are painted green. Can the rings be placed on top of each other so that the dividing lines on the side of the rings align, and the overlapping side parts are of the same ... | 18 | 87 | 2 |
math | You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth 1, 2, 5, 10, and 20 florins, but you have no idea which coin is which and you don't speak the local language. You find a vending ma... | 4 | 166 | 1 |
math | Example 5.7. Using the D'Alembert's criterion, determine the convergence of the series
$$
\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\ldots+\frac{n}{3^{n}}+\ldots
$$ | \frac{1}{3} | 66 | 7 |
math | In the garden, there were three boxes of apples. Altogether, there were more than 150 apples, but less than 190. Maruška moved apples from the first box to the other two boxes so that the number of apples in each of these two boxes doubled compared to the previous state. Similarly, Marta moved apples from the second bo... | 91,49,28 | 163 | 8 |
math | 3. Two females with PPF $M=40-2K$. PPFs are linear with identical opportunity costs. By adding individual PPFs, we get that the PPF of the two females: M=80-2K, $\mathrm{K} \leq 40$. | \mathrm{M}=104-\mathrm{K}^{\wedge}2,\mathrm{~K}\leq1\\\mathrm{M}=105-2\mathrm{~K},1<\mathrm{K}\leq21\\\mathrm{M}=40\mathrm{~K}-\mathrm{K}\wedge2-336,21<\mathrm{} | 64 | 87 |
math | 5. Determine the smallest possible value of the expression $4 x^{2}+4 x y+4 y^{2}+12 x+8$. For which $x$ and $y$ will this expression have the smallest value?
Each task is scored out of 10 points.
The use of a pocket calculator or any reference materials is not allowed. | -4 | 75 | 2 |
math | A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer. | 17 | 71 | 2 |
math | Find natural numbers $a$ and $b$ such that $7^{3}$ is a divisor of $a^{2}+a b+b^{2}$, but 7 is not a divisor of either $a$ or $b$. | =1,b=18 | 50 | 6 |
math | 5. Let $a$ and $b$ be real numbers such that
$$
a^{3}-3 a b^{2}=44 \text{ and } b^{3}-3 a^{2} b=8
$$
Determine $a^{2}+b^{2}$. | 10\sqrt[3]{2} | 63 | 9 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-4 ; 2 ; 6)$
$M_{2}(2 ;-3 ; 0)$
$M_{3}(-10 ; 5 ; 8)$
$M_{0}(-12 ; 1 ; 8)$ | 4 | 94 | 1 |
math | 7. Given a regular 2019-gon, then, the maximum number of diagonals such that any two of them are either perpendicular or do not intersect except at endpoints. | 2016 | 38 | 4 |
math | 3. Let $f(x)=\frac{4^{x}}{4^{x}+2}$, then the sum
$$
f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+f\left(\frac{3}{1001}\right)+\cdots+f\left(\frac{1000}{1001}\right)
$$
is equal to . $\qquad$ | 500 | 105 | 3 |
math | 1.1. Five non-negative numbers are written in a row. The sum of any two adjacent numbers does not exceed 1. What is the maximum value that the sum of all five numbers can take? | 3 | 42 | 1 |
math | Determine the number of all positive ten-digit integers with the following properties:
- The number contains each of the digits 0, 1, 2, ..., 8, and 9 exactly once.
- Each digit, except for the 9, has a neighboring digit that is greater than it.
(Note. For example, in the number 1230, the digits 1 and 3 are the neigh... | 256 | 139 | 3 |
math | Example 3.7. Is the function $z=$ $=f(x, y)=\sqrt{4-x^{2}-y^{2}}$ bounded above (below)? | 0\leqslantf(x,y)\leqslant2 | 37 | 15 |
math | The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$. | \frac{289}{259} | 110 | 11 |
math | Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$. | (p, q) = (n+1, n^2 + n + 1) | 55 | 21 |
math | 2. For all natural numbers $n$ not less than 2, the two roots of the quadratic equation in $x$, $x^{2}-(n+2) x-2 n^{2}=0$, are denoted as $a_{n} 、 b_{n}(n \geqslant 2)$. Then
$$
\begin{array}{c}
\frac{1}{\left(a_{2}-2\right)\left(b_{2}-2\right)}+\frac{1}{\left(a_{3}-2\right)\left(b_{3}-2\right)}+ \\
\cdots+\frac{1}{\le... | -\frac{1003}{4016} | 171 | 13 |
math | [ Measurement of lengths of segments and measures of angles. Adjacent angles. ] [ Central angle. Length of an arc and circumference. ]
Determine the angle between the hour and minute hands of a clock showing 1 hour and 10 minutes, given that both hands move at constant speeds. | 25 | 60 | 2 |
math | Think about Question 1 The sequence $a_{1}, a_{2}, \cdots$ is defined as follows: $a_{n}=2^{n}+3^{n}+6^{n}-1, n=1,2,3, \cdots$ Find all positive integers that are coprime to every term of this sequence. | 1 | 74 | 1 |
math | Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$. | 200 | 107 | 3 |
math | 7. For any $n$-element set $S_{n}$, if its subsets $A_{1}$, $A_{2}, \cdots, A_{k}$ satisfy $\bigcup_{i=1}^{k} A_{i}=S_{n}$, then the unordered set group $\left(A_{1}, A_{2}, \cdots, A_{k}\right)$ is called a “$k$-stage partition” of the set $S_{n}$. Therefore, the number of 2-stage partitions of $S_{n}$ is $\qquad$ | \frac{1}{2}(3^{n}+1) | 124 | 14 |
math | Problem 3. The lengths of the sides of a triangle are in a geometric progression with a common ratio $r$. Find the values of $r$ for which the triangle is, respectively, acute, right, or obtuse. | 1\leqr<\sqrt{\frac{1+\sqrt{5}}{2}} | 47 | 19 |
math | Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two plan... | 450 | 126 | 3 |
math | 11. Given the parabola $y=x^{2}+m x+n$ passes through the point $(2,-1)$, and intersects the $x$-axis at points $A(a, 0)$ and $B(b, 0)$. If $P$ is the vertex of the parabola, find the equation of the parabola that minimizes the area of $\triangle P A B$. | y=x^{2}-4 x+3 | 88 | 9 |
math | We call a number [i]perfect[/i] if the sum of its positive integer divisors(including $1$ and $n$) equals $2n$. Determine all [i]perfect[/i] numbers $n$ for which $n-1$ and $n+1$ are prime numbers. | 6 | 65 | 1 |
math | 10.2. What is the smallest positive number that the leading coefficient of a quadratic trinomial $P(x)$, which takes integer values for all integer $x$, can equal? | \frac{1}{2} | 39 | 7 |
math | 19. In $\triangle A B C$, $A B=A C, \angle A=100^{\circ}, I$ is the incenter, $D$ is a point on $A B$ such that $B D=B I$. Find the measure of $\angle B C D$.
(Problem 1073 from Mathematical Bulletin) | 30 | 75 | 2 |
math | C2
(a) Find the distance from the point $(1,0)$ to the line connecting the origin and the point $(0,1)$.
(b) Find the distance from the point $(1,0)$ to the line connecting the origin and the point $(1,1)$.
(c) Find the distance from the point $(1,0,0)$ to the line connecting the origin and the point $(1,1,1)$. | \frac{\sqrt{2}}{2} | 91 | 10 |
math | 5. Let $x, y, z \in \mathbf{R}_{+}$, satisfying $x+y+z=x y z$. Then the function
$$
\begin{array}{l}
f(x, y, z) \\
=x^{2}(y z-1)+y^{2}(z x-1)+z^{2}(x y-1)
\end{array}
$$
has the minimum value of $\qquad$ | 18 | 93 | 2 |
math | ## Task A-2.1.
Which number has more divisors in the set of natural numbers, $2013^{2}$ or 20480? | 2013^{2} | 37 | 7 |
math | 5. Given $n$ points $A_{1}, A_{2}, \cdots, A_{n}(n \geqslant 3)$ in the plane, no three of which are collinear. By selecting $k$ pairs of points, determine $k$ lines (i.e., draw a line through each pair of the $k$ pairs of points), such that these $k$ lines do not form a triangle with all three vertices being given poi... | \frac{n^2}{4} \text{ if } n \text{ is even; } \frac{n^2-1}{4} \text{ if } n \text{ is odd.} | 119 | 44 |
math | 8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 2343 | 48 | 4 |
math | IMO 1968 Problem A2 Find all natural numbers n the product of whose decimal digits is n 2 - 10n - 22. | 12 | 34 | 2 |
math | ## 8. Wooden Numbers
Little Marko is playing with numbers made of wood. He has two number 1s, seven number 2s, and one number 3. He wants to string them together one by one so that the numbers 1 and 3 are not adjacent. How many different ten-digit numbers can Marko get this way?
Result: $\quad \mathbf{2 2 4}$ | 224 | 87 | 3 |
math | ## Condition of the problem
Find the derivative.
$$
y=\operatorname{ctg}(\cos 5)-\frac{1}{40} \cdot \frac{\cos ^{2} 20 x}{\sin 40 x}
$$ | \frac{1}{4\sin^{2}20x} | 56 | 15 |
math | Problem 7. For what least value of the parameter $a$ will the coefficient of $x^{4}$ in the expansion of the polynomial $\left(1-3 x+a x^{2}\right)^{8}$ be equal to $70 ?$ | -50 | 54 | 3 |
math | 1. [5 points] $S$ is the sum of the first 15 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{7} a_{16}>S-24, a_{11} a_{12}<S+4$. Determine all possible values of $a_{1}$. | -5;-4;-2;-1 | 89 | 8 |
math | Four, (Full marks 20 points, Sub-question (1) 6 points, Sub-question (2) 14 points) Let the two real roots of $x^{2}-p x+q=0$ be $\alpha, \beta$.
(1) Find the quadratic equation whose roots are $\alpha^{3}, \beta^{3}$;
(2) If the quadratic equation whose roots are $\alpha^{3}, \beta^{3}$ is still $x^{2}-p x$ $+q=0$, fi... | x^{2}=0, x^{2}-x=0, x^{2}+x=0, x^{2}-2 x+1=0, x^{2}+2 x+1=0, x^{2}-1=0 | 118 | 53 |
math | 4. Points $A, B, C, D, E$ are sequentially located on a line, such that $A B=4, B C=D E=2, C D=3$. Circles $\Omega$ and $\omega$, touching each other, are such that $\Omega$ passes through points $D$ and $E$, and $\omega$ passes through points $B$ and $C$. Find the radii of circles $\Omega$ and $\omega$, given that the... | r=\frac{9\sqrt{3}}{2\sqrt{17}},R=\frac{8\sqrt{3}}{\sqrt{17}} | 153 | 34 |
math | Determine the integers $m \geqslant 2, n \geqslant 2$ and $k \geqslant 3$ having the following property: $m$ and $n$ each have $k$ positive divisors and, if we denote $d_{1}<\ldots<d_{k}$ the positive divisors of $m$ (with $d_{1}=1$ and $d_{k}=m$) and $d_{1}^{\prime}<\ldots<d_{k}^{\prime}$ the positive divisors of $n$ ... | (4,9,3)(8,15,4) | 194 | 14 |
math | [ Arithmetic. Mental arithmetic, etc.] $[$ Work problems $]$
Three diggers dug three holes in two hours. How many holes will six diggers dig in five hours?
# | 15 | 38 | 2 |
math | 3. [5 points] Solve the system of equations
$$
\left\{\begin{array}{l}
\left(\frac{x^{4}}{y^{2}}\right)^{\lg y}=(-x)^{\lg (-x y)} \\
2 y^{2}-x y-x^{2}-4 x-8 y=0
\end{array}\right.
$$ | (-4;2),(-2;2),(\frac{\sqrt{17}-9}{2};\frac{\sqrt{17}-1}{2}) | 81 | 34 |
math | 32. (USA 3) The vertex \( A \) of the acute triangle \( ABC \) is equidistant from the circumcenter \( O \) and the orthocenter \( H \). Determine all possible values for the measure of angle \( A \). | 60^{\circ} | 56 | 6 |
math | Three. (50 points) Given a finite set of planar vectors $M$, for any three elements chosen from $M$, there always exist two elements $\boldsymbol{a}, \boldsymbol{b}$ such that $\boldsymbol{a}+\boldsymbol{b} \in M$. Try to find the maximum number of elements in $M$.
| 7 | 75 | 1 |
math | 24. C4 (BUL) Let \( T \) be the set of ordered triples \((x, y, z)\), where \( x, y, z \) are integers with \( 0 \leq x, y, z \leq 9 \). Players \( A \) and \( B \) play the following guessing game: Player \( A \) chooses a triple \((x, y, z)\) in \( T \), and Player \( B \) has to discover \( A \)'s triple in as few m... | 3 | 212 | 1 |
math | In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled w... | \left\lceil \frac{n}{2} \right\rceil | 180 | 16 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(x ; 0 ; 0)$
$B(1 ; 5 ; 9)$
$C(3 ; 7 ; 11)$ | A(18;0;0) | 62 | 9 |
math | 12. There are 8 black, 8 white, and 8 yellow chopsticks mixed together. In the dark, you want to take out two pairs of chopsticks of different colors. How many chopsticks do you need to take out to ensure you meet the requirement? | 11 | 57 | 2 |
math | 10.362 The hypotenuse of a right triangle is equal to $m$, the radius of the inscribed circle is $r$. Determine the legs. Under what relation between $r$ and $\boldsymbol{m}$ does the problem have a solution? | \frac{2r+\\sqrt{^{2}-4r^{2}-4r}}{2} | 56 | 23 |
math | 6. Players divide the chips. The first player takes $m$ chips and a sixth of the remainder; the second $-2 m$ chips and a sixth of the new remainder; the third $-3 m$ chips and a sixth of the new remainder, and so on. It turned out that the chips were divided equally in this way. How many players were there? | 5 | 77 | 1 |
math | Determine the number of triples $(a, b, c)$ of three positive integers with $a<b<c$ whose sum is 100 and whose product is 18018 . | 2 | 41 | 1 |
math | Example 1 Find all integer values of $a$ such that the equation $(a+1) x^{2}-\left(a^{2}+1\right) x+2 a^{3}-6=0$ has integer roots. (1996, Huanggang Region, Hubei Junior High School Mathematics Competition) | a=-1,0,1 | 69 | 7 |
math | 9. Buratino the Statistician (from 7th grade, 2 points). Every month, Buratino plays in the "6 out of 45" lottery organized by Karabas-Barabas. In the lottery, there are 45 numbered balls, and in each draw, 6 random winning balls are drawn.
Buratino noticed that in each subsequent draw, there are no balls that appeare... | 7 | 180 | 1 |
math | 5. Solve the equation $a^{b}+a+b=b^{a}$ in natural numbers.
(O. A. Pyayve, E. Yu. Voronetsky) | =5,b=2 | 38 | 5 |
math | Problem 2. Given an isosceles triangle $ABC$ with base $AB$, where the length of the base $a$ is half the length of the leg $b$.
a) Calculate the lengths of the base $a$ and the leg $b$, if the perimeter of the triangle is $150 \text{~mm}$.
b) Calculate the length of the side of an equilateral triangle whose perimete... | 30 | 99 | 2 |
math | Problem 8. For what values of the parameter $a$ does the equation
$$
3^{x^{2}-2 a x+a^{2}}=a x^{2}-2 a^{2} x+a^{3}+a^{2}-4 a+4
$$
have exactly one solution? | 1 | 65 | 1 |
math | $ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$? | \frac{1}{3} | 48 | 7 |
math | 7. Two identical resistors with resistance $R$ each are connected in series and connected to a source of constant voltage $U$. An ideal voltmeter is connected in parallel to one of the resistors. Its readings were $U_{v}=10 B$. After that, the voltmeter was replaced with an ideal ammeter. The ammeter readings were $-I_... | 2 | 102 | 1 |
math | 6. For $n$ an integer, evaluate
$$
\lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^{2}-0^{2}}}+\frac{1}{\sqrt{n^{2}-1^{2}}}+\cdots+\frac{1}{\sqrt{n^{2}-(n-1)^{2}}}\right)
$$ | \frac{\pi}{2} | 81 | 7 |
math | 8. Find all prime numbers $p$ such that there exists an integer-coefficient polynomial
$$
f(x)=x^{p-1}+a_{p-2} x^{p-2}+\cdots+a_{1} x+a_{0} \text {, }
$$
satisfying that $f(x)$ has $p-1$ consecutive positive integer roots, and $p^{2} \mid f(\mathrm{i}) f(-\mathrm{i})$, where $\mathrm{i}$ is the imaginary unit.
(Yang X... | p \equiv 1(\bmod 4) | 116 | 11 |
math | 2. From place $A$ to place $B$, two groups of tourists start simultaneously. The first group starts by bus, traveling at an average speed of $20 \mathrm{~km} / \mathrm{s}$ and reaches place $C$, which is halfway between $A$ and $B$, and then walks. The second group starts walking, and after 1 hour, they board a bus tra... | 30 | 202 | 2 |
math | Let \[T_0=2, T_1=3, T_2=6,\] and for $n\ge 3$, \[T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.\] The first few terms are \[2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.\] Find a formula for $T_n$ of the form \[T_n=A_n+B_n,\] where $\{A_n\}$ and $\{B_n\}$ are well known sequences. | T_n = n! + 2^n | 163 | 10 |
math | LVII OM - I - Problem 1
Determine all non-negative integers $ n $ for which the number
$ 2^n +105 $ is a perfect square of an integer. | 4,6,8 | 41 | 5 |
math | [ $\quad$ Number of divisors and their sum of a number [Fundamental Theorem of Arithmetic. Factorization into prime factors ]
a) Find the number $k$, which is divisible by 2 and 9 and has exactly 14 divisors (including 1 and $k$ ).
b) Prove that if 14 is replaced by 15, the problem will have multiple solutions, and i... | 1458 | 105 | 4 |
math | $a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$ | 628 | 91 | 3 |
math | Find the sum of the two smallest odd primes $p$ such that for some integers $a$ and $b$, $p$ does not divide $b$, $b$ is even, and $p^2=a^3+b^2$.
[i]2021 CCA Math Bonanza Individual Round #13[/i] | 122 | 71 | 3 |
math | 7.1. The angle formed by the bisector of angle $A B C$ with its sides is 6 times smaller than the angle adjacent to angle $A B C$. Find angle $A B C$. | 45 | 44 | 2 |
math | 9. (16 points) Given that $f(x)$ is a function defined on the set of real numbers $\mathbf{R}$, $f(0)=2$, and for any $x \in \mathbf{R}$, we have
$$
\begin{array}{l}
f(5+2 x)=f(-5-4 x), \\
f(3 x-2)=f(5-6 x) .
\end{array}
$$
Find the value of $f(2012)$. | 2 | 113 | 1 |
math | # Task 5. Maximum 20 points
In Moscow, a tennis tournament is being held. Each team consists of 3 players. Each team plays against every other team, with each participant of one team playing against each participant of the other exactly one match. Due to time constraints, a maximum of 150 matches can be played in the ... | 6 | 89 | 1 |
math | Example 4.26. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{(x-2)^{n}}{n^{2} 2^{n}}$. | x\in[0,4] | 47 | 8 |
math | 13.27 Given a precise balance scale where weights can be placed on both the left and right pans, to measure the masses of $1,2, \cdots, n(n \in N)$ grams of $n$ steel balls respectively, what is the minimum number of weights needed? | {\begin{pmatrix}\log_{3}(2n+1),\text{when}2n+1\text{ispowerof}3,\{[\log_{3}(2n+1)]+1,\quad\text{otherwise.}}\end{pmatrix}.} | 61 | 61 |
math | Let $p$ be a prime number. Determine the remainder of the Euclidean division of $1^{k}+\cdots+(p-1)^{k}$ for any integer $k \geq 0$ by $p$ (without primitive root). | p-1 | 54 | 3 |
math | 7. If $\alpha, \beta, \gamma$ are acute angles, and $\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1$, then the maximum value of $\frac{\sin \alpha+\sin \beta+\sin \gamma}{\cos \alpha+\cos \beta+\cos \gamma}$ is
$\qquad$ . | \frac{\sqrt{2}}{2} | 84 | 10 |
math | Kalinin D.A.
On a line, several points were marked. After that, between each pair of adjacent points, one more point was marked. This "densification" was repeated two more times (a total of three times). As a result, 113 points were marked on the line. How many points were initially marked? | 15 | 70 | 2 |
math | 430. A man, on his death, left a will according to which the eldest son receives 100 francs and one-tenth of the remainder, the second son - 200 francs and one-tenth of the new remainder, the third - 300 francs and one-tenth of the new remainder, and so on until the last. In this way, the shares of all the sons should ... | 8100, | 118 | 5 |
math | 19. [11] Let $P$ be a polynomial with $P(1)=P(2)=\cdots=P(2007)=0$ and $P(0)=2009$ !. $P(x)$ has leading coefficient 1 and degree 2008. Find the largest root of $P(x)$. | 4034072 | 76 | 7 |
math | 7. The solution set of the inequality $\sqrt{2 x+5}>x+1$ is | -\frac{5}{2}\leqslantx<2 | 21 | 14 |
math | \section*{Problem 6}
Find all integers \(x, y\) satisfying \(x^{2}+x=y^{4}+y^{3}+y^{2}+y\).
| x,y=-1,1;0,-1;-1,0;0,0;-6,2;5,2 | 42 | 26 |
math | 5. If $a>b>c, a+b+c=0$, and $x_{1}, x_{2}$ are the two real roots of $a x^{2}+b x+c=0$. Then the range of $\left|x_{1}^{2}-x_{2}^{2}\right|$ is $\qquad$ | [0,3) | 70 | 5 |
math | # 8. Variant 1.
101 natural numbers are written in a circle. It is known that among any 5 consecutive numbers, there will be at least two even numbers. What is the minimum number of even numbers that can be among the written numbers? | 41 | 55 | 2 |
math | 2. Find all real numbers $x$ for which
$$
\left(x^{2}-7 x+11\right)^{x^{2}+5 x-6}=1
$$ | 1,2,3,4,5,-6 | 42 | 11 |
math | 6. In $\triangle A B C$, $\sin ^{2} A+\sin ^{2} C=2018 \sin ^{2} B$, then $\frac{(\tan A+\tan C) \tan ^{2} B}{\tan A+\tan B+\tan C}=$ | \frac{2}{2017} | 66 | 10 |
math | 6. Cheburashka bought as many mirrors from Galina in the store as Gen bought from Shapoklyak. If Gen had bought from Galina, he would have 27 mirrors, and if Cheburashka had bought from Shapoklyak, he would have 3 mirrors. How many mirrors would Gen and Cheburashka buy together if Galina and Shapoklyak agreed and set t... | 18 | 133 | 2 |
math | 34 chameleons live on an island. At the beginning, there are 7 yellow, 10 red, and 17 green ones. When two chameleons of different colors meet, they simultaneously adopt the third color. One day, Darwin arrives on the island and observes that all the chameleons are of the same color. What is this color? | green | 79 | 1 |
math | 4-181 Find all integer pairs $(a, b)$, where $a \geqslant 1, b \geqslant 1$, and satisfy the equation $a^{b^{2}}=b^{a}$. | (,b)=(1,1),(16,2),(27,3) | 52 | 18 |
math | \section*{Problem 1 - 041041}
The 30 prize winners of a student competition are to be awarded with newly published specialist books.
Three different types of books, valued at \(30 \mathrm{M}, 24 \mathrm{M}\) and \(18 \mathrm{M}\) respectively, are available. At least one book of each type must be purchased.
What are... | \begin{pmatrix}30\mathrm{M}&24\mathrm{M}&18\mathrm{M}\\4&2&24\\3&4&23\\2&6&22\\1&8&21\end{pmatrix} | 123 | 60 |
math | ## Task A-1.1.
Determine all three-digit numbers with the sum of digits 11, such that by swapping the units and hundreds digits, a number 594 greater is obtained. | 137,218 | 43 | 7 |
math | 14. The function $f(x)$ defined on the interval $[1,2017]$ satisfies $f(1)=f(2017)$, and for any $x, y \in [1,2017]$, it holds that $|f(x)-f(y)| \leqslant 2|x-y|$. If the real number $m$ satisfies that for any $x, y \in [1,2017]$, it holds that $|f(x)-f(y)| \leqslant m$, find the minimum value of $m$. | 2016 | 128 | 4 |
math | Example 1 Find the range of the function $y=\frac{x^{2}-x}{x^{2}-x+1}$. | [-\frac{1}{3},1) | 28 | 10 |
math | [ [ equations in integers ]
Solve the equation $12 a+11 b=2002$ in natural numbers.
# | (11,170),(22,158),(33,146),(44,134),(55,122),(66,110),(77,98),(88,86),(88,74),(99,62),(110,50),(12 | 29 | 76 |
math | Let's find the integer solutions for \(x\) and \(y\) in the following equation:
$$
2 x^{2}+8 y^{2}=17 x y-423
$$ | 11,19-11,-19 | 42 | 11 |
math | 2. Non-zero numbers $a$ and $b$ are roots of the quadratic equation $x^{2}-5 p x+2 p^{3}=0$. The equation $x^{2}-a x+b=0$ has a unique root. Find $p$. | 3 | 56 | 1 |
math | Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define $$f(A)=\sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.$$If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the... | 441 | 174 | 3 |
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