task_type stringclasses 1 value | problem stringlengths 46 985 | answer stringlengths 1 146 | problem_tokens int64 20 349 | answer_tokens int64 1 92 |
|---|---|---|---|---|
math | Example 9. A coin is tossed 10 times. Find the probability that heads will appear: a) from 4 to 6 times; b) at least once. | \frac{21}{32} | 37 | 9 |
math | How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips. | 14 | 61 | 2 |
math | [ Processes and operations [ Examples and counterexamples. Constructions ]
Several boxes together weigh 10 tons, and each of them weighs no more than one ton. How many three-ton trucks are definitely enough to haul this cargo?
# | 5 | 48 | 1 |
math | Friends Jarda, Premek, and Robin were playing marbles. Jarda wasn't doing well, so after the game, he had the fewest marbles of all. The boys felt sorry for him, so Robin gave Jarda half of all his marbles, and Premek gave him a third of his. Then Jarda had the most marbles, so he returned seven marbles to each of his friends. After these exchanges, they all had the same number, which was 25 marbles.
How many marbles did Jarda have after the game (before the exchanges)?
(M. Petrová) | 12 | 127 | 2 |
math | 307. Given two equations:
$$
x^{2}+a x+1=0, \quad x^{2}+x+a=0
$$
Determine all values of the coefficient $a$, for which these equations have at least one common root. | a_{1}=-2,a_{2,3}=1 | 57 | 13 |
math | 10. Given a sequence $\left\{a_{n}\right\}$ with all terms being positive integers, it satisfies: for any positive integers $m, k$ there are $a_{m^{2}}=a_{m}^{2}$ and $a_{m^{2}+k^{2}}=$ $a_{m} a_{k}$, find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=1 | 97 | 5 |
math | 6. What are the acute angles of a right triangle if for its legs $a, b$ and hypotenuse (20) $c$ it holds that $4 a b=c^{2} \sqrt{3}$? | 30 | 48 | 2 |
math | 3.9 Given numbers $\alpha$ and $\beta$ satisfy the following two equations: $\alpha^{3}-3 \alpha^{2}+5 \alpha=1, \beta^{3}-3 \beta^{2}+5 \beta=5$, try to find $\alpha+\beta$. | \alpha+\beta=2 | 62 | 6 |
math | 381. Given the curve $y=-x^{2}+4$. Draw a tangent to it at the point where the abscissa is $x=-1$. | 2x+5 | 36 | 4 |
math | 4. Let $a$ be a real number. If there exists a real number $t$, such that $\frac{a-\mathrm{i}}{t-\mathrm{i}}+\mathrm{i}$ is a real number (where $\mathrm{i}$ is the imaginary unit), then the range of values for $a$ is . $\qquad$ | (-\infty,-\frac{3}{4}] | 70 | 12 |
math | Example 2 Given a positive number $p$ and a parabola $C: y^{2}=2 p x$ $(p>0)$, $A\left(\frac{p}{6}, 0\right)$ is a point on the axis of symmetry of the parabola $C$, and $O$ is the vertex of the parabola $C$. $M$ is any point on the parabola $C$. Find the maximum value of $\frac{|O M|}{|A M|}$. ${ }^{[2]}$
(2013, Xin Zhi Cup Shanghai High School Mathematics Competition) | \frac{3 \sqrt{2}}{4} | 134 | 12 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ with the general term formula $a_{n}=\log _{3}\left(1+\frac{2}{n^{2}+3 n}\right)$, then $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)=$ | 1 | 83 | 1 |
math | 9. (6 points) In isosceles $\triangle A B C$, the ratio of the measures of two interior angles is 1:2. Then, the largest angle in $\triangle A B C$ can be $\qquad$ degrees. | 90 | 52 | 2 |
math | 2. Solve the equation $\frac{1}{4^{x}+1}+\frac{1}{2^{x} \cdot 3^{x}-1}=\frac{2^{x}}{4^{x} \cdot 3^{x}-2 \cdot 2^{x}-3^{x}}$. | x_{1}=-1x_{2}=1 | 67 | 11 |
math | Problem 4. The number zero is written on the board. Peter is allowed to perform the following operations:
- apply a trigonometric (sin, $\cos$, $\operatorname{tg}$, or ctg) or inverse trigonometric (arcsin, arccos, $\operatorname{arctg}$, or arcctg) function to one of the numbers written on the board and write the result on the board;
- write the quotient or product of two already written numbers on the board.
Help Peter write $\sqrt{3}$ on the board. | \sqrt{3} | 118 | 5 |
math | 7.14. In a quarry, 120 granite slabs weighing 7 tons each and 80 slabs weighing 9 tons each are prepared. Up to 40 tons can be loaded onto a railway platform. What is the minimum number of platforms required to transport all the slabs? | 40 | 64 | 2 |
math | 11. Let real numbers $x_{1}, x_{2}, \cdots, x_{2011}$ satisfy
$$
\left|x_{1}\right|=99,\left|x_{n}\right|=\left|x_{n-1}+1\right| \text {, }
$$
where, $n=2,3, \cdots, 2014$. Find the minimum value of $x_{1}+x_{2}+\cdots+x_{2014}$. | -5907 | 113 | 5 |
math | Let $\sigma (n)$ denote the sum and $\tau (n)$ denote the amount of natural divisors of number $n$ (including $1$ and $n$). Find the greatest real number $a$ such that for all $n>1$ the following inequality is true: $$\frac{\sigma (n)}{\tau (n)}\geq a\sqrt{n}$$ | \frac{3 \sqrt{2}}{4} | 82 | 12 |
math | Problem 7.3. (15 points) Several boxes are stored in a warehouse. It is known that there are no more than 60 boxes, and each of them contains either 59 apples or 60 oranges. After a box with a certain number of oranges was brought to the warehouse, the number of fruits in the warehouse became equal. What is the smallest number of oranges that could have been in the brought box? | 30 | 90 | 2 |
math | 4. 47. Let $a, b$ be real numbers, and $x^{4}+a x^{3}+b x^{2}+a x+1=0$ has at least one real root. Try to find the minimum value of $a^{2}+b^{2}$. | \frac{4}{5} | 67 | 7 |
math | 2. $[\mathbf{2 0}]$ The four sides of quadrilateral $A B C D$ are equal in length. Determine the perimeter of $A B C D$ given that it has area 120 and $A C=10$. | 52 | 56 | 2 |
math | 1. How many numbers with at least four digits are divisible by 9 and can be formed from the digits $1,9,0,1,2,0,1,9$ (each digit can be used as many times as it is listed)? | 3 | 53 | 1 |
math | 1. $\arcsin (\sin \sqrt{3})-\arcsin (\cos 5)=$ | \frac{5\pi}{2}-5-\sqrt{3} | 23 | 15 |
math | 16 In the sequence $\left\{a_{n}\right\}$, $a_{1}, a_{2}$ are given non-zero integers, $a_{n+2}=\left|a_{n+1}-a_{n}\right|$.
(1) Let $a_{1}=2, a_{2}=-1$, find $a_{2008}$;
(2) Prove: From $\left\{a_{n}\right\}$, it is always possible to select infinitely many terms to form two different constant sequences. | 0 | 118 | 1 |
math | Dudeney, Amusements in Mathematics Problem 16 Mr Morgan G Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil". He therefore proposed to distribute a million dollars among them and watch them go rapidly to the bad. But he as a man of strange fancies and superstitions, and it was an inviolable rule with him never to make a gift that was not either one dollar or some power of seven - such as 7, 49, 343, 2401, which numbers of dollars are produced by simply multiplying sevens together. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the 1,000,000 dollars? You may distribute the money among as many people as you like, under the conditions given. | 823543,117649,16807,2401,343,49,7,1 | 241 | 35 |
math | Let $t$ be TNYWR.
Suppose that
$$
\frac{1}{2^{12}}+\frac{1}{2^{11}}+\frac{1}{2^{10}}+\cdots+\frac{1}{2^{t+1}}+\frac{1}{2^{t}}=\frac{n}{2^{12}}
$$
(The sum on the left side consists of $13-t$ terms.)
What is the value of $n$ ? | 127 | 103 | 3 |
math | N2) Find all triplets $(a, b, p)$ of strictly positive integers where $p$ is prime and the equation
$$
(a+b)^{p}=p^{a}+p^{b}
$$
is satisfied. | (1,1,2) | 50 | 7 |
math | Four points lying on one circle Auxiliary circle $\quad]$
In a convex quadrilateral $A B C D$, it is known that $\angle A C B=25^{\circ}, \angle A C D=40^{\circ}$ and $\angle B A D=115^{\circ}$. Find the angle $A D B$ | 25 | 75 | 2 |
math | Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$ | \left(\frac{7}{8}, \sqrt{5} - 1\right) | 57 | 21 |
math | ## Task Condition
Approximately calculate using the differential.
$y=\sqrt[3]{x^{2}+2 x+5}, x=0.97$ | 1.99 | 35 | 4 |
math | In the triangle $ABC$, point $D$ is given on the extension of side $CA$ beyond $A$, and point $E$ is given on the extension of side $CB$ beyond $B$, such that $AB = AD = BE$. The angle bisectors from $A$ and $B$ of triangle $ABC$ intersect the opposite sides at points $A_1$ and $B_1$, respectively. What is the area of triangle $ABC$ if the area of triangle $DCE$ is 9 units, and the area of triangle $A_1CB_1$ is 4 units? | 6 | 129 | 1 |
math | The Nováks baked wedding cakes. They took a quarter of them to their relatives in Moravia, gave a sixth to their colleagues at work, and gave a ninth to their neighbors. If they had three more cakes left, it would be half of the original number. How many cakes did they bake? | 108 | 62 | 3 |
math | We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle? | 3\sqrt{3} - \pi | 35 | 9 |
math | Find $ \int_{ - 1}^1 {n\choose k}(1 + x)^{n - k}(1 - x)^k\ dx\ (k = 0,\ 1,\ 2,\ \cdots n)$. | \frac{2^{n+1}}{n+1} | 52 | 14 |
math | 9.2. Solve the system of equations: $\left\{\begin{array}{c}x^{4}+x^{2} y^{2}+y^{4}=481 \\ x^{2}+x y+y^{2}=37\end{array}\right.$. | (4;3),(3;4),(-3;-4),(-4;-3) | 62 | 19 |
math | Example 3 In the Cartesian coordinate system $x O y$, given two points $M(-1,2)$ and $N(1,4)$, point $P$ moves on the $x$-axis. When $\angle M P N$ takes the maximum value, find the x-coordinate of point $P$.
(2004 National High School Mathematics Competition Problem) | 1 | 79 | 1 |
math | [ Decimal numeral system ]
What is the smallest integer of the form 111...11 that is divisible by 333...33 (100 threes)? | 1\ldots1 | 38 | 5 |
math | Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$, all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\] | n = 2^3 \cdot 3 \cdot 5 \cdot 17 | 87 | 20 |
math | 10. $[8]$ Evaluate the integral $\int_{0}^{1} \ln x \ln (1-x) d x$. | 2-\frac{\pi^{2}}{6} | 30 | 11 |
math | For each positive integer $n$, let $f(n)$ represent the last digit of $1+2+\cdots+n$. For example, $f(1)=$
$$
\begin{aligned}
1, f(2)=3, f(5) & =5 \text {. Find } \\
f(1)+f(2) & +f(3)+\cdots+f(2004) .
\end{aligned}
$$ | 7010 | 95 | 4 |
math | For example, $1 k$ is a natural number, and $\frac{1001 \cdot 1002 \cdot \cdots \cdot 2005 \cdot 2006}{11^{k}}$ is an integer, what is the maximum value of $k$? | 101 | 67 | 3 |
math | 6. Three fractions with numerators of 1 and different denominators (natural numbers) sum up to 1. Find these fractions. | \frac{1}{2},\frac{1}{3},\frac{1}{6} | 28 | 21 |
math | 80.
This case is more interesting than the previous ones. There are four defendants: A, B, C, D. The following has been established:
1) If A and B are both guilty, then C was an accomplice.
2) If A is guilty, then at least one of the accused B, C was an accomplice.
3) If C is guilty, then D was an accomplice.
4) If A is not guilty, then D is guilty.
Who among the four defendants is undoubtedly guilty, and whose guilt remains in doubt? | D | 115 | 1 |
math | In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
If the rules are followed, in how many different orders can the eight targets be broken? | 560 | 96 | 3 |
math | 7. Given the function $f(x)=x^{3}-2 x^{2}-3 x+4$, if $f(a)=f(b)=f(c)$, where $a<b<c$, then $a^{2}+b^{2}+c^{2}=$ | 10 | 58 | 2 |
math | In a group of $2020$ people, some pairs of people are friends (friendship is mutual). It is known that no two people (not necessarily friends) share a friend. What is the maximum number of unordered pairs of people who are friends?
[i]2020 CCA Math Bonanza Tiebreaker Round #1[/i] | 1010 | 73 | 4 |
math | A permutation of the set $\{1, \ldots, 2021\}$ is a sequence $\sigma=\left(\sigma_{1}, \ldots, \sigma_{2021}\right)$ such that each element of the set $\{1, \ldots, 2021\}$ is equal to exactly one term $\sigma_{i}$. We define the weight of such a permutation $\sigma$ as the sum
$$
\sum_{i=1}^{2020}\left|\sigma_{i+1}-\sigma_{i}\right|
$$
What is the greatest possible weight of permutations of $\{1, \ldots, 2021\}$? | 2042219 | 152 | 7 |
math | Find all positive integers $n$ for which both numbers \[1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\!\!\!\!\quad\text{and}\quad 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\] are prime. | n = 1 | 86 | 5 |
math | Let's find the sum of the following series:
$$
S_{n}=x^{\lg a_{1}}+x^{\lg a_{2}}+\ldots+x^{\lg a_{n}}
$$
where $x$ is a positive number different from 1, and $a_{1}, a_{2}, \ldots, a_{n}$ are distinct positive numbers that form a geometric sequence. | S_{n}=a_{1}^{\lgx}\frac{(\frac{a_{2}}{a_{1}})^{n\lgx}-1}{(\frac{a_{2}}{a_{1}})^{\lgx}-1} | 87 | 55 |
math | Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity. | \frac{|G|}{|Z(G)|} | 103 | 11 |
math | $$
\begin{array}{l}
\text { 3. } 1+\cos ^{2}(2 x+3 y-1) \\
=\frac{x^{2}+y^{2}+2(x+1)(1-y)}{x-y+1}
\end{array}
$$
Then the minimum value of $xy$ is | \frac{1}{25} | 75 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{3-\sqrt{10-x}}{\sin 3 \pi x}$ | -\frac{1}{18\pi} | 40 | 10 |
math | Place a triangle under a magnifying glass that enlarges by 5 times, the perimeter is $\qquad$ times that of the original triangle, and the area is $\qquad$ times that of the original triangle. | 5,25 | 45 | 4 |
math | Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen. | (m, n) = (2k+1, 2t(2k+1)) | 103 | 22 |
math | 6. Given the function $f(x)=\frac{(\sqrt{2})^{x}-1}{(\sqrt{2})^{x}+1}$, the smallest positive integer $n$ such that $f(n)>\frac{n}{n+1}$ is $\qquad$ | 9 | 60 | 1 |
math | 16. (20 points) Given $a>0$, the function
$$
f(x)=\ln x-a(x-1), g(x)=\mathrm{e}^{x} \text {. }
$$
(1) Draw the tangents $l_{1} 、 l_{2}$ to the curves $y=f(x) 、 y=$ $g(x)$ through the origin, respectively. If the slopes of the two tangents are reciprocals of each other, prove: $\frac{\mathrm{e}-1}{\mathrm{e}}<a<\frac{\mathrm{e}^{2}-1}{\mathrm{e}}$;
(2) Let $h(x)=f(x+1)+g(x)$, when $x \geqslant 0$, $h(x) \geqslant 1$ always holds, try to find the range of the real number $a$. | \in(-\infty,2] | 196 | 9 |
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