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 | [u]Set 1
[/u]
[b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, ... | 9 | 337 | 1 |
math | Find all integers $n$ such that $n(n+1)$ is a perfect square. | n=0orn=-1 | 19 | 6 |
math | Example 1. 6 students line up, where one of them does not stand at the head or the tail of the line, how many ways are there to arrange them? (Question 4, page 157) | 480 | 47 | 3 |
math | 2. Find all triples ( $a, b, c$ ) where $a, b, c$ are the lengths of the sides of triangle $ABC$ with angles $\alpha, \beta, \gamma$, such that the numbers $\cos \alpha, \cos \beta, \cos \gamma$ are the lengths of the sides of a triangle congruent to triangle $ABC$. | (\frac{1}{2},\frac{1}{2},\frac{1}{2}) | 79 | 21 |
math | 22. Let $f(x)=x^{2}-2 a x+2$, when $x \in[-1,+\infty]$, $f(x) \geqslant a$, find the range of values for $a$.
| \in[-3,1] | 52 | 7 |
math | 8. Solve the system $\left\{\begin{array}{l}3^{y} \cdot 81=9^{x^{2}} ; \\ \lg y=\lg x-\lg 0.5 .\end{array}\right.$ | 2,4 | 53 | 3 |
math | II. (40 points) Given that there exist integers $x_{1}, x_{2}, \cdots, x_{n}$ $\left(n \in \mathbf{Z}_{+}\right)$, such that
$$
(4 n+1)\left(\sum_{i=1}^{n} x_{i}^{2}-1\right)=4\left(\sum_{i=1}^{n} x_{i}\right)^{2} \text {. }
$$
Find all possible values of $n$ and the corresponding integer solutions $\left(x_{1}, x_{2}... | 2or6 | 140 | 3 |
math | 8.1. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved two seats to the right, Galia had moved one seat to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. Wh... | 2 | 94 | 1 |
math | 16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888. | 192 | 37 | 3 |
math | Example 1 Let real numbers $x_{1}, x_{2}, \cdots, x_{1}, \cdots 77$ satisfy the following two conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$.
Try to find the maximum value of $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$. (1... | 189548 | 170 | 6 |
math | ## Task 2.
A frog is located at the origin $O$ of a number line and needs to perform 2022 jumps, each of a different length from $1, 2, \ldots, 2022$. The jumps must be performed in such an order that the following rules are respected:
- if the frog is currently at point $O$ or to the left of it, it must jump to the ... | 1010 | 153 | 4 |
math | 8. The sequence $\left\{a_{n}\right\}$ with all terms being positive integers is defined as follows: $a_{0}=m, a_{n+1}=a_{n}^{5}+487(n \in \mathbf{N})$, then the value of $m$ that makes the sequence $\left\{a_{n}\right\}$ contain the most perfect squares is $\qquad$. | 9 | 92 | 1 |
math | Problem 4. Students from a school were supposed to go on a trip. $\frac{2}{9}$ more students registered than the planned number. Before departure, $\frac{3}{11}$ of the registered students canceled due to illness, so 5 fewer students went on the trip than the planned number. How many students went on the trip? | 40 | 73 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{-2 x}}{2 \arcsin x-\sin x}$ | 5 | 45 | 1 |
math | 4. On a plane, there are 10 different points. We considered the midpoints of all segments connecting all pairs of points. What is the smallest number of midpoints that could have arisen? | 17 | 41 | 2 |
math | 9. In a regular tetrahedron $S-ABC$, $M$ and $N$ are the midpoints of $SB$ and $SC$ respectively. If the section $AMN \perp$ side face $SBC$, then the dihedral angle between the side face and the base of the tetrahedron is $\qquad$ | \arccos \frac{\sqrt{6}}{6} | 75 | 14 |
math | ## Task B-3.2.
In a triangle with sides of length $a, b, c$ and area $P$, the following equality holds:
$$
\sqrt{3}\left(b^{2}+a^{2}-c^{2}\right)=2 a b-4 P
$$
Calculate the measure of the angle opposite the side of length $c$. | 90 | 78 | 2 |
math | 97. The number $A$ is written in the decimal system using 666 threes, and the number $B$ is written using 666 sixes. From which digits does the product $A \cdot B$ consist? | 22\ldots2177\ldots78 | 52 | 14 |
math | 6.45 Let $\varphi(n)$ denote the number of natural numbers less than $n$ that are coprime to $n$.
(1) $p, q$ are two distinct prime numbers. Prove that
$$
\varphi(p q)=(p-1)(q-1) .
$$
(2) Using the result from (1), find the values of $p, q$ that satisfy $\varphi(p q)=3 p+q$. (Anhui Province, Anqing City Junior High Sch... | p=3,q=11 | 121 | 7 |
math | Let $ m \equal{} 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n \plus{} 1)(5n \plus{} 2)$ is divisible by $ m$ (which means that $ m \mid n(2n \plus{} 1)(5n \plus{} 2)$) ? | 8 | 86 | 3 |
math | 13.341. Three workers participated in a competition. The first and third of them produced twice as much product as the second, while the second and third produced three times as much as the first. What place did each worker take in the competition? | 5:4:3 | 53 | 5 |
math | 1. Can we obtain the triplet of numbers $2,6,9$ in some order from the triplet with numbers $2,4,7$? | no | 32 | 1 |
math | 11. (20 points) Let $\{a_n\}$ be an arithmetic sequence with a non-zero common difference, satisfying $a_3 + a_6 = a_9$ and $a_5 + a_7^2 = 6a_9$. Let the sequence $\{b_n\}$ have the sum of its first $n$ terms denoted by $S_n$, and $4S_n + 2b_n = 3$. For any $i \in \mathbf{Z}_+$, insert $i$ numbers $x_{i1}, x_{i2}, \cdo... | (9,2),(3,3) | 270 | 9 |
math | Solve the following equation:
$$
3 \sin ^{2} x-4 \cos ^{2} x=\frac{\sin 2 x}{2} .
$$ | x_{1}=135\k\cdot180,\quadx_{2}=537^{\}\k\cdot180(k=0,1,2\ldots) | 37 | 43 |
math | Exercise 6. We have 102 distinct gifts. We want to distribute them to 100 winners of a contest, so that each winner receives at least one gift. Let $N$ be the number of ways to do this. Calculate
$$
\frac{N \times 48}{1 \times 2 \times 3 \times \cdots \times 102}
$$ | 60200 | 87 | 5 |
math | 6. If acute angles $A, B, C$ satisfy $\sin ^{2} A+\sin ^{2} B+\sin ^{2} C=2$, then the minimum value of $\frac{1}{\sin ^{2} A \cos ^{4} B}+\frac{1}{\sin ^{2} B \cos ^{4} C}+\frac{1}{\sin ^{2} C \cos ^{4} A}$ is $\qquad$ | \frac{81}{2} | 107 | 8 |
math | 13. Find the sum of all the real numbers $x$ that satisfy the equation
$$
\left(3^{x}-27\right)^{2}+\left(5^{x}-625\right)^{2}=\left(3^{x}+5^{x}-652\right)^{2} .
$$ | 7 | 75 | 1 |
math | 18. (16 points) In $\triangle A B C$ with a fixed perimeter, it is known that $|A B|=6$, and when vertex $C$ is at a fixed point $P$, $\cos C$ has a minimum value of $\frac{7}{25}$.
(1) Establish an appropriate coordinate system and find the equation of the locus of vertex $C$;
(2) Draw a line through point $A$ that in... | 16 | 130 | 2 |
math | [b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\s... | 1 | 142 | 3 |
math | Question 85: Find all pairs of positive integers $(m, a)$ such that the sequence $\left\{a_{n}\right\}_{n \in Z^{+}}: a_{1}=a, a_{n+1}=$ $\left\{\begin{array}{c}a_{n}^{2}+2^{m}, \text { when } a_{n}<2^{m} \\ \frac{a_{n}}{2}, \text { when } a_{n} \geq 2^{m}\end{array}\right.$ consists of integers for all terms. | (2,2^{\mathrm{k}}),\mathrm{k}\in\mathrm{Z}^{+} | 128 | 23 |
math | Example 4.13 Find the number of 7-combinations of the multiset $S=\{4 \cdot a, 4 \cdot b, 3 \cdot c, 3 \cdot d\}$. | 60 | 47 | 2 |
math | Exercise 2. In a game, a strictly positive integer $n$ can be replaced by the integer $a b$ if $n=a+b$, with strictly positive integers $a$ and $b$. Can we obtain the number 2011 starting from $n=5$? | 2011 | 60 | 4 |
math | 3.260. $\frac{4 \sin ^{4}\left(\alpha-\frac{3}{2} \pi\right)}{\sin ^{4}\left(\alpha-\frac{5 \pi}{2}\right)+\cos ^{4}\left(\alpha+\frac{5 \pi}{2}\right)-1}$. | -2\operatorname{ctg}^{2}\alpha | 74 | 13 |
math | 9. In the sequence $\left\{a_{n}\right\}$, $a_{4}=1, a_{11}=9$, and the sum of any three consecutive terms is 15, then $a_{2016}=$ $\qquad$ | 5 | 58 | 1 |
math | 38.2. Solve the equation
$$
\left(x^{2}+x+1\right)\left(3-x-x^{2}\right)=3
$$
$$
\text { (8-10 grades) }
$$ | x_{1}=0,x_{2}=-1,x_{3}=1,x_{4}=-2 | 52 | 22 |
math | Find all pairs of positive integers $(p; q) $such that both the equations $x^2- px + q = 0 $ and $ x^2 -qx + p = 0 $ have integral solutions. | (p, q) = (4, 4), (6, 5), (5, 6) | 46 | 25 |
math | 6.64*. In a regular $n$-gon ( $n \geqslant 3$ ), the midpoints of all sides and diagonals are marked. What is the maximum number of marked points that can lie on one circle? | n | 52 | 1 |
math | Example 2 Given that $a$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Then the value of $\frac{a^{3}-1}{a^{5}+a^{4}-a^{3}-a^{2}}$ is $\qquad$ .
(1995, National Junior High School Mathematics League) | 20 | 80 | 2 |
math | 1. The last four digits of the number $7^{355}$ are $\qquad$ | 1943 | 21 | 4 |
math | 24th ASU 1990 Problem 21 For which positive integers n is 3 2n+1 - 2 2n+1 - 6 n composite? | all\n\neq1 | 41 | 6 |
math | B1. Given is a square $A B C D$. You start at vertex $A$. On each turn, you may walk along a side from one vertex to another.
How many walks of 10 turns are there such that you are back at vertex $A$ after the 10 turns? During a walk, you may pass through $A$ on the way. | 512 | 78 | 3 |
math | [ Properties and characteristics of a parallelogram ] [ Congruent triangles. Criteria for congruence ]
Side $BC$ of parallelogram $ABCD$ is twice as long as side $AB$. The bisectors of angles $A$ and $B$ intersect line $CD$ at points $M$ and $N$, respectively, and $MN=12$.
Find the sides of the parallelogram. | 4,8,4,8 | 88 | 7 |
math | Example $\mathbf{1}$ Given $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+}$ and $\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=\pi$, find the minimum value of $\left(2 \sin ^{2} \theta_{1}+\frac{1}{\sin ^{2} \theta_{1}}\right)\left(2 \sin ^{2} \theta_{2}+\frac{1}{\sin ^{2} \theta_{2}}\right)\left(2 \si... | 3^{4} | 202 | 4 |
math | 31. How many natural numbers not exceeding 500 and not divisible by 2, 3, or 5 are there? | 134 | 29 | 3 |
math | Which is the geometric mean pair, where the sum of the outer terms is 24, the sum of the inner terms is 16, and the sum of the squares of all terms is 580? | 21:7=9:3 | 45 | 8 |
math | 2. $975 \times 935 \times 972 \times(\quad)$, to make the last four digits of this product all "0", what is the smallest number that should be filled in the parentheses?
To make the last four digits of this product all "0", what is the smallest number that should be filled in the parentheses? | 20 | 76 | 2 |
math | Adva van $A B C$ háromszögnek egyik oldala $A B$. Emeljünk a $B C$ oldalra, ennek középpontjában merólegest, mely $A C$-t $M$-ben metszi. Mi az $M$ pont mértani helye, ha a háromszög $C$ csúcsa $A$ körül forog?
Given that one side of triangle $ABC$ is $AB$. Let's raise a perpendicular from the midpoint of side $BC$, w... | \frac{x^{2}}{b^{2}}+\frac{y^{2}}{b^{2}-^{2}}=\frac{1}{4} | 160 | 33 |
math | [Example 3.6.4] There are $h+s$ lines on a plane, where $h$ lines are horizontal, and $s$ lines satisfy:
(1) None of them are horizontal lines;
(2) No two of them are parallel;
(3) No three of the $h+s$ lines are concurrent, and these $h+s$ lines exactly divide the plane into 1992 regions.
Find all pairs of positive in... | (995,1),(176,10),(80,21) | 101 | 20 |
math | Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation. | n = 1 \text{ and } n = 5 | 30 | 13 |
math | 9. In the Cartesian coordinate system, $F_{1}, F_{2}$ are the two foci of the hyperbola $\Gamma: \frac{x^{2}}{3}-y^{2}=1$. A point $P$ on $\Gamma$ satisfies $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}=1$. Find the sum of the distances from point $P$ to the two asymptotes of $\Gamma$. | \frac{3\sqrt{2}}{2} | 98 | 12 |
math | (14 Given real numbers $a$, $b$, $c$, $d$ satisfy $ab=c^2+d^2=1$, then $(a-c)^2+$ $(b-d)^2$ the minimum value is $\qquad$ . | 3-2\sqrt{2} | 52 | 8 |
math | How many twin prime pairs are there whose sum is a prime power? (We call two prime numbers twin primes if their difference is 2.) | (3,5) | 29 | 5 |
math | ## Problem 4
Determine the primitives of the function $f:[0, \pi] \rightarrow \mathbf{R}$, which is primitive-able, and satisfies the relation $f(x) \sin x - f(\pi - x) = \cos^2 x, \forall x \in [0, \pi]$. | F(x)=-x+\cosx+C | 72 | 9 |
math | 11. In tetrahedron $ABCD$, $AB=CD=a, BC=AD=b, CA=BD=c$. If the angle between the skew lines $AB$ and $CD$ is $\theta$, then, $\cos \theta=$ $\qquad$ . | \frac{\left|b^{2}-c^{2}\right|}{a^{2}} | 58 | 20 |
math | 6、Taking the train from City A to City B, it initially took 19.5 hours at the beginning of 1998. In 1998, the train's speed was increased by $30\%$ for the first time, in 1999 it was increased by $25\%$ for the second time, and in 2000 it was increased by $20\%$ for the third time. After these three speed increases, th... | 10 | 122 | 2 |
math | Ten consecutive positive integers have a sum that is a divisor of the sum of their squares. Which are these ten numbers? | 1,2,3,\ldots,10or12,13,14,\ldots,21 | 24 | 26 |
math | Let $\alpha$ and $\beta$ be positive integers such that
$$
\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} .
$$
Find the smallest possible value of $\beta$. | 23 | 53 | 2 |
math | Example 1. Let $z$ be a complex number, solve the equation
$$
\frac{1}{2}(z-1)=\frac{\sqrt{3}}{2}(1+z) \text { i. }
$$ | z=-\frac{1}{2}+\frac{\sqrt{3}}{2} i | 50 | 20 |
math | 1st Brazil 1979 Problem 2 The remainder on dividing the polynomial p(x) by x 2 - (a+b)x + ab (where a and b are unequal) is mx + n. Find the coefficients m, n in terms of a, b. Find m, n for the case p(x) = x 200 divided by x 2 - x - 2 and show that they are integral. | =\frac{2^{200}-1}{3},n=\frac{2^{200}+2}{3} | 90 | 28 |
math | Write the result of the following expressions in decimal form:
(a) $7 \times \frac{2}{3}+16 \times \frac{5}{12}$
(b) $5-\left(2 \div \frac{5}{3}\right)$
(c) $1+\frac{2}{1+\frac{3}{1+4}}$ | 11.3333\ldots,3.8,2.25 | 78 | 19 |
math | 1. Given $y z \neq 0$, and the set $\{2 x, 3 z, x y\}$ can also be represented as $\left\{y, 2 x^{2}, 3 x z\right\}$, then $x=$ | 1 | 58 | 1 |
math | Example: Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$. | 402005 | 86 | 6 |
math | Example 4 Find the equation of the curve $E^{\prime}$ symmetric to the curve $E: 2 x^{2}+4 x y+5 y^{2}-22=0$ with respect to the line $l: x-2 y+1=0$. | 146x^{2}-44xy+29y^{2}+152x-64y-494=0 | 60 | 32 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{\cos (2 \pi x)}{2+\left(e^{\sqrt{x-1}}-1\right) \operatorname{arctg} \frac{x+2}{x-1}}$ | \frac{1}{2} | 67 | 7 |
math | ## 1. Jaja
Baka Mara has four hens. The first hen lays one egg every day. The second hen lays one egg every other day. The third hen lays one egg every third day. The fourth hen lays one egg every fourth day. If on January 1, 2023, all four hens laid one egg, how many eggs in total will Baka Mara's hens lay throughout... | 762 | 108 | 3 |
math | 6. $\sum_{i=0}^{50} \sum_{j=0}^{50} \mathrm{C}_{50}^{i} \mathrm{C}_{50}^{j}$ modulo 31 is
$\qquad$ . | 1 | 57 | 1 |
math | In the sum $+1+3+9+27+81+243+729$, any addends can be crossed out and the signs before some of the remaining numbers can be changed from "+" to "-". Masha wants to use this method to first get an expression whose value is 1, then, starting over, get an expression whose value is 2, then (starting over again) get 3, and ... | 1093 | 111 | 4 |
math | Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72.
a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$.
b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $... | m_0 = 200 | 143 | 9 |
math | Let $n_0$ be the product of the first $25$ primes. Now, choose a random divisor $n_1$ of $n_0$, where a choice $n_1$ is taken with probability proportional to $\phi(n_1)$. ($\phi(m)$ is the number of integers less than $m$ which are relatively prime to $m$.) Given this $n_1$, we let $n_2$ be a random divisor of $n_1$, ... | \frac{256}{5929} | 138 | 12 |
math | Problem 4. Determine the natural numbers $a$ for which there exist exactly 2014 natural numbers $b$ that satisfy the relation $2 \leq \frac{a}{b} \leq 5$. | 6710,6712,6713 | 48 | 14 |
math | 6. Choose any two non-adjacent numbers from $1,2, \cdots, 10$ and multiply them. The sum of all such products is $\qquad$ | 990 | 39 | 3 |
math | Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer. | 600 | 75 | 3 |
math | 11. Find all values of $b$ for which the equation
$$
a^{2-2 x^{2}}+(b+4) a^{1-x^{2}}+3 b+4=0
$$
has no solutions for any $a>1$. | [-\frac{4}{3};+\infty) | 58 | 12 |
math | Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a [i]grade[/i] is a nonnegative decimal number with finitely many digits after the decimal point.
Given a grade with digits $a_1a_2 \dots a_m.b_1b_2 \dots b_n$, Joe first rounds the number to the nearest $10^{-n... | 814 | 275 | 3 |
math | 8. In $\triangle A B C$,
$$
\frac{\sin \frac{A}{2} \cdot \sin \frac{B}{2}+\sin \frac{B}{2} \cdot \sin \frac{C}{2}+\sin \frac{C}{2} \cdot \sin \frac{A}{2}}{\sin A+\sin B+\sin C}
$$
the maximum value is | \frac{\sqrt{3}}{6} | 90 | 10 |
math | $3 \cdot 10$ Find all odd natural numbers $n$ such that $(n-1)!$ is not divisible by $n^2$.
(4th All-Russian Mathematical Olympiad, 1964) | nbeinganoddn=9 | 49 | 7 |
math | 4. Given the function $f(x)=4 \sin x \cdot \sin ^{2}$ $\left(\frac{\pi}{4}+\frac{x}{2}\right)+\cos 2 x$, if the constant $w>0, y=f(w x)$ is an increasing function on the interval $\left[-\frac{\pi}{2}, \frac{2 \pi}{3}\right]$, then the range of $w$ is_ _. $\qquad$ | (0,\frac{3}{4}] | 100 | 9 |
math | 9. (15 points) 12 children jointly buy a set of books, and the cost of buying the books is shared equally among them. Since 2 of the children did not bring money when purchasing, the remaining 10 children each paid an extra 10 yuan. How much does it cost to buy the set of books in total? | 600 | 73 | 3 |
math | 1765. To determine the percentage of non-standard items in a batch with a volume of 10000, 500 items were selected, among which 10 were found to be non-standard. Find the sampling error in determining the proportion in cases of with-replacement and without-replacement sampling. | 0.00626 | 67 | 7 |
math | # Problem 6. (3 points)
Positive numbers $x, y, z$ are such that $x y + y z + x z = 12$.
Find the smallest possible value of $x + y + z$.
# | 6 | 51 | 1 |
math | 8. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is known that $A B=$ $A A_{1}=4, A D=3$. Then the distance between the skew lines $A_{1} D$ and $B_{1} D_{1}$ is $\qquad$ | \frac{6 \sqrt{34}}{17} | 78 | 14 |
math | 18. How many two-digit numbers can be formed from the five digits $1,2,3,4,5$ if no digit is repeated? | 20 | 32 | 2 |
math | 4. If $p, q \in N^{+}$ and $p+q>2017, 0<p<q \leq 2017, (p, q)=1$, then the sum of all fractions of the form $\frac{1}{p q}$ is $\qquad$ | \frac{1}{2} | 67 | 7 |
math | 7. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{0}=0$,
$$
a_{n+1}=\frac{8}{5} a_{n}+\frac{6}{5} \sqrt{4^{n}-a_{n}^{2}}(n \geqslant 0, n \in \mathbf{N}) \text {. }
$$
Then $a_{10}=$ . $\qquad$ | \frac{24576}{25} | 103 | 12 |
math | Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$ | 1999000 \text{ and } 2000 | 35 | 17 |
math | ## 7. Bats, bears, poodles, and elephants
Attentive naturalists have established that the daily diet of 17 bears matches the diet of 170 poodles, the diet of 100000 bats matches the diet of
50 poodles, and 10 bears consume as much in a day as 4 elephants. How many bats can handle the diet of a dozen elephants? | 600000x | 91 | 7 |
math | 10.1 A weirdo didn't mind and recorded 2022 numbers in a circle, and it turned out that each number coincides with the absolute difference of its two neighbors. Determine which numbers were recorded if their sum is 2022. | \frac{3}{2} | 55 | 7 |
math | 16. (6 points) If $2^{a} \times 3^{b} \times 5^{c} \times 7^{d}=252000$, then the probability that a three-digit number formed by randomly selecting 3 out of the natural numbers $a$, $b$, $c$, $d$ is divisible by 3 and less than 250 is $\qquad$ . | \frac{1}{4} | 90 | 7 |
math | Example 9. Find the range of $y=\frac{4 \sin x+5 \cos x}{\sin x+\cos x+3}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \left[\frac{-9-4 \sqrt{23}}{7}, \frac{-9+4 \sqrt{23}}{7}\right] | 58 | 34 |
math | 3. [6 points] In the plane $O x y$, the equation $5 a^{2}-4 a y+8 x^{2}-4 x y+y^{2}+12 a x=0$ defines the coordinates of point $A$, and the equation $a x^{2}-2 a^{2} x-a y+a^{3}+3=0$ defines a parabola with vertex at point $B$. Find all values of the parameter $a$ for which points $A$ and $B$ lie on the same side of th... | (-\frac{5}{2};-\frac{1}{2})\cup(0;3) | 142 | 22 |
math | 13.355. A car, having traveled a distance from $A$ to $B$, equal to 300 km, turned back and after 1 hour 12 minutes from leaving $B$, increased its speed by 16 km/h. As a result, it spent 48 minutes less on the return trip than on the trip from $A$ to $B$. Find the original speed of the car. | 60 | 91 | 2 |
math | Ask 12 Let $f(x)=\sqrt{x^{2}+1}-1$. Find all real solutions to the equation $f(\underbrace{f(\cdots(f(x))}_{260} \cdots$
1)) $=x$. | 0 | 55 | 1 |
math | Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$.
[I]Proposed by Italy[/i] | \{\pm 1, \pm 2, \pm 3, \pm 6, \pm x, \pm 2x\} | 45 | 32 |
math | Task A-4.5. (8 points)
What is the sum of all natural numbers $n$ for which the number $\frac{2009-n}{99}$ is a natural number? | 19390 | 43 | 5 |
math | Find all positive integers $a\in \{1,2,3,4\}$ such that if $b=2a$, then there exist infinitely many positive integers $n$ such that $$\underbrace{aa\dots aa}_\textrm{$2n$}-\underbrace{bb\dots bb}_\textrm{$n$}$$ is a perfect square. | \{1, 4\} | 82 | 9 |
math | 8. Given positive real numbers $a, b, c$ satisfy $2(a+b)=a b$, and $a+b+c=a b c$, then the maximum value of $c$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{8}{15} | 66 | 8 |
math | 6. Given three points $A, B, C$ on a plane satisfying $|\overrightarrow{A B}|=3,|\overrightarrow{B C}|=4,|\overrightarrow{C A}|=5$, then $\overrightarrow{A B} \cdot \overrightarrow{B C}+\overrightarrow{B C} \cdot \overrightarrow{C A}+\overrightarrow{C A} \cdot \overrightarrow{A B}$ is equal to | -25 | 99 | 3 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-8 ; 8]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 272 | 54 | 3 |
math | 2. [4] Let $A, B, C, D, E, F$ be 6 points on a circle in that order. Let $X$ be the intersection of $A D$ and $B E$, $Y$ is the intersection of $A D$ and $C F$, and $Z$ is the intersection of $C F$ and $B E$. $X$ lies on segments $B Z$ and $A Y$ and $Y$ lies on segment $C Z$. Given that $A X=3, B X=2, C Y=4, D Y=10, E ... | \frac{77}{6} | 154 | 8 |
math | The student population at one high school consists of freshmen, sophomores, juniors, and seniors. There are 25 percent more freshmen than juniors, 10 percent fewer sophomores than freshmen, and 20 percent of the students are seniors. If there are 144 sophomores, how many students attend the school? | 540 | 74 | 3 |
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