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 | 1089. Given an infinite arithmetic progression $3 ; 16 ; 29 ; 42 ; \ldots$. Find:
a) the smallest term of the sequence that can be written using only sevens;
b) all such terms. | 777\ldots7,wheretheofsevensis6k+4(k=0,1,2,\ldots) | 54 | 30 |
math | Given two positive integers $n$ and $k$, we say that $k$ is [i]$n$-ergetic[/i] if:
However the elements of $M=\{1,2,\ldots, k\}$ are coloured in red and green, there exist $n$ not necessarily distinct integers of the same colour whose sum is again an element of $M$ of the same colour. For each positive integer $n$, determine the least $n$-ergetic integer, if it exists. | n^2 + n - 1 | 106 | 9 |
math | One, (40 points) Find the smallest integer $c$, such that there exists a sequence of positive integers $\left\{a_{n}\right\}(n \geqslant 1)$ satisfying:
$$
a_{1}+a_{2}+\cdots+a_{n+1}<c a_{n}
$$
for all $n \geqslant 1$. | 4 | 84 | 1 |
math | N3 (4-1, Poland) Find the smallest positive integer $n$ with the following properties:
(1) The unit digit of $n$ is 6;
(2) If the unit digit 6 of $n$ is moved to the front of the other digits, the resulting new number is 4 times $n$.
| 153846 | 71 | 6 |
math | Example 4. Find (1) the principal period of $f(x)=\operatorname{tg} 2 x-\operatorname{ctg} 2 x$; (2) the principal period of $g(x)=12 \sin ^{2} 15 x+15 \cos ^{2} 12 x$. | \frac{\pi}{4} | 74 | 7 |
math | 9. (This question is worth 16 points) If real numbers $a, b, c$ satisfy $2^{a}+4^{b}=2^{c}, 4^{a}+2^{b}=4^{c}$, find the minimum value of $c$.
| \log_{2}3-\frac{5}{3} | 61 | 13 |
math | 15 Let vectors $\vec{i}$ and $\vec{j}$ be the unit vectors in the positive directions of the $x$-axis and $y$-axis, respectively, in a rectangular coordinate plane. If $\vec{a}=(x+2) \vec{i}+y \vec{j}$, $\vec{b}=(x-2) \vec{i}+y \vec{j}$, and $|\vec{a}|-|\vec{b}|=2$.
(1) Find the equation of the locus of point $P(x, y)$ that satisfies the above conditions;
(2) Let $A(-1,0)$ and $F(2,0)$. Is there a constant $\lambda(\lambda>0)$ such that $\angle P F A=\lambda \angle P A F$ always holds? Prove your conclusion. | \lambda=2 | 180 | 4 |
math | Two math students play a game with $k$ sticks. Alternating turns, each one chooses a number from the set $\{1,3,4\}$ and removes exactly that number of sticks from the pile (so if the pile only has $2$ sticks remaining the next player must take $1$). The winner is the player who takes the last stick. For $1\leq k\leq100$, determine the number of cases in which the first player can guarantee that he will win. | 71 | 106 | 2 |
math | Task B-2.6. Two cyclists started simultaneously from two places $A$ and $B$ towards each other. After one hour, the first cyclist had traveled $10 \mathrm{~km}$ more than the second cyclist. The first cyclist arrived 50 minutes earlier at place $B$ than the second cyclist at place $A$. What is the distance between places $A$ and $B$? | 50\mathrm{~} | 86 | 7 |
math | 2. We started the Mathematical Olympiad punctually at 9:00, as I checked on my watch, which was working correctly at that time. When I finished, at 13:00, I looked at the watch again and saw that the hands had come off their axis but maintained the position they were in when the watch was working. Curiously, the hour and minute hands were exactly superimposed, one on top of the other, forming a non-zero angle less than $120^{\circ}$ with the second hand. At what time did my watch break? (Give the answer in hours, minutes, and seconds with a maximum error of one second; assume that, when it was working, the hands of the watch moved continuously.) | 09:49:05 | 157 | 8 |
math | Example 1 When $a^{3}-a-1=0$, $a+\sqrt{2}$ is a root of some integer-coefficient polynomial. Find the integer-coefficient polynomial with the lowest degree and the leading coefficient of 1 that satisfies the given condition.
(1997 Japan Mathematical Olympiad Preliminary Exam) | x^{6}-8x^{4}-2x^{3}+13x^{2}-10x-1 | 68 | 26 |
math | 12.338. The side of the base of a regular quadrilateral prism is equal to $a$, its volume is equal to $V$. Find the cosine of the angle between the diagonals of two adjacent lateral faces. | \frac{V^{2}}{V^{2}+^{6}} | 48 | 16 |
math | 4. Determine all natural numbers $n$ for which:
$$
[\sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+[\sqrt[3]{n}]=2 n
$$ | 33 | 45 | 2 |
math | Example 1 Given the sequence $\left\{a_{n}\right\}$,
$$
\begin{array}{l}
a_{1}=1, a_{2}=2, a_{3}=-1, \\
a_{n+2} a_{n+1}=4 a_{n+1} a_{n}-6 a_{n+1} a_{n-1}+9 a_{n} a_{n-1}-6 a_{n}^{2} .
\end{array}
$$
Find $a_{n}$. | a_{n}=\frac{5}{4}+\frac{1}{12}(-3)^{n} | 118 | 25 |
math |
N7. Find all perfect squares $n$ such that if the positive integer $a \geqslant 15$ is some divisor of $n$ then $a+15$ is a prime power.
| 1,4,9,16,49,64,196 | 47 | 18 |
math | ## Problem Statement
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(-1 ; 0 ; 1)$
$\frac{x+0.5}{0}=\frac{y-1}{0}=\frac{z-4}{2}$ | M^{\}(0;2;1) | 66 | 10 |
math | Problem 4. At the exchange office, there are two types of operations: 1) give 2 euros - get 3 dollars and a candy as a gift; 2) give 5 dollars - get 3 euros and a candy as a gift.
When the wealthy Pinocchio came to the exchange office, he only had dollars. When he left, he had fewer dollars, no euros appeared, but he received 50 candies. How much did such a "gift" cost Pinocchio in dollars?
[6 points] (I.V. Raskina) | 10 | 119 | 2 |
math | # 6. Option 1
Dasha poured 9 grams of feed into the aquarium for the fish. In the first minute, they ate half of the feed, in the second minute - a third of the remaining feed, in the third minute - a quarter of the remaining feed, and so on, in the ninth minute - a tenth of the remaining feed. How many grams of feed are left floating in the aquarium? | 0.9 | 87 | 3 |
math | Problem 1. Determine for how many triples $(m, n, p)$ of non-zero natural numbers less than or equal to 5 the number
$$
A=2^{m}+3^{n}+5^{p}
$$
is divisible by 10. | 35 | 58 | 2 |
math | 4. If $x, y, z$ are real numbers, satisfying
$$
x+\frac{1}{y}=2 y+\frac{2}{z}=3 z+\frac{3}{x}=k \text{, and } x y z=3 \text{, }
$$
then $k=$ | 4 | 67 | 1 |
math | -、(Full marks 10 points) There are two decks of playing cards, each deck arranged in such a way that the first two cards are the Big Joker and the Small Joker, followed by the four suits of Spades, Hearts, Diamonds, and Clubs, with each suit arranged in the order of $1,2,3$, $\cdots, J, Q, K$. Someone stacks the two decks of cards in the above order, then discards the first card, places the second card at the bottom, discards the third card, places the fourth card at the bottom, …, and continues this process until only one card remains. What is the last remaining card? | 6 \text{ of Diamonds} | 140 | 7 |
math | Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions:
(a) $P(2017)=2016$ and
(b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$. | P(x) = x - 1 | 63 | 8 |
math | a) How many zeros are at the end of the number $A=2^{5} \times 3^{7} \times 5^{7} \times 11^{3}$?
b) How many zeros are at the end of the number $B=1 \times 2 \times 3 \times 4 \times \cdots \times 137$? | 33 | 83 | 2 |
math | 14. Given the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$ with its left and right foci being $F_{1}, F_{2}$ respectively, let $P$ be a point on the ellipse $C$ in the first quadrant. The extensions of $P F_{1}, P F_{2}$ intersect the ellipse $C$ at points $Q_{1}, Q_{2}$ respectively. Then the maximum value of the difference in areas between $\triangle P F_{1} Q_{2}$ and $\triangle P F_{2} Q_{1}$ is $\qquad$. | \frac{2\sqrt{2}}{3} | 133 | 12 |
math | $4 \cdot 232$ If $a, b, c$ are positive integers, satisfying $c=(a+b i)^{3}-107 i$, find $c$ (where $i^{2}=-1$ ). | 198 | 52 | 3 |
math | 3. Given $x_{1}=x_{2011}=1$, $\left|x_{n+1}\right|=\left|x_{n}+1\right|(n=1,2, \cdots, 2010)$.
Then $x_{1}+x_{2}+\cdots+x_{2010}=$ $\qquad$ | -1005 | 81 | 5 |
math | Find the smallest real constant $p$ for which the inequality holds $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$ with any positive real numbers $a, b$. | p = 1 | 61 | 5 |
math | [ The inscribed angle is half the central angle ]
In an acute-angled triangle $A B C$, altitudes $C H$ and $A H_{1}$ are drawn. It is known that $A C=2$, and the area of the circle circumscribed around triangle $H B H_{1}$ is $\pi / 3$. Find the angle between the altitude $C H$ and the side $B C$.
# | 30 | 92 | 2 |
math | 1. When $3 x^{2}$ was added to the quadratic trinomial $f(x)$, its minimum value increased by 9, and when $x^{2}$ was subtracted from it, its minimum value decreased by 9. How will the minimum value of $f(x)$ change if $x^{2}$ is added to it? | \frac{9}{2} | 73 | 7 |
math | Determine all positive integers $d$ such that whenever $d$ divides a positive integer $n$, $d$ will also divide any integer obtained by rearranging the digits of $n$. | d \in \{1, 3, 9\} | 39 | 14 |
math | Let $ n \geq 3$ be an odd integer. Determine the maximum value of
\[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\]
where $ x_{i}$ are positive real numbers from the interval $ [0,1]$. | n - 2 + \sqrt{2} | 111 | 10 |
math | 2. Let planar vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy $|\boldsymbol{a}+\boldsymbol{b}|=3$, then the maximum value of $\boldsymbol{a} \cdot \boldsymbol{b}$ is $\qquad$ | \frac{9}{4} | 59 | 7 |
math | (12) If for all positive real numbers $x, y$, the inequality $\frac{y}{4}-\cos ^{2} x \geqslant a \sin x- \frac{9}{y}$ holds, then the range of the real number $a$ is $\qquad$. | [-3,3] | 65 | 5 |
math | Example 1 Find the positive integer solutions of the equation $3 x^{2}+7 x y-2 x-5 y-35=0$. | 1,17or2,3 | 33 | 8 |
math | 5. (10 points) The 3rd term of an arithmetic sequence is 14, and the 18th term is 23. How many terms in the first 2010 terms of this sequence are integers. | 402 | 51 | 3 |
math | 1. Find a polynomial of degree less than 4, such that it satisfies $f(0)=1, f(1)=1, f(2)=5, f(3)=11$. | -\frac{1}{3}(x^{3}-9x^{2}+8x-3) | 42 | 22 |
math | Example 4 The sports meet lasted for $n$ days $(n>1)$, and $m$ medals were awarded. On the first day, 1 medal plus $\frac{1}{7}$ of the remaining $m-1$ medals were awarded, on the second day, 2 medals plus $\frac{1}{7}$ of the remaining medals were awarded, and so on, until on the $n$th day, the remaining medals were all awarded. How many days did the sports meet last? How many medals were awarded in total? | 6, 36 | 114 | 5 |
math | 1. A tractor is pulling a very long pipe on sled runners. Gavrila walked along the entire pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters. The speed of the tractor is constant. | 108 | 102 | 3 |
math | Let $a, b, c, d, e, f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$$
a b c+b c d+c d e+d e f+e f a+f a b
$$
and determine all 6-tuples $(a, b, c, d, e, f)$ for which this maximal value is achieved.
Answer: 8 . | 8 | 95 | 1 |
math | 3. If three numbers are taken simultaneously from the 14 integers $1,2, \cdots, 14$, such that the absolute difference between any two numbers is not less than 3, then the number of different ways to choose is | 120 | 52 | 3 |
math | 10. (10 points) This year, the sum of Jia and Yi's ages is 70 years. Several years ago, when Jia was as old as Yi is now, Yi's age was exactly half of Jia's age. How old is Jia this year?
保留源文本的换行和格式如下：
10. (10 points) This year, the sum of Jia and Yi's ages is 70 years. Several years ago, when Jia was as old as Yi is now, Yi's age was exactly half of Jia's age. How old is Jia this year? | 42 | 132 | 2 |
math | Let $n$ and $k$ be integers, $1\le k\le n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:
(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$.
(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$.
(iii) For all distinct $a,a'$ in $A$, $a$ does not divide $a'$. | b = 2^k | 110 | 7 |
math | Task 3. Sasha chose five numbers from the numbers 1, 2, 3, 4, 5, 6, and 7 and told Anna their product. Based on this information, Anna realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha tell Anna? (20 points) | 420 | 75 | 3 |
math | Example 8. How many roots of the equation
$$
z^{4}-5 z+1=0
$$
lie in the annulus $1<|z|<2 ?$ | 3 | 41 | 1 |
math | 5. Among the teachers of a certain middle school, the total number of people who can speak English and Russian is 110. According to statistics, 75 people can speak English, and 55 people can speak Russian. The number of teachers in the school who can speak English but not Russian is . $\qquad$ | 55 | 69 | 2 |
math | 8. Given that the perimeter of the regular pentagon square $A B C D E$ is $2000 \mathrm{~m}$, two people, A and B, start from points $A$ and $C$ respectively at the same time, walking around the square in the direction of $A \rightarrow B \rightarrow C \rightarrow D \rightarrow E \rightarrow A \rightarrow \cdots$. Person A's speed is $50 \mathrm{~m} / \mathrm{min}$, and person B's speed is $46 \mathrm{~m} / \mathrm{min}$. Then, after $\qquad$ $\min$, A and B will first start walking on the same side. | 104 | 151 | 3 |
math | Let's calculate the sum $S_{n}=1 \cdot 2^{2}+2 \cdot 3^{2}+3 \cdot 4^{2}+\ldots+n(n+1)^{2}$. | \frac{n(n+1)(n+2)(3n+5)}{12} | 48 | 20 |
math | 9.6. Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing in a circle around a Christmas tree. Later, each of them was asked if their right neighbor was in a blue dress. It turned out that those who answered correctly were only the girls standing between girls in dresses of the same color. How many girls could have answered affirmatively?
(R. Zhenodarov) | 17 | 86 | 2 |
math | 4B. For which natural number $n$ does the expression $\frac{\lg 2 \cdot \lg 3 \cdot \lg 4 \ldots \ldots \lg n}{10^{n}}$ attain its smallest value? | 10^{10}10^{10}-1 | 52 | 13 |
math | 6.1. In the numerical example АБВ $+9=$ ГДЕ, the letters А, Б, В, Г, Д, and Е represent six different digits. What digit is represented by the letter Д? | 0 | 46 | 1 |
math | 8. There are three points $A$, $B$, and $C$ on the ground. A frog is located at point $P$, which is 0.27 meters away from point $C$. The frog's first jump is from $P$ to the symmetric point $P_{1}$ with respect to $A$, which we call the frog making a "symmetric jump" from $P$ about $A$; the second jump is from $P_{1}$ to the symmetric point $P_{2}$ with respect to $B$; the third jump is from $P_{2}$ to the symmetric point $P_{3}$ with respect to $C$; the fourth jump is from $P_{3}$ to the symmetric point $P_{1}$ with respect to $A$, and so on. If the frog reaches $P_{2009}$ after 2009 symmetric jumps, what is the distance between this point and the starting point $P$ in centimeters? | 54 | 209 | 2 |
math | 2. For any subset $A(|A|>1)$ of $X_{n}=\{1,2, \cdots, n\}$, let the sum of the products of every two elements in $A$ be $f(A)$, the sum of the sums of every two elements in $A$ be $g(A)$, and the sum of the absolute values of the differences of every two elements in $A$ be $h(A)$. Find $\sum_{A \leq X_{n}} f(A), \sum_{A \leq X_{n}} g(A), \sum_{A \leq X_{n}} h(A)$. | \sum_{1 \leq x_{n}} f(A)=2^{n-5} \cdot \frac{n(n+1)(3 n+2)(n-1)}{3}, \sum_{1 \leq x_{n}} g(A)=2^{n-3} n\left(n^{2}-1\right), \sum_{1 \leq x_{n}} h(A)=2^{n-2} C_{n+1} | 140 | 98 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f(f(x) f(y))=f(x) y
$$ | f(x)=0,f(x)=x,f(x)=-x | 50 | 13 |
math | Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit. | 34 | 80 | 2 |
math | 1. (8 points) Calculate: $(18 \times 23-24 \times 17) \div 3+5$, the result is | 7 | 35 | 1 |
math | Point $M$ divides the side $B C$ of triangle $A B C$ in the ratio $B M: M C=2: 5$, It is known that $\overrightarrow{A B}=\vec{a}, \overrightarrow{A C}=$ $\vec{b}$. Find the vector $\overrightarrow{A M}$. | \frac{5}{7}\vec{}+\frac{2}{7}\vec{b} | 74 | 20 |
math | 9.2. Solve the system of equations
$$
\left\{\begin{aligned}
10 x^{2}+5 y^{2}-2 x y-38 x-6 y+41 & =0 \\
3 x^{2}-2 y^{2}+5 x y-17 x-6 y+20 & =0
\end{aligned}\right.
$$ | 2,1 | 85 | 3 |
math | 15. Given the sequence
$$
b_{n}=\frac{1}{3 \sqrt{3}}\left[(1+\sqrt{3})^{n}-(1-\sqrt{3})^{n}\right](n=0,1, \cdots) \text {. }
$$
(1) For what values of $n$ is $b_{n}$ an integer?
(2) If $n$ is odd and $2^{-\frac{2 n}{3}} b_{n}$ is an integer, what is $n$? | 3 | 118 | 1 |
math | 2. Each cell of a $100 \times 100$ board is painted blue or white. We will call a cell balanced if among its neighbors there are an equal number of blue and white cells. What is the maximum number of balanced cells that can be on the board? (Cells are considered neighbors if they share a side.) | 9608 | 71 | 4 |
math | Example 11. Let $\mathrm{a}$ be a real number, find the minimum value of the quadratic function
$$
y=x^{2}-4 a x+5 a^{2}-3 a
$$
denoted as $\mathrm{m}$.
When $a$ varies in $0 \leqslant a^{2}-4 a-2 \leqslant 10$, find the maximum value of $m$. | 18 | 93 | 2 |
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