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 | 8
Two touching circles with centers $O_{1}$ and $O_{2}$ touch internally a circle of radius $R$ with center $O$. Find the perimeter of triangle $O O_{1} O_{2}$. | 2R | 48 | 2 |
math | 6.106. $\left\{\begin{array}{l}u^{2}+v^{2}=u v+13 \\ u+v=\sqrt{u v}+3\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
u^{2}+v^{2}=u v+13 \\
u+v=\sqrt{u v}+3
\end{array}\right.
\] | (1;4),(4;1),(2-\sqrt{3};2+\sqrt{3}),(2+\sqrt{3};2-\sqrt{3}) | 102 | 34 |
math | Amálka observed squirrels in the garden of the nursery, where these three trees grew: spruce, beech, and fir. The squirrels were sitting quietly on the trees, so she could count them - there were 34 of them. When 7 squirrels jumped from the spruce to the beech, there were as many on the beech as on both conifers combin... | 13 | 182 | 2 |
math | Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i] | 6039 | 84 | 4 |
math | ## Task 3 - 211213
A set $M$ contains exactly 55 elements. For each natural number $k$ with $0 \leq k \leq 55$, let $A_{k}$ denote the number of all those subsets of $M$ that contain exactly $k$ elements.
Determine all those natural numbers $k$ for which $A_{k}$ is the largest! | k=27k=28 | 91 | 8 |
math | 767. Solve the equation in natural numbers $x$ and $y$
$$
65 x - 43 y = 2
$$ | 4-43,\quad6-65,where=0,-1,-2,\ldots | 32 | 21 |
math | 5. (Average) The sum of the terms of an infinite geometric series is 2 and the sum of the squares of the corresponding terms of this series is 6 . Find the sum of the cubes of the corresponding terms. | \frac{96}{7} | 46 | 8 |
math | Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$P(Q(x))=(x-1)(x-3)(x-5)(x-7)$$$$Q(R(x))=(x-2)(x-4)(x-6)(x-8)$$
for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$
[i]Proposed by Kyle Lee[/i] | 129 | 107 | 3 |
math | 2. The inequality $x^{2}+|2 x-6| \geqslant a$ holds for all real numbers $x$. Then the maximum value of the real number $a$ is $\qquad$ | 5 | 47 | 1 |
math | 4. Let $H$ be the orthocenter of acute triangle $A B C$, given that $\angle A=60^{\circ}, B C=3$, then $A H=$ | \sqrt{3} | 41 | 5 |
math | Example 3. Given the function $z=\arcsin \frac{y}{x^{2}}$, point $A(-2,-1)$, and vector $\vec{a}=3 \vec{i}-4 \vec{j}$. Find:
1) $\overrightarrow{\operatorname{grad}} z$ at point $A$;
2) the derivative at point $A$ in the direction of vector $\vec{a}$. | -\frac{1}{\sqrt{15}}\vec{i}+\frac{1}{\sqrt{15}}\vec{j} | 91 | 30 |
math | Let 5 be a point inside an acute triangle $\triangle ABC$ such that $\angle ADB = \angle ACB + 90^{\circ}$, and $AC \cdot BD = AD \cdot BC$. Find the value of $\frac{AB \cdot CD}{AC \cdot BD}$. | \sqrt{2} | 64 | 5 |
math | 14th Mexico 2000 Problem A2 A triangle is constructed like that below, but with 1, 2, 3, ... , 2000 as the first row. Each number is the sum of the two numbers immediately above. Find the number at the bottom of the triangle. 1 2 3 4 5 3 5 7 9 8 12 16 20 28 48 | 2^{1998}\cdot2001 | 100 | 12 |
math | Comparing the fractions ${ }^{111110} / 111111,{ }^{222221 / 222223},{ }^{333331} /{ }_{333334}$, arrange them in ascending order.
# | 111110/111111<333331/333334<222221/222223 | 68 | 41 |
math | 11. Let $f(z)=\frac{a z+b}{c z+d}$ for $a, b, c, d \in \mathbb{C}$. Suppose that $f(1)=i, f(2)=i^{2}$, and $f(3)=i^{3}$. If the real part of $f(4)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m^{2}+n^{2}$. | 34 | 109 | 2 |
math | Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd. | \text{Odd} | 80 | 6 |
math | In a certain job, 3 workers worked; the first for 6 days, the second for 3 days, and the third for 8 days. Per head wages were $36 \mathrm{~K}$ for the first, $12 \mathrm{~K}$ for the second, and $24 \mathrm{~K}$ for the third. How many days would it take for each to complete the job alone? | 12,18,24 | 90 | 8 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-3 ;-5 ; 6)$
$M_{2}(2 ; 1 ;-4)$
$M_{3}(0 ;-3 ;-1)$
$M_{0}(3 ; 6 ; 68)$ | \sqrt{573} | 88 | 7 |
math | 1. Find all functions $f: \mathbf{Q} \rightarrow \mathbf{Q}$ that satisfy $f(1)=2$ and $f(x y)=f(x) \cdot f(y)-f(x+y)+1, x, y \in \mathbf{Q}$
(1980 Luxembourg Competition Problem) | f(x)=x+1 | 72 | 6 |
math | Example 11 (1988 National High School League Question) Team A and Team B each send out 7 players to participate in a Go chess competition in a pre-arranged order. Both sides start with the No. 1 player competing. The loser is eliminated, and the winner then competes with the No. 2 player of the losing side... until all... | 3432 | 104 | 4 |
math | 8. 1b.(TUR 5) Find the smallest positive integer $n$ such that
(i) $n$ has exactly 144 distinct positive divisors, and
(ii) there are ten consecutive integers among the positive divisors of $n$. | 110880 | 56 | 6 |
math | 9.20 Find the largest binomial coefficient in the expansion of $\left(n+\frac{1}{n}\right)^{n}$, if the product of the fourth term from the beginning and the fourth term from the end is 14400. | 252 | 55 | 3 |
math | 8.5. The placement of chess kings on a board is called "correct" if no king attacks another, and every square of the board is either under attack or occupied by one of the kings. What is the minimum and maximum number of kings that can be correctly placed on an $8 \times 8$ chessboard? | 916 | 67 | 3 |
math | 5. $[\mathbf{5}]$ Find all real values of $x$ for which
$$
\frac{1}{\sqrt{x}+\sqrt{x-2}}+\frac{1}{\sqrt{x+2}+\sqrt{x}}=\frac{1}{4} .
$$ | \frac{257}{16} | 62 | 10 |
math | 11.1. Solve the inequality: $\sqrt{(x-2)^{2}\left(x-x^{2}\right)}<\sqrt{4 x-1-\left(x^{2}-3 x\right)^{2}}$. | 2 | 50 | 1 |
math | The base $4$ repeating decimal $0.\overline{12}_4$ can be expressed in the form $\frac{a}{b}$ in base 10, where $a$ and $b$ are relatively prime positive integers. Compute the sum of $a$ and $b$.
[i]2020 CCA Math Bonanza Team Round #2[/i] | 7 | 81 | 1 |
math | Example 1.18. Form the equations of lines parallel to the line $3 x+4 y-1=0(l)$ and at a distance of 1 from it. | 3x+4y-6=0\quad\text{}\quad3x+4y+4=0 | 38 | 24 |
math | Yaishchenko I.V.
In Mexico, environmentalists have succeeded in passing a law according to which each car must not be driven at least one day a week (the owner reports to the police the car's number and the "day off" for the car). In a certain family, all adults wish to drive daily (each for their own business!). How ... | 6;10 | 100 | 4 |
math | 11.5. Find all pairs of natural numbers $a$ and $b$ such that both numbers $\frac{a^{2}+b}{b^{2}-a}$ and $\frac{b^{2}+a}{a^{2}-b}$ are integers. | (2,2),(3,3),(1,2),(2,1),(2,3),(3,2) | 58 | 25 |
math | 88. Form the equation of the line passing through the point $A(3, -2)$ and having the direction vector $\vec{n}=(-5, 3)$. | 3x+5y+1=0 | 37 | 9 |
math | [ Rebus $]$
Decode the rebus: KIS + KSI = ISK. Identical letters correspond to identical digits, different ones to different digits.
# | 495+459=954 | 35 | 11 |
math | 4.99 Solve the system of equations $\left\{\begin{array}{l}x+y=4, \\ \left(x^{2}+y^{2}\right)\left(x^{3}+y^{3}\right)=280 .\end{array}\right.$
(Kyiv Mathematical Olympiad, 1935) | {\begin{array}{}{x_{1}=3,}\{y_{1}=1;}\end{pmatrix}\quad\text{or}\quad{\begin{pmatrix}x_{2}=1,\\y_{2}=30\end{pmatrix}..}} | 75 | 61 |
math | ## Problem 1
Calculate the sum
$$
S=[\lg 1]+[\lg 2]+[\lg 3]+\cdots+\left[\lg 10^{2014}\right]
$$ | \frac{10}{9}(2013\cdot10^{2014}-2014\cdot10^{2013}+1)+2014 | 46 | 43 |
math | Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence? | 5 | 36 | 1 |
math | ## Problem Statement
Find the derivative.
$y=\frac{2 x-1}{4 x^{2}-4 x+3}+\frac{1}{\sqrt{2}} \cdot \operatorname{arctg} \frac{2 x-1}{\sqrt{2}}$ | \frac{8}{(4x^{2}-4x+3)^{2}} | 62 | 19 |
math | [ Relationships between the sides and angles of triangles (other).]
In a right triangle $ABC$ with a right angle at $A$, a circle is constructed on the altitude $AD$ as a diameter, intersecting side $AB$ at point $K$ and side $AC$ at point $M$. Segments $AD$ and $KM$ intersect at point $L$. Find the acute angles of tr... | 15 | 101 | 2 |
math | 2. At the front of the cycling race column are cyclists Aco and Boro, riding side by side. They have so far outpaced the other cyclists that no one can catch them before the finish line. At $20 \mathrm{~km}$ before the finish, Aco's tire goes flat. Boro continues to the finish line at an average speed of $40 \mathrm{~k... | \frac{2}{9}\mathrm{~} | 136 | 11 |
math | B1. The solution to each clue of this crossnumber is a two-digit number, that does not begin with a zero.
Find all the different ways in which the crossnumber can be completed correctly.
Across
Down
1. A prime
1. A square
3. A square
2. A square | 83 | 64 | 2 |
math | Example 1. Find the approximate value of the smallest characteristic number of the kernel by the Ritz method
$$
K(x, t)=x t ; \quad a=0, b=1
$$ | 3 | 43 | 1 |
math | Question 3. If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive integers, satisfying $\mathrm{c}=$ $(a+b i)^{3}-107 i$, find $c$. (where $i^{2}=-1$) | 198 | 59 | 3 |
math | Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$. | 84 | 74 | 2 |
math | Tolpygo A.K.
Two players are involved. The first player has 1000 even cards (2, 4, ..., 2000), and the second player has 1001 odd cards (1, 3, ..., 2001). They take turns, with the first player starting. A move consists of the following: the player whose turn it is lays down one of their cards, and the other player, a... | First\player\-\499\points,\\player\-\501\points | 173 | 20 |
math | 10.4 Given a triangle $A B C$. Point $P$ is the incenter. Find the angle $B$, if it is known that $R_{A B C}=R_{A P C}$, where $R_{A B C}, R_{A P C}$ are the radii of the circumcircles of triangles $ABC$ and $APC$ respectively. | 60 | 81 | 2 |
math | 1. Determine all two-digit numbers with the following property: the number and the number written with the same digits, but in reverse order, are both prime. | {11,13,17,31,37,71,73} | 32 | 22 |
math | Example 6 Given that $x, y, z$ are two non-negative rational numbers, and satisfy $3x+2y+z=5, x+y-z=2$. If $S=2x+y-z$, then what is the sum of the maximum and minimum values of $S$?
(1996, Tianjin City Junior High School Mathematics Competition) | 5 | 77 | 1 |
math | 3. Let $x>0$, and $x^{2}+\frac{1}{x^{2}}=7$. Then $x^{5}+\frac{1}{x^{5}}=$ | 123 | 42 | 3 |
math | ## Task B-1.4.
Among all natural numbers divisible by 8, determine those for which the sum of the digits is 7, and the product of the digits is 6. | 16,1312,3112 | 40 | 12 |
math | 3. At a basketball tournament, 16 teams participate, playing in a double round-robin format, meaning each team plays every other team twice. The top 8 teams advance to the next tournament. The ranking of the teams is determined based on the number of wins, and if multiple teams have the same number of wins, their relat... | 23 | 90 | 2 |
math | 47. a) How many different squares, in terms of size or position, consisting of whole cells, can be drawn on a chessboard of 64 cells?
b) The same question for a chessboard of $n^{2}$ cells. | 204 | 52 | 3 |
math | One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have ob... | 99000 | 101 | 5 |
math | Triangle $ ABC$ obeys $ AB = 2AC$ and $ \angle{BAC} = 120^{\circ}.$ Points $ P$ and $ Q$ lie on segment $ BC$ such that
\begin{eqnarray*}
AB^2 + BC \cdot CP = BC^2 \\
3AC^2 + 2BC \cdot CQ = BC^2
\end{eqnarray*}
Find $ \angle{PAQ}$ in degrees. | 30^\circ | 106 | 4 |
math | 1. Let the polynomial $f(x)$ satisfy: for any $x \in \mathbf{R}$, we have
$$
f(x+1)+f(x-1)=2 x^{2}-4 x .
$$
Then the minimum value of $f(x)$ is $\qquad$ | -2 | 63 | 2 |
math | 2. In the rectangular paper piece $A B C D$, $A B=6, B C=8$. If the paper is folded so that $A$ coincides with $C$, then the length of the fold line $E F$ is | \frac{15}{2} | 52 | 8 |
math | [ Arithmetic. Mental calculation, etc.]
When the barrel is empty by $30 \%$, it contains 30 liters more honey than when it is filled to $30 \%$. How many liters of honey are in a full barrel? | 75 | 49 | 2 |
math | 5.1.3 ** Given a complex number $z$ satisfying $|z|-1$, find the maximum value of $\left|z^{3}-3 z-2\right|$.
untranslated portion:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The untranslated portion is not part of the translation request and is provided for context. | 3\sqrt{3} | 85 | 6 |
math | 26th Putnam 1965 Problem A1 How many positive integers divide at least one of 10 40 and 20 30 ? Solution | 2301 | 37 | 4 |
math | 4. Ivan Ivanovich approached a source with two empty cans, one with a capacity of 10 liters, and the other - 8 liters. Water from the source flowed in two streams - one stronger, the other weaker. Ivan Ivanovich simultaneously placed the cans under the streams and, when half of the smaller can was filled, he switched t... | 2 | 102 | 1 |
math | Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$
$ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$.
$ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit.
$ (c)$ Determine $ \displaystyle\lim_{n\t... | 1 | 151 | 3 |
math | 7. In $\triangle A B C, \tan \angle C A B=\frac{22}{7}$ and the altitude from $A$ to $B C$ divides $B C$ into segments of lengths 3 and 17. Find the area of $\triangle A B C$.
(1 mark)
在 $\triangle A B C$ 中, $\tan \angle C A B=\frac{22}{7}$, 且從 $A$ 到 $B C$ 的高把 $B C$ 分成長度 3 和 17 的兩段。求 $\triangle A B C$ 的面積。
（１分） | 110 | 139 | 3 |
math | Solve the following equation:
$$
\lg \left(3^{x^{\sqrt{x}}-(\sqrt{x})^{x}+2}+1\right)=1
$$ | x_1=1,x_2=4 | 40 | 10 |
math | Subject I
Determine the remainder of the division of the number $1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 69 \cdot 70 + 1234$ by 2013. | 1234 | 56 | 4 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$ | 0 | 40 | 1 |
math | In triangle $A B C$, angle $C$ is twice angle $A$ and $b=2 a$. Find the angles of this triangle.
# | 30,90,60 | 32 | 8 |
math | 8. Given the sequence $\left\{a_{n}\right\}$, where $a_{n}$ is a real number, and for $n \geqslant 3, n \in$ $\mathbf{N}$, we have $a_{n}=a_{n-1}-a_{n-2}$. If the sum of the first 1985 terms is 1000, and the sum of the first 1995 terms is 4000, then the sum of the first 2002 terms is $\qquad$. | 3000 | 126 | 4 |
math | 3-rd 5. When dividing the polynomial \(x^{1951}-1\) by \(x^{4}+x^{3}+2 x^{2}+x+1\), a quotient and a remainder are obtained. Find the coefficient of \(x^{14}\) in the quotient. | -1 | 66 | 2 |
math | $\mathrm{B}-1$. Let $S$ be a set of $n$ distinct real numbers, and $A_{s}$ be the set of all distinct averages of pairs of elements from $S$. For a given $n \geqslant 2$, what is the minimum number of elements that $A_{s}$ can have? | 2n-3 | 72 | 4 |
math | $$
\begin{array}{l}
\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta \\
<3 a+6
\end{array}
$$
For $\theta \in\left[0, \frac{\pi}{2}\right]$, the inequality always holds. Find the range of values for $a$.
(Zhang Daocheng, provided) | >3 | 111 | 2 |
math | Example 1. Find $\int \frac{7 x^{3}-4 x^{2}-32 x-37}{(x+2)(2 x-1)\left(x^{2}+2 x+3\right)} d x$. | 3\ln|x+2|-\frac{5}{2}\ln|2x-1|+\frac{3}{2}\ln(x^{2}+2x+3)-\frac{4}{\sqrt{2}}\operatorname{arctg}\frac{x+1}{\sqrt{2}}+C | 53 | 69 |
math | 4. A train $110 \mathrm{~m}$ long is moving at a speed of $\frac{25}{3} \mathrm{~m} / \mathrm{s}$. On its way to the track at $09.10 \mathrm{~h}$, it encountered a pedestrian walking in the same direction and passed him in $15 \mathrm{sec}$. At $09.16 \mathrm{~h}$, it met a pedestrian walking towards it and passed him ... | 9:40 | 125 | 4 |
math | 9. If $P(x, y)$ is a point on the hyperbola $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$, then the minimum value of $|x-y|$ is $\qquad$ . | 2 | 54 | 1 |
math | 5. (10 points) An electronic clock always displays the date as an eight-digit number, such as 20110101 for January 1, 2011. Therefore, the last day of 2011 that can be divided by 101 is $\overline{2011 \mathrm{ABCD}}$, so $\overline{\mathrm{ABCD}}=$ $\qquad$ | 1221 | 93 | 4 |
math | Example 2. Let's calculate the sum
$$
a^{2000}+\frac{1}{a^{2000}}
$$
if $a^{2}-a+1=0$. | -1 | 45 | 2 |
math | Example 2-33 Put $n$ labeled balls into $m$ distinct boxes, with no box empty. How many different ways are there to do this? | a_{}=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(n-k)^{} | 34 | 29 |
math | There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two square... | 3 | 80 | 1 |
math | Quadrilateral $ABCD$ has both an inscribed and a circumscribed circle and sidelengths $BC = 4, CD = 5, DA = 6$. Find the area of $ABCD$. | 10\sqrt{6} | 46 | 7 |
math | 7. Let $x, y, z$ be real numbers, $3 x, 4 y, 5 z$ form a geometric sequence, and $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ form an arithmetic sequence, then the value of $\frac{x}{z}+\frac{z}{x}$ is $\qquad$ . | \frac{34}{15} | 81 | 9 |
math | 10. let $m$ be any natural number. As a function of $m$, determine the smallest natural number $k$ for which the following applies: If $\{m, m+1, \ldots, k\}=A \cup B$ is any decomposition into two sets $A$ and $B$, then $A$ or $B$ contains three elements $a, b, c$ (which do not necessarily have to be different) with $... | ^{^{+2}} | 106 | 5 |
math | 5. In a Cartesian coordinate system, points with both coordinates being integers are called integer points, and the set of all integer points is denoted as $X$. Find the largest real number $\alpha$, such that there exists a function $f: X \rightarrow \mathbf{N}^{*}$, satisfying: (1) There exists a positive real number... | \alpha=1 | 174 | 4 |
math | Example 4. Calculate: $\frac{3.6 \times 11.74 \times 138.4}{6 \times 2437}$.
(6 is an exact number) | 0.40 | 46 | 4 |
math | Let's find arithmetic sequences of natural numbers
$$
\begin{array}{llllll}
a_{1}, & a_{2}, & a_{3}, & \ldots, & a_{k}, & \ldots \\
b_{1}, & b_{2}, & b_{3}, & \ldots, & b_{k}, & \ldots \\
c_{1}, & c_{2}, & c_{3}, & \ldots, & c_{k}, & \ldots \\
d_{1}, & d_{2}, & d_{3}, & \ldots, & d_{k}, & \ldots
\end{array}
$$
such t... | 20 | 238 | 2 |
math | 50. If the sum of the digits of a four-digit number added to the number itself equals 2017, then the four-digit number is . $\qquad$ | 1994or2012 | 37 | 9 |
math | 9. (12 points) Optimus Prime and Bumblebee start from locations $A$ and $B$ respectively at the same time, heading towards each other. When they start as robots, their speed ratio is 4:3. After they meet, they transform into Autobots, with Optimus Prime's speed increasing by $25\%$ and Bumblebee's speed increasing by $... | 350 | 128 | 3 |
math | 5 . Find the largest positive integer $n$, such that $n^{3}+100$ can be divided by $n+10$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
5 . Find the largest positive integer $n$, such that $n^{3}+100$ can be divided by $n+1... | 890 | 91 | 3 |
math | An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face? | 6 | 51 | 1 |
math | Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
| 627 | 60 | 3 |
math | 2.148. $\frac{3+\sqrt{2}+\sqrt{3}}{3-\sqrt{2}-\sqrt{3}}$. | -\frac{(4+3\sqrt{2})(5+3\sqrt{3})}{2} | 33 | 22 |
math | ## Task A-2.5.
How many elements can the largest subset of the set $\{1,2,3, \ldots, 2017\}$ have such that for any two elements $a$ and $b$ of this subset, the number $a+b$ is not divisible by $a-b$? | 673 | 70 | 3 |
math | A convex polyhedron has $m$ triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of $m$. | 8 | 40 | 1 |
math | 3. Let $A B C$ be a triangle such that $A B=7$, and let the angle bisector of $\angle B A C$ intersect line $B C$ at $D$. If there exist points $E$ and $F$ on sides $A C$ and $B C$, respectively, such that lines $A D$ and $E F$ are parallel and divide triangle $A B C$ into three parts of equal area, determine the numbe... | 13 | 107 | 2 |
math | A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are r... | 1333 | 114 | 4 |
math | 3 [ Completing the square. Sums of squares ]
Find all real solutions of the equation with 4 unknowns: $x^{2}+y^{2}+z^{2}+t^{2}=x(y+z+t)$. | (0,0,0,0) | 51 | 9 |
math | Let $m$ be a given positive integer. Find all positive integer solutions of the equation $\left(x^{2}+y^{2}\right)^{m}=(x y)^{z}$. | 2^{k},(1+2k)n,=2kn | 42 | 14 |
math | 10. Let $x_{1}, x_{2}, \ldots, x_{1970}$ be positive integers satisfying $x_{1}+x_{2}+\cdots+x_{1970}=2007$. Determine the largest possible value of $x_{1}^{3}+x_{2}^{3}+\cdots+x_{1970}^{3}$. | 56841 | 89 | 5 |
math | 26 Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y$, we have
$$
f(f(x)+y)=2 x+f(f(y)-x) .
$$ | f(x)=x+ | 55 | 5 |
math | ## Task B-3.6.
Determine all values of the parameter $m \in \mathbb{R}$ such that the equation
$$
\log _{x+m}\left(x^{3}-9 x+8\right) \cdot \log _{x-1}(x+m)=3
$$
has a unique solution. | >-3\neq-2 | 73 | 7 |
math | 12 The range of negative values of $a$ that make the inequality
$$
\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x
$$
hold for all $x \in \mathbf{R}$ is $\qquad$ | \leqslant-2 | 64 | 7 |
math | 23rd VMO 1985 Problem A2 Find all real-valued functions f(n) on the integers such that f(1) = 5/2, f(0) is not 0, and f(m) f(n) = f(m+n) + f(m-n) for all m, n. | f(n)=2^n+\frac{1}{2^n} | 69 | 13 |
math | 6. Find the minimum value of the expression $x^{2}+y^{2}+3 z^{2}+2 x y$ given that $x y+z=1$ and $x, y, z>0$. | \frac{8}{3} | 49 | 7 |
math | Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies:
(i) $f(1995) = 1996$
(ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$.
Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ . | k = -1 | 129 | 5 |
math | G7.1 There are $a$ zeros at the end of the product $1 \times 2 \times 3 \times \ldots \times 100$. Find $a$. | 24 | 42 | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.