task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-4 ; 0 ; 4), B(-1 ; 6 ; 7), C(1 ; 10 ; 9)$ | 1 | 60 | 1 |
math | 12.432 The side of the lower base of a regular truncated quadrilateral pyramid is 5 times the side of the upper base. The lateral surface area of the pyramid is equal to the square of its height. Find the angle between the lateral edge of the pyramid and the plane of the base. | \operatorname{arctg}\sqrt{9+3\sqrt{10}} | 63 | 19 |
math | Example 2 Find all integer triples $(x, y, z)$ such that:
$$
x^{3}+y^{3}+z^{3}-3 x y z=2003 .
$$
(2003 17th Nordic Mathematical Contest) | (668,668,667),(668,667,668),(667,668,668) | 58 | 37 |
math | Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
[color=#0000FF]Alice:[/color] Are you going to cover your keys?
[color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys.
[color=... | 2 | 240 | 1 |
math | The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers.
Translating the text as requested, while maintaining the original formatting and line breaks. | 3:2 | 44 | 3 |
math | A1. John had a box of candies. On the first day he ate exactly half of the candies and gave one to his little sister. On the second day he ate exactly half of the remaining candies and gave one to his little sister. On the third day he ate exactly half of the remaining candies and gave one to his little sister, at whic... | 14 | 87 | 2 |
math | 4. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for $m, n \in \mathbb{N}$ and $m>n$ we have
$$
f(f(m+n)+f(m-n))=8 m
$$ | f(n)=2n | 61 | 5 |
math | 2. A 200-digit natural number had one of its digits erased. As a result, the number decreased by 5 times. Find all numbers for which this is possible. | 125\cdot10^{197}for=1,2,3 | 38 | 19 |
math | Example 9 Find the greatest common divisor of $198,252,924$, and express it as an integer linear combination of 198,252 and 924. | 6 = 924 - 204 \cdot 252 + 255 \cdot 198 | 45 | 28 |
math | 2. (3 points) On the Island of Misfortune, there live knights who always tell the truth, and liars who always lie. One day, $n$ islanders gathered in a room.
The first one said: "Exactly 1 percent of those present in this room are liars."
The second one said: "Exactly 2 percent of those present in this room are liars... | 100 | 138 | 3 |
math | A number, when divided by 7, gives a remainder of 2, and when divided by 8, gives a remainder of 4. In the former case, the quotient is 7 greater than in the latter case. What is the number? | 380 | 52 | 3 |
math | Five. (20 points) Given
$$
f(x)=\frac{x^{4}+k x^{2}+1}{x^{4}+x^{2}+1}, k, x \in \mathbf{R} \text {. }
$$
(1) Find the maximum and minimum values of $f(x)$;
(2) Find all real numbers $k$ such that for any three real numbers $a, b, c$, there exists a triangle with side lengths $f(a)$, $f(b)$, and $f(c)$. | -\frac{1}{2}<k<4 | 120 | 10 |
math | 7. In the tetrahedron $P-ABC$, $PB \perp AC$, $PH$ $\perp$ plane $ABC$ at point $H$, $H$ is inside $\triangle ABC$, $PB$ makes a $30^{\circ}$ angle with plane $ABC$, the area of $\triangle PAC$ is 1. When the dihedral angle $P-AC-B$ is $\qquad$, $S_{\triangle ABC}$ is maximized. | 60 | 102 | 2 |
math | For the school birthday party, Ana, Pedro, Miriam, and Fábio brought a total of 90 sweets together. Their teacher observed that:
- if Ana had brought 2 more sweets;
- if Pedro had brought 2 fewer sweets;
- if Miriam had brought twice as many;
- if Fábio had brought half as many;
all four friends would have brought th... | A=18,P=22,M=10,F=40 | 93 | 16 |
math | ## Task Condition
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(2, -1, 1)$
$$
\frac{x-4.5}{1}=\frac{y+3}{-0.5}=\frac{z-2}{1}
$$ | M^{\}(3;-3;-1) | 72 | 10 |
math | . For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16 n)}$.
## Answer: 13 | 13 | 49 | 2 |
math | 11.175. A right parallelepiped is described around a sphere, with the diagonals of the base being $a$ and $b$. Determine the total surface area of the parallelepiped. | 3ab | 44 | 2 |
math | 13. A circle is inscribed in $\triangle A B C$ with sides $A B=4, B C=6$, and $A C=8$. If $P$ and $Q$ are the respective points of tangency of $\overline{A B}$ and $\overline{A C}$ with the circle, determine the length of chord $P Q$. | \frac{3\sqrt{10}}{4} | 79 | 13 |
math | Example 5 Find the number of integers $n$ greater than 1, such that for any integer $a$, $n \mid a^{25}-a$.
Find the number of integers $n$ greater than 1, such that for any integer $a$, $n \mid a^{25}-a$. | 31 | 68 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{x^{3}-2 x-1}{x^{4}+2 x+1}$ | -\frac{1}{2} | 42 | 7 |
math | Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geqslant 2 + u_k$ for all $k \geqslant 0$; and $u_{\ell+5} \leqslant 5 + u_\ell$ for all $\ell \geqslant 0$. Find all possible values for the integer $u_{2023}$. | 2122, 2123, 2124, 2125 | 113 | 22 |
math | Dorichenko S.A.
Which are there more of: rectangles with integer sides and a perimeter of 1996, or rectangles with integer sides and a perimeter of $1998$?
(Rectangles $a \times b$ and $b \times a$ are considered the same.) | 499 | 63 | 3 |
math | 86. The opposite sides of a quadrilateral inscribed in a circle intersect at points $P$ and $Q$. Find the length of the segment $|P Q|$, if the tangents to the circle drawn from $P$ and $Q$ are equal to $a$ and $b$. | |PQ|=\sqrt{^{2}+b^{2}} | 63 | 15 |
math | Let $n>1$ be an integer. An $n \times n \times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \times n \times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed... | \frac{n(n+1)(2 n+1)}{6} | 177 | 15 |
math | During a math class, the teacher wrote a number on the board. One student said, "The number is divisible by 31." The second student said, "The number is also divisible by 30." A third student claimed the number was divisible by 29, a fourth said it was divisible by 28, and so on, until the thirtieth student said the nu... | 1617 | 119 | 4 |
math | It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b... | 14 | 222 | 2 |
math | 2. [6 points] Solve the equation $\sqrt{x+4}-\sqrt{6-x}+4=2 \sqrt{24+2 x-x^{2}}$. | 5,\frac{2-3\sqrt{11}}{2} | 38 | 16 |
math | Problem 10.2. A real number $a$ is such that of the two equations
$$
5+|x-2|=a \text { and } 7-|2 x+6|=a
$$
one has exactly one solution, and the other has exactly two solutions. What can $a$ be? List all possible options. | 5,7 | 75 | 3 |
math | 6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\triangle P F_{1} F_{2}$ opposite to $\angle P F_{1} F_{2}$ is $I$. The line $P I$ intersects the $x$-axis at point $Q$... | 4 | 169 | 1 |
math | 4. For a given positive integer $n$. Try to find the smallest positive integer $d_{n}$ that cannot be expressed in the form $\sum_{i=1}^{n}(-1)^{a_{i}} \times 2^{b_{i}}$, where $a_{i} 、 b_{i}(i=1,2, \cdots, n)$ are all non-negative integers. | \frac{2^{2 n+1}+1}{3} | 87 | 15 |
math | Task B-3.8. In a basketball tournament, 8 teams participated and each played one match against each other. For a win, 2 points are awarded, and for a loss, 0 points (no match ended in a draw). The teams accumulated 14, 12, 8, 8, 6, 4, 2, 2 points respectively. How many matches did the last 4 teams lose to the first 4 t... | 15 | 99 | 2 |
math | 7. Given prime numbers $p$ and $q$ satisfy $q^{5}-2 p^{2}=1$. Then $p+q=$ $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 14 | 60 | 2 |
math | Determine all solutions $(n, k)$ of the equation $n!+An = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$. | (2, 4) | 55 | 7 |
math | 3. Throw a die three times, find the probability of getting exactly two ones.
The above text is translated into English, keeping the original text's line breaks and format. | \frac{5}{72} | 35 | 8 |
math | ## Task Condition
Find the derivative.
$$
y=x(\arcsin x)^{2}+2 \sqrt{1-x^{2}} \arcsin x-2 x
$$ | (\arcsinx)^{2} | 41 | 8 |
math | 13. In $\triangle A B C$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, $b=1$, and $\cos C+(2 a+c) \cos B=0$.
(1) Find $B$;
(2) Find the maximum area of $\triangle A B C$. | \frac{\sqrt{3}}{12} | 75 | 11 |
math | 5. In $\triangle A B C$, if $\tan A \tan B=\tan A \tan C+\tan C \tan B$, then $\frac{a^{2}+b^{2}}{c^{2}}=$ | 3 | 48 | 1 |
math | 7.001. $\sqrt{25^{\frac{1}{\log _{6} 5}}+49^{\frac{1}{\log _{8} 7}}}$ | 10 | 45 | 2 |
math | 1. $A, B, C, D, E$ five people participate in a chess tournament together and find that their average age is exactly 28 years old. One year later, $A, B, C, D, F$ participate together and find that their average age is exactly 30 years old. How many years older is $F$ than $E$? | 5 | 79 | 1 |
math | 5 (1133). For what value of $a$ does the sum of the squares of the roots of the quadratic trinomial $x^{2}-(a-2) x-a-1$ take the smallest value? | 1 | 49 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 8} \frac{\sqrt{9+2 x}-5}{\sqrt[3]{x}-2}$ | \frac{12}{5} | 42 | 8 |
math | ## Task A-2.1.
Determine the radius of the base of a cone whose slant height is 1, so that the difference between the area of its lateral surface and the area of its base is maximized. | \frac{1}{2} | 47 | 7 |
math | 1082. How many terms of the sum
$$
1+2+3+\ldots+n
$$
are needed to get a three-digit number that is written with identical digits? | 36 | 41 | 2 |
math | 6.24 In a geometric sequence with a common ratio greater than 1, what is the maximum number of terms that are integers between 100 and 1000.
(4th Canadian Mathematics Competition, 1972) | 6 | 52 | 1 |
math | Solve the following system of equations:
$$
\begin{gathered}
\frac{x^{2}+x y+y^{2}}{x^{2}-x y+y^{2}}=3 \ldots \\
x^{3}+y^{3}=2 \ldots
\end{gathered}
$$ | 1,1 | 67 | 3 |
math | 8. Randomly select three different vertices from a regular 11-sided polygon, the probability that they form an acute triangle is $\qquad$ . | \frac{1}{3} | 31 | 7 |
math | 3. Given $x^{2}-x-1=0$. Then, the value of the algebraic expression $x^{3}$ $-2 x+1$ is $\qquad$ . | 2 | 41 | 1 |
math | 6・119 Every natural number $n$ greater than 2 can be expressed as the sum of several pairwise distinct natural numbers. Let the maximum number of terms be $A(n)$. Find $A(n)$ (expressed in terms of $n$). | A(n)=\left[\frac{\sqrt{8 n+1}-1}{2}\right] | 55 | 21 |
math | 1. (10 points) Chess clubs from Moscow, Saint Petersburg, and Kazan agreed to hold a tournament. Each Muscovite played exactly 9 Saint Petersburg residents and $n$ Kazan residents. Each Saint Petersburg resident played exactly 6 Muscovites and 2 Kazan residents. Each Kazan resident played exactly 8 Muscovites and 6 Sai... | 4 | 85 | 1 |
math | Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could po... | \frac{\pi}{4} | 78 | 7 |
math | 6.2. Find all solutions to the puzzle: ARKA + RKA + KA + A = 2014. (Different letters correspond to different digits, and the same letters correspond to the same digits.) | 1471+471+71+1=2014 | 45 | 18 |
math | 3. Solve the equation
$$
2 x+2+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+2) \cdot \sqrt{x^{2}+4 x+5}=0
$$ | -1 | 61 | 2 |
math | 10. (1991, 2nd "Hope" Math Competition) Let $\{x\}$ denote the smallest integer not less than the real number $x$. Then the value of $\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}$ is what? | 19854 | 110 | 5 |
math | In a bus with $n$ seats, all tickets are sold to $n$ passengers. The first to enter the bus is the Absent-Minded Scholar, who, without looking at the ticket, takes the first available seat. Subsequently, passengers enter one by one. If an entering passenger sees that their seat is free, they take their seat. If their s... | \frac{1}{2} | 106 | 7 |
math | Example 2 Find all integers $n>1$, such that $\frac{2^{n}+1}{n^{2}}$ is an integer. | n=3 | 32 | 3 |
math | 2. (12 points) In a family, there are four children of different ages. Their total age is 31 years. Four years ago, the total age of all the children in the family was 16 years, 7 years ago it was 8 years, and 11 years ago it was 1 year. How old are the children at present? (Age is always expressed as a whole number of... | 3,6,10,12 | 90 | 9 |
math | 3. Let the midpoint of the chord $AB$ of the circle $x^{2}+y^{2}=9$ be $(1,1)$, and the line $AB$ intersects the $x$-axis at point $P$, then $|PA| \cdot |PB|=$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 5 | 88 | 1 |
math | 7. Given positive real numbers $a, b$ satisfy $a b(a+b)=4$. Then the minimum value of $2 a+b$ is $\qquad$ .
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2\sqrt{3} | 61 | 6 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\sqrt{1-2 x+x^{2}}-(1+x)}{x}
$$ | -2 | 44 | 2 |
math | Murashkin M.V.
On each cell of a $10 \times 10$ board, there is a chip. It is allowed to choose a diagonal with an even number of chips and remove any chip from it.
What is the maximum number of chips that can be removed from the board using such operations? | 90 | 65 | 2 |
math | Example 17. Given as shown, in quadrilateral $ABCD$, $AD=DC=1, \angle DAB=$ $\angle DCB=90^{\circ}, BC, AD$ extended intersect at $P$. Find the minimum value of $AB \cdot S_{\triangle PAB}$.
(1994, Sichuan Province Junior High School Mathematics League Competition) | 4 | 83 | 1 |
math | 7. Let the four vertices of a regular tetrahedron be $A, B, C, D$, with each edge length being 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to that vertex with equal probability, and crawls all the way to the end of ... | \frac{182}{729} | 105 | 11 |
math | Example 7 A positive integer $n$ cannot be divisible by $2$ or $3$, and there do not exist non-negative integers $a$, $b$ such that $\left|2^{a}-3^{b}\right|=n$. Find the minimum value of $n$.
(2003, National Training Team Problem) | 35 | 71 | 2 |
math | $6 \cdot 156$ Find all continuous functions $f:(1,+\infty) \rightarrow R$ satisfying
$$
f(x y) \equiv x f(y)+y f(x), x, y>1
$$ | f(x)=\lnx | 51 | 6 |
math | Example 12 Find the minimum value of $\frac{3}{\cos x}+\frac{2}{\sin x}\left(0<x<\frac{\pi}{2}\right)$. | \sqrt{(\sqrt[3]{9}+\sqrt[3]{4})^{3}} | 42 | 20 |
math | A4. The equations $|x|^{2}-3|x|+2=0$ and $x^{4}-a x^{2}+4=0$ have the same roots. Determine the value of $a$. | 5 | 48 | 1 |
math | 815. Find the differentials of the functions:
1) $y=x^{3}$ at the point $x=0$, if $\left.\Delta x=0.3 ; 2\right) y=x^{3}-x^{2}+$ $+3 x-1$; 3) $r=\varphi^{4}+2^{\sin 3 \varphi}$; 4) $y=\ln \left(x^{2}+1\right)+\operatorname{arctg} \sqrt{x}$ at the point $x=1$, if $\Delta x=0.1$. | 0.125 | 132 | 5 |
math | Example 5 Given a positive integer $n$ that satisfies the following condition: In any $n$ integers greater than 1 and not exceeding 2009 that are pairwise coprime, at least one is a prime number. Find the minimum value of $n$. ${ }^{[2]}$ | 15 | 64 | 2 |
math | Solve the following system of equations:
$$
x^{2}+x y+y^{2}=7
$$
$$
x^{2}+x z+z^{2}=13
$$
$$
y^{2}+y z+z^{2}=19
$$ | \begin{aligned}&y_{1}=2,y_{2}=-2,y_{3}=\frac{1}{3}\sqrt{3},y_{4}=-\frac{1}{3}\sqrt{3}\\&z_{1}=3,z_{2}=-3,z_{3}=\frac{7}{3}\sqrt{3},z_{4}=-\frac{7}{3}\sqrt | 60 | 87 |
math | ## Task 1 - 230621
From a dairy farm, 2200 crates each containing 25 containers of $\frac{1}{4}$ liter of milk, 600 crates each containing 24 bottles of $\frac{1}{2}$ liter, and 800 crates each containing 12 bags of 1 liter of milk are to be delivered in one day.
The total amount of milk required for this was deliver... | 4 | 159 | 1 |
math | In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the... | n-2 | 112 | 4 |
math | Example 23 (2003 Anhui Provincial Competition Question) Given that $x$, $y$, and $z$ are all positive integers, the equation $x+y+z=15$ has $\qquad$ groups of solutions. | 91 | 52 | 2 |
math | For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition. | h(r) = 2r | 104 | 8 |
math | A pair of positive integers $m, n$ is called [i]guerrera[/i], if there exists positive integers $a, b, c, d$ such that $m=ab$, $n=cd$ and $a+b=c+d$. For example the pair $8, 9$ is [i]guerrera[/i] cause $8= 4 \cdot 2$, $9= 3 \cdot 3$ and $4+2=3+3$. We paint the positive integers if the following order:
We start paintin... | \mathbb{Z}^+ \setminus \{1, 2\} | 182 | 19 |
math | 33 Find the largest real number $\lambda$, such that for a real-coefficient polynomial $f(x)=x^{3}+a x^{2}+c$ with all roots being non-negative real numbers, if $x \geqslant 0$, then $f(x) \geqslant \lambda(x-a)^{3}$, and find the condition for equality. | -\frac{1}{27} | 81 | 8 |
math | 2. On an island, there live liars and knights. Knights always tell the truth, and liars always lie. Each inhabitant of the island knows whether each of the others is a knight or a liar. One day, 19 islanders met. Three of them said: "Exactly three of us are liars," then six of the others said: "Exactly six of us are li... | 9,18,19 | 123 | 7 |
math | 8.6. Let the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a x+b, a, b \in \mathbb{R}^{*}$. If $m, n$ are distinct natural numbers such that
$$
\begin{gathered}
n>2, m>2, f(1)+f(2)+\cdots+f(m)=-n \text { and } f(1)+f(2)+\cdots+f(n)=-m \text {, calculate } \\
\frac{f(1)+f(2)+\cdots+f(m+n)}{m+n}
\end{gathered}
$$ | 1 | 148 | 1 |
math | 10. $P$ is a point inside $\triangle ABC$, and $D, E, F$ are the feet of the perpendiculars from $P$ to $BC, CA, AB$ respectively. Find all points $P$ that minimize $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$. (22nd IMO Problem) | P \text{ is the incenter of } \triangle ABC | 82 | 13 |
math | There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines?
We say that a plane is determined if it contains at least two of the lines. | \frac{n(n-1)}{2} | 60 | 10 |
math | Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\tfrac{p}{q}... | 89 | 108 | 2 |
math | ## Task 6 - 191236
Given a positive real number $a$.
Determine (for each value of this given number $a$) all real solutions $(x ; y)$ of the system of equations
$$
x^{5}+y^{5}=1 \quad, \quad x+y=a
$$ | (x,y)=(\frac{1}{2}\\frac{1}{2}\sqrt{2\sqrt{\frac{^{5}+4}{5}}-^{2}},\frac{1}{2}\\frac{1}{2}\sqrt{2\sqrt{\frac{^{5}+4}{5}}-^{2}}) | 72 | 70 |
math | Find the sum of the prime factors of $67208001$, given that $23$ is one.
[i]Proposed by Justin Stevens[/i] | 781 | 37 | 3 |
math | ## Task B-4.6.
Let $f_{1}(x)=\frac{1}{2-x}, f_{n}(x)=\left(f_{1} \circ f_{n-1}\right)(x), n \geqslant 2$, for all real numbers $x$ for which the given functions are defined. What is $f_{2021}(4)$? | \frac{6059}{6062} | 85 | 13 |
math | \section*{Problem 1 - 031041}
A cyclist is riding over a bridge at a constant speed. When he has covered \(\frac{3}{8}\) of the distance, he meets another cyclist coming towards him at the same speed.
At what speed were both cycling if a car traveling at \(80 \frac{\mathrm{km}}{\mathrm{h}}\) on the same road met one ... | 20\frac{\mathrm{}}{} | 103 | 9 |
math | Let $A$ and $B$ be points on circle $\Gamma$ such that $AB=\sqrt{10}.$ Point $C$ is outside $\Gamma$ such that $\triangle ABC$ is equilateral. Let $D$ be a point on $\Gamma$ and suppose the line through $C$ and $D$ intersects $AB$ and $\Gamma$ again at points $E$ and $F \neq D.$ It is given that points $C, D, E, F$ are... | \frac{38\pi}{15} | 138 | 11 |
math | 14. A ball was added to an urn containing one white ball - either white or black (with equal probabilities of selection). After this, one ball was randomly drawn from the urn. It turned out to be white. What is the conditional probability that the remaining ball in the urn is also white? | \frac{2}{3} | 61 | 7 |
math | 2. Each of the islanders is either a knight, who always tells the truth, or a liar, who always lies (both types are present on the island). Every resident knows whether each other is a knight or a liar. Some of the islanders stated that there is an even number of knights on the island, while all the others claimed that... | No | 97 | 1 |
math | Given a positive integer $n$, consider a triangular array with entries $a_{ij}$ where $i$ ranges from $1$ to $n$ and $j$ ranges from $1$ to $n-i+1$. The entries of the array are all either $0$ or $1$, and, for all $i > 1$ and any associated $j$ , $a_{ij}$ is $0$ if $a_{i-1,j} = a_{i-1,j+1}$, and $a_{ij}$ is $1$ otherwi... | 2^{\left\lceil \frac{n}{2} \right\rceil} | 209 | 18 |
math | Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$.
If $x_{5} = 43$, what is the sum of digits of the largest prime factors of $x_{6}$? | 13 | 86 | 2 |
math | 4. 55 Try to find real numbers \(x, y, z\) greater than 1, satisfying the equation
$$x+y+z+\frac{3}{x-1}+\frac{3}{y-1}+\frac{3}{z-1}=2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2}) .$$ | x=y=z=\frac{3+\sqrt{13}}{2} | 80 | 16 |
math | 4. Six numbers are written in a row on the board. It is known that each number, starting from the third, is equal to the product of the two preceding numbers, and the fifth number is equal to 108. Find the product of all six numbers in this row. | 136048896 | 59 | 9 |
math | Example 7 Let $x=(15+\sqrt{220})^{19}+(15+\sqrt{200})^{22}$. Find the unit digit of the number $x$.
| 9 | 46 | 1 |
math | 8.3. Given a parallelogram $A B C D, \angle D=100^{\circ}, B C=12$. On side $A D$ there is a point $L$ such that $\angle A B L=50^{\circ}, L D=4$. Find the length of $C D$. | 8 | 72 | 1 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=6 \\
\operatorname{LCM}(a ; b ; c)=2^{15} \cdot 3^{16}
\end{array}\right.
$$ | 7560 | 89 | 4 |
math | 1. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
& |1-x|=y+1 \\
& |1+y|=z-2 \\
& |2-z|=x-x^{2}
\end{aligned}
$$ | (1;-1;2) | 56 | 7 |
math | 8. In the tetrahedron $S-ABC$, $SA=4$, $SB \geqslant 7$, $SC \geqslant 9$, $AB=5$, $BC \leqslant 6$, $AC \leqslant 8$. Then the maximum volume of the tetrahedron is $\qquad$. | 8 \sqrt{6} | 78 | 6 |
math | (a) Determine the largest possible value $M$ that $x+y+z$ can take if $x$, $y$, and $z$ are positive real numbers such that
$$
16 x y z=(x+y)^{2}(x+z)^{2}
$$
(b) Show that there are infinitely many triples $(x, y, z)$ of positive rational numbers for which
$$
16 x y z=(x+y)^{2}(x+z)^{2} \text { and } x+y+z=M
$$
hol... | 4 | 122 | 1 |
math | 12. Maximum 15 points. Let $[x]$ denote the integer part of the number $x$ (i.e., the greatest integer not exceeding $x$).
Solve the equation.
$$
[\sin x+\cos x]=1
$$
# | 1\leq\sinx+\cosx<2 | 56 | 12 |
math | $1^{\circ}$. Show that the roots of the equation
$$
x^{2}+(\lambda-2) x-(\lambda+3)=0
$$
are real for any (real) value of $\lambda$.
$2^{\circ}$. Express the sum of the squares of the roots as a function of $\lambda$!
$3^{\circ}$. Determine the value of $\lambda$ so that the sum of the squares of the roots is equal... | 9 | 119 | 1 |
math | Example 2. Find the integral $\int \sqrt{x+4} d x$. | \frac{2}{3}(x+4)\sqrt{x+4}+C | 18 | 18 |
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