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 | 6. A die is rolled four times in succession, and from the second roll onwards, the number of points that appear each time is not less than the number of points that appeared in the previous roll. The probability of this happening is $\qquad$ . | \frac{7}{72} | 52 | 8 |
math | Example 3 Let the equation of the ellipse be $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$, the major axis be $A A^{\prime}, P$ be any point on the ellipse, draw $A Q \perp A P$, $A^{\prime} Q \perp A P, A Q$ and $A^{\prime} Q$ intersect at $Q$, find the locus of $Q$. | \frac{x^{2}}{16}+\frac{y^{2}}{\frac{256}{9}}=1 | 99 | 28 |
math | The expressions $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$ and $ abc$ are called the elementary symmetric expressions on the three letters $ a, b, c;$ symmetric because if we interchange any two letters, say $ a$ and $ c,$ the expressions remain algebraically the same. The common degree of its terms is called ... | \binom{9891}{1989} | 181 | 14 |
math | Let $a$ and $b$ be positive integers that are relatively prime, and let $A=8a+3b$ and $B=3a+2b$. The greatest common divisor of $A$ and $B$ is not 1. What is the greatest common divisor of $A$ and $B$? | 7 | 69 | 1 |
math | 28. In response to a question about his age, the grandfather answered: “The number expressing my age in years is a two-digit number equal to the sum of the number of its tens and the square of its units.” How old is the grandfather? | 89 | 52 | 2 |
math | Example 21 Let $x_{1}, x_{2}, \cdots, x_{n}$ all be natural numbers, and satisfy $x_{1}+x_{2}+\cdots+x_{n}=x_{1} x_{2} \cdots x_{n}$.
Find the maximum value of $x_{1}, x_{2}, \cdots, x_{n}$ $(n \geqslant 2)$. | n | 95 | 1 |
math | 3. Given $\left\{a_{n}\right\}$ is a geometric sequence, and $a_{1} a_{2017}=1$. If $f(x)=\frac{2}{1+x^{2}}$, then $\sum_{i=1}^{2017} f\left(a_{i}\right)=$ $\qquad$ | 2017 | 78 | 4 |
math | 3 [ Quadrilateral: calculations, metric relations.]
Find the angles of the convex quadrilateral $A B C D$, in which $\angle B A C=30^{\circ}, \angle A C D=40^{\circ}, \angle A D B=50^{\circ}, \angle C B D$ $=60^{\circ}$ and $\angle A B C+\angle A D C=180^{\circ}$. | \angleABC=100,\angleADC=80,\angleBAD=\angleBCD=90 | 96 | 23 |
math | $\begin{array}{l}\text { 6. Given }\left(1+x+x^{2}+x^{3}\right)^{400} \\ =\sum_{k=0}^{300} c_{4 k} x^{4 k}+\sum_{k=0}^{299} c_{4 k+1} x^{4 k+1}+ \\ \sum_{k=0}^{299} c_{4 k+2} x^{4 k+2}+\sum_{k=0}^{299} c_{4 k+3} x^{4 k+3} \text {. } \\ \text { then } \sum_{k=0}^{300} c_{4 k}=\end{array}$ | 4^{399} | 170 | 6 |
math | Problem 6. Find the equation of the plane $ABF$, if $F(-4; 8; -3), A(-1; 2; -3), B(3; 4; 1)$. | 5z-2y-4x+15=0 | 47 | 13 |
math | 2. (12 points) In a family, there are four children of different ages. Their total age is 33 years. Three years ago, the total age of all the children in the family was 22 years, 7 years ago it was 11 years, and 13 years ago it was 1 year. How old are the children at present? (Age is always expressed as a whole number ... | 2,6,11,14 | 91 | 9 |
math | Example 4.25. Find a particular solution of the equation
$$
y^{\prime \prime}+2 y^{\prime}-8 y=(12 x+20) e^{2 x}
$$
satisfying the initial conditions $y(0)=0, y^{\prime}(0)=1$. | \frac{1}{3}e^{-4x}-\frac{1}{3}e^{2x}+(x^{2}+3x)e^{2x} | 70 | 37 |
math | A square $ABCD$ with side length $1$ is inscribed in a circle. A smaller square lies in the circle with two vertices lying on segment $AB$ and the other two vertices lying on minor arc $AB$. Compute the area of the smaller square.
| \frac{1}{25} | 55 | 8 |
math | 3-ча 1. Find all positive rational solutions of the equation \(x^{y}=y^{x}(x \neq y)\). | (\frac{p+1}{p})^{p},(\frac{p+1}{p})^{p+1} | 30 | 26 |
math | 110) Let $p$ be a given odd prime, and let the positive integer $k$ be such that $\sqrt{k^{2}-p k}$ is also a positive integer. Then $k=$ $\qquad$ | \frac{(p+1)^{2}}{4} | 48 | 13 |
math | 9. Arrange all positive integers that are coprime with 143 in ascending order to form a sequence. The 2022nd term of this sequence is $\qquad$ | 2410 | 40 | 4 |
math | Example 3.2.5 Let the sequence $A=a_{1}, a_{2}, \cdots, a_{2005}, a_{i} \in \mathbf{N}^{+}, m(A)$ be the number of $\left(a_{i}, a_{j}, a_{k}\right)(1 \leqslant i<j<k \leqslant 2005)$ satisfying $a_{j}=$ $a_{i}+1, a_{k}=a_{j}+1$. For all possible $A$, find the maximum value of $m(A)$. | 668^{2}\times669 | 131 | 10 |
math | 6. Let $x$ be an acute angle. Then the maximum value of the function $y=\sin x \cdot \sin 2 x$ is . $\qquad$ | \frac{4\sqrt{3}}{9} | 37 | 12 |
math | Initial 275 Given $\qquad$
$$
\begin{array}{l}
23 \times \overline{a b c}=4 \times \overline{p q r}, 16 \times \overline{r p q}=5 \times \overline{x y z}, \\
22 \times \overline{z x y}=7 \times c a b,
\end{array}
$$
where $\overline{a b c} 、 \overline{p r} 、 \overline{x y z}$ represent three-digit numbers, and the sam... | \begin{array}{l}
23 \times 148=4 \times 851, \\
16 \times 185=5 \times 592, \\
22 \times 259=7 \times 814
\end{array} | 138 | 65 |
math | 2. Find the minimum value of the expression $\frac{1+x^{2}}{1+x}$, for $x \geq 0$. | 2(\sqrt{2}-1) | 31 | 8 |
math | 2. For what values of $a, b, c$ is the polynomial $x^{4}+a x^{2}+b x+c$ divisible by $(x-1)^{3}$? | =-6,b=8,=-3 | 43 | 8 |
math | (a) In how many ways can $1003$ distinct integers be chosen from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers di ffer by $10?$
(b) Show that there are $(3(5151) + 7(1700)) 101^7$ ways to choose $1002$ distinct integers from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers dif... | 101^7 | 129 | 5 |
math | 6.27 In triangle $A B C$, $\angle A=2 \angle B$, $\angle C$ is an obtuse angle, and the three side lengths $a$, $b$, $c$ are all integers. Find the minimum perimeter and provide a proof.
(20th United States of America Mathematical Olympiad, 1991) | 77 | 75 | 2 |
math | (1) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following two conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1977}=-318 \sqrt{3}$.
Find: $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$'s maximum value, and explain the reason. | 189548 | 159 | 6 |
math | Example 1. Let $F=x^{3}-3 x^{2}-9 x+5$, find the extremum of $F$. | F_{\text {min }} = -22, F_{\text {max }} = 10 | 29 | 23 |
math | 1. Use $1,2,3,4,5$ to form a five-digit number, such that the difference between any two adjacent digits is at least 2. Then the number of such five-digit numbers is $\qquad$ . | 14 | 50 | 2 |
math | Let's determine those twin prime numbers $p$ and $q$ for which $p^{2}-p q+q^{2}$ is also prime. (The prime numbers $p$ and $q$ are twin primes if $|p-q|=2$.
---
The translation is provided as requested, maintaining the original formatting and structure. | p=3,q=5 | 70 | 6 |
math | Let $n$ be a positive integer such that $\lfloor\sqrt n\rfloor-2$ divides $n-4$ and $\lfloor\sqrt n\rfloor+2$ divides $n+4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x\rfloor$ refers to the greatest integer less than or equal to $x$.) | 956 | 87 | 3 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-1}{1}=\frac{y+1}{0}=\frac{z-1}{-1}$
$3 x-2 y-4 z-8=0$ | (2,-1,0) | 58 | 7 |
math | [Theorem of Three Perpendiculars]
The base of the pyramid is a rectangle with an area of S. Two lateral faces are perpendicular to the base plane, while the other two are inclined to it at angles of $30^{\circ}$ and $60^{\circ}$. Find the volume of the pyramid. | \frac{1}{3}S\sqrt{S} | 67 | 13 |
math | Example 2. Find the maximum and minimum values of the function $y=\frac{x^{2}+x-1}{x^{2}+x+1}$. | y_{\mathrm{min}}=-\frac{5}{3} | 36 | 15 |
math | What is the probability when rolling 3 dice that
a) exactly one die shows a 6?
b) at least one die shows a 6?
c) at most one die shows a 6?
(For the example, see the article “>> A” Elements of Probability Calculation <<” in this issue.) | \frac{25}{72},\frac{91}{216},\frac{25}{27} | 65 | 28 |
math | 1. How many tiles with dimensions $15 \times 15$ cm are needed to tile a wall that is 3 m 6 dm long and 27 dm wide $?$ | 432 | 40 | 3 |
math | $:$ Govanov A.S.
Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya sequentially called out natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice and wrote down the obtained values on the board from left to right. It tur... | 50 | 106 | 2 |
math | 5. Given $[x]$ denotes the greatest integer not exceeding $x$, $a, b, c \in$ $R^{+}, a+b+c=1$, let $M=\sqrt{3 a+1}+\sqrt{3 b+1}+$ $\sqrt{3 c+1}$. Then the value of $[M]$ is $($. | 4 | 76 | 1 |
math | Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smal... | 89 | 149 | 2 |
math | Example 16 Find all real-coefficient binary polynomials $f(x, y)$ of degree $n$, such that for any real numbers $x, y$, we have $f(x+1, y+1)=f(x, y)$. | f(x,y)=\sum_{k=0}^{n}a_{k}(x-y)^{k} | 52 | 24 |
math | 7. Given a regular 2019-gon, then, the maximum number of diagonals such that any two of them are either perpendicular or do not intersect except at endpoints. | 2016 | 38 | 4 |
math | 5. In a sports park, over 4 days, Jurica could have earned 760 kuna and a ball. He only worked one day for which he received 40 kuna and a ball. What is the value of the ball? | 200 | 53 | 3 |
math | Positive integers $a$ and $b$ satisfy $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$. Find $a^2 + b^2 + c^2$. | 226 | 65 | 3 |
math | 13. [25] Four circles with radii $1,2,3$, and $r$ are externally tangent to one another. Compute $r$. (No proof is necessary.) | \frac{6}{23} | 40 | 8 |
math | H5511 ** If a number can be decomposed into the product of $k$ consecutive natural numbers greater than 1, then we say this number has property $p(k)$.
(1) Find a $k$, for which there exists a number that has both properties $p(k)$ and $p(k+2)$;
(2) Prove: A number that has both properties $p(2)$ and $p(4)$ does not ex... | 3 | 97 | 1 |
math | 2. Find the last three digits of $7^{2014}$.
. | 849 | 18 | 3 |
math | 14. Given the functions $f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}+\sqrt{x+\frac{1}{x}+1}(x>0), g(x)=\sqrt{x}+$ $\frac{1}{\sqrt{x}}-\sqrt{x+\frac{1}{x}+1}(x>0)$, find the minimum value of $f(x)$ and the maximum value of $g(x)$. | 2-\sqrt{3} | 94 | 6 |
math | [b]p1.[/b] You have bought a box which contains six unsharpened pencils. Is it possible to arrange them so that every pair of pencils touch.
[b]p2.[/b] Is it possible to put $54$ rabbits in $10$ cages so that every pair of cages have a different number of rabbits, and each cage contains at least one rabbit?
[b]p3.[... | 2^7 > 100 | 212 | 10 |
math | Example 1 Solve the inequality $\log _{8}\left(x^{3}+x+3\right)>\log _{2}(x+1)$. | -1<x<\frac{-1+\sqrt{7}}{3} | 35 | 16 |
math | Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$ and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. ... | 770 | 122 | 3 |
math | 457. As a result of five measurements of the length of a rod with one instrument (without systematic errors), the following results (in mm) were obtained: $92 ; 94 ; 103 ; 105 ; 106$. Find: a) the sample mean length of the rod; b) the sample and corrected variances of the instrument errors. | 100,34,42.5 | 82 | 11 |
math | 3. How many three-digit positive numbers $x$ exist that are divisible by 3 and satisfy the equation $GCD(15, GCD(x, 20))=5$? Find the largest one. | 60 | 46 | 2 |
math | $1 \cdot 36$ Find all such four-digit numbers, when 400 is written to their left, the result is a perfect square.
Find all such four-digit numbers, when 400 is written to their left, the result is a perfect square. | 4001or8004 | 58 | 9 |
math | 4th Irish 1991 Problem A4 8 people decide to hold daily meetings subject to the following rules. At least one person must attend each day. A different set of people must attend on different days. On day N for each 1 ≤ k < N, at least one person must attend who was present on day k. How many days can the meetings be hel... | 128 | 79 | 3 |
math | ## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(2 ; 3 ; 1) \)
\( A_{2}(4 ; 1 ;-2) \)
\( A_{3}(6 ; 3 ; 7) \)
\( A_{4}(7 ; 5 ;-3) \) | 5 | 125 | 1 |
math | Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?
[i](Walther Janous)[/i] | 0 < \frac{1}{x + y + z + 1} - \frac{1}{(x + 1)(y + 1)(z + 1)} \leq \frac{1}{8} | 87 | 50 |
math | 4. Solve the system of equations:
$$
\begin{aligned}
& x^{2}+2 y z=1 \\
& y^{2}+2 x z=2 \\
& z^{2}+2 x y=1
\end{aligned}
$$
in the set of real numbers. | (1,0,1),(\frac{1}{3},\frac{4}{3},\frac{1}{3}),(-1,0,-1),(-\frac{1}{3},-\frac{4}{3},-\frac{1}{3}) | 65 | 57 |
math | 2. (16 points) Find the minimum value of the function
$$
f(x)=7 \sin ^{2} x+5 \cos ^{2} x+2 \sin x
$$ | 4.5 | 44 | 3 |
math | 10. In an oblique $\triangle A B C$, $\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=\sin ^{2} B$, then $\tan A \tan C=$ | 3 | 50 | 1 |
math | 1. Given $a^{2}(b+c)=b^{2}(a+c)=2010$, and $a \neq b$. Then $c^{2}(a+b)=$ | 2010 | 41 | 4 |
math | Condition of the problem
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{\operatorname{tg} x-\sin x}{x(1-\cos 2 x)}$ | \frac{1}{4} | 45 | 7 |
math | A rock travelled through an n x n board, stepping at each turn to the cell neighbouring the previous one by a side, so that each cell was visited once. Bob has put the integer numbers from 1 to n^2 into the cells, corresponding to the order in which the rook has passed them. Let M be the greatest difference of the numb... | 2n-1 | 87 | 6 |
math | [ Intersecting lines, angle between them]
In rectangle $A B C D$, the sides are given as $A B=3, B C=4$. Point $K$ is at distances $\sqrt{10}$, 2, and 3 from points $A, B$, and $C$ respectively. Find the angle between the lines $C K$ and $B D$. | \arcsin\frac{4}{5} | 82 | 11 |
math | ## problem statement
Write the canonical equations of the line.
$$
\begin{aligned}
& 2 x-3 y-2 z+6=0 \\
& x-3 y+z+3=0
\end{aligned}
$$ | \frac{x+3}{9}=\frac{y}{4}=\frac{z}{3} | 51 | 22 |
math | 5. If $1 \frac{5}{100}$ is subtracted from a number, and the resulting difference is multiplied by $\frac{4}{5}$, then the product is increased by $2 \frac{21}{25}$, and the resulting sum is divided by 0.01, the result is 1400. Determine the initial number.
Each problem is scored out of 10 points.
The use of a pocket... | 15 | 116 | 2 |
math | 6. Each of the three children was bought candies by their parents. Vitya was given 5 candies, while Masha received fewer candies than Vitya, and Sasha received as many candies as Vitya and Masha combined. How many candies could the parents have bought for all the children? | 18,16,14,12 | 62 | 11 |
math | Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three ... | 47 | 106 | 2 |
math | Find all solutions, in the set of positive real numbers, of the system of equations:
$$
\left\{\begin{array}{l}
x(x+y+z)=26 \\
y(x+y+z)=27 \\
z(x+y+z)=28
\end{array}\right.
$$ | (26/9,3,28/9) | 62 | 13 |
math | \section*{Exercise 6 - 021116}
Determine all real numbers \(x\) that satisfy the inequality
\[
\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}
\]
Verify the result! | -1\leqx<1-\frac{\sqrt{31}}{8} | 56 | 18 |
math | Example 1 Find all triples of integers $(x, y, z)$ such that
$$
x^{3}+y^{3}+z^{3}-3 x y z=2003^{[1]} \text {. }
$$
$(2003$, Nordic Mathematical Contest) | (668,668,667),(668,667,668),(667,668,668) | 63 | 37 |
math | Exercise 3. Consider a number $N$ that is written in the form $30 x 070 y 03$, with $x, y$ being digits between 0 and 9. For which values of $(x, y)$ is the integer $N$ divisible by 37? | (x,y)=(8,1),(4,4),(0,7) | 65 | 15 |
math | 4.1. All natural numbers from 1 to 2017 inclusive were written in a row. How many times was the digit 7 written? | 602 | 33 | 3 |
math | We say that a natural number $n$ is interesting if it can be written in the form
\[
n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor,
\] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$
Determine all interesting ... | n \geq 7 | 129 | 7 |
math | 10. Solve the system of equations $\left\{\begin{array}{l}x+y=\frac{\pi}{2} \\ \cos 2 x=\cos x+\cos y\end{array}\right.$ | {\begin{pmatrix}k\pi-\frac{\pi}{4}\\-k\pi+\frac{3\pi}{4}\end{pmatrix},\quad{\begin{pmatrix}-2k\pi\\2k\pi+\frac{\pi}{2}\end{pmatrix},\quad{\begin{pmatrix}-2k\pi-\frac{\pi} | 46 | 80 |
math | Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible. | n = 10 | 68 | 6 |
math | 7. If a small ball with a radius of 1 can move freely in all directions inside a regular tetrahedron container with an edge length of $6 \sqrt{6}$, then the area of the container's inner wall that the ball can never touch is $\qquad$ . | 120\sqrt{3} | 60 | 8 |
math | Task 4.1.3 Two mothers with their children want to sit on a bench with 4 seats. In how many ways can they sit so that each mother sits next to her child? Each mother is walking with one child.
# | 8 | 49 | 1 |
math | 1. Given that $x$ is a four-digit number, and the sum of its digits is $y$. When the value of $\frac{x}{y}$ is the smallest, $x=$ $\qquad$ | 1099 | 44 | 4 |
math | Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$. We are also given that $\angle ABC = \angle CDA = 90^o$. Determine the length of the diagonal $BD$. | \frac{2021 \sqrt{2}}{2} | 59 | 15 |
math | 17. Let $x=\sin ^{4}\left(\frac{\pi}{8}\right)+\cos ^{4}\left(\frac{\pi}{8}\right)+\sin ^{4}\left(\frac{7 \pi}{8}\right)+\cos ^{4}\left(\frac{7 \pi}{8}\right)$. Find the value of $36 x$. | 54 | 84 | 2 |
math | 14 Labeled as $1,2, \cdots, 100$, there are some matches in the matchboxes. If each question allows asking about the parity of the sum of matches in any 15 boxes, then to determine the parity of the number of matches in box 1, at least how many questions are needed? | 3 | 71 | 1 |
math | Find all triplets $(x, y, \ell) \in \mathbb{N}^{3}$ such that
$$
x^{3}+y^{3}-53=7^{\ell}
$$ | (3,3,0) | 46 | 7 |
math | 2. The coefficient of the $x$ term in the expansion of $\left(2 x+\frac{1}{\sqrt{x}}\right)^{7}$ is | 280 | 35 | 3 |
math | 11.10. What is the maximum area of the projection of a regular tetrahedron with edge length a onto a plane? | \frac{^2}{2} | 29 | 8 |
math | 11. Let the sequence of positive numbers $a_{n}, a_{1}, a_{2}, \cdots$ satisfy $a_{0}=a_{1}=1$ and $\sqrt{a_{n} a_{n-2}}-\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1}, n=2,3, \cdots$, find the general term formula of the sequence. | a_{n}=\prod_{k=1}^{n}(2^{k}-1)^{2} | 94 | 23 |
math | Example 8.17 (2009 Vietnam) Determine the minimum value of $k$ such that the following inequality holds for all positive real numbers $a, b, c$
$$\left(k+\frac{a}{b+c}\right)\left(k+\frac{b}{c+a}\right)\left(k+\frac{c}{a+b}\right) \geqslant\left(k+\frac{1}{2}\right)^{3}$$ | \frac{\sqrt{5}-1}{4} | 98 | 11 |
math | ## Task 3/70
It is $x^{0}=1$ and $0^{x}=0$ for $x \neq 0$. What is the value of $\lim _{x \rightarrow 0} x^{x}$ ? | 1 | 54 | 1 |
math | Task 4. (20 points) Find the smallest natural solution of the inequality $\left(\frac{2023}{2022}\right)^{36+24+16+\ldots+36\left(\frac{2}{3}\right)^{n}}>\left(\frac{2023}{2022}\right)^{96}$. | 5 | 85 | 1 |
math | IS. 4 If $\frac{d}{114}=\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right) \cdots\left(1-\frac{1}{c^{2}}\right)$, find the value of $d$. | \frac{58}{114} | 72 | 10 |
math | David, when submitting a problem for CMIMC, wrote his answer as $100\tfrac xy$, where $x$ and $y$ are two positive integers with $x<y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andr... | 299 | 88 | 3 |
math | Shelly-Ann normally runs along the Laurel Trail at a constant speed of $8 \mathrm{~m} / \mathrm{s}$. One day, onethird of the trail is covered in mud, through which Shelly-Ann can only run one-quarter of her normal speed, and it takes her 12 s to run the entire length of the trail. How long is the trail, in metres? | 48 | 85 | 2 |
math | Example 3. Determine the shape of the mirror reflecting the rays from a point source into a parallel beam. | y^{2}=2Cx+C^{2} | 22 | 10 |
math | 30. It is given that $a$ and $b$ are positive integers such that $a$ has exactly 9 positive divisors and $b$ has exactly 10 positive divisors. If the least common multiple (LCM) of $a$ and $b$ is 4400 , find the value of $a+b$. | 276 | 74 | 3 |
math | 2. Calculate
$$
\frac{\frac{1}{2}-\frac{1}{3}}{\frac{1}{3}-\frac{1}{4}} \cdot \frac{\frac{1}{4}-\frac{1}{5}}{\frac{1}{5}-\frac{1}{6}} \cdot \ldots \cdot \frac{\frac{1}{98}-\frac{1}{99}}{\frac{1}{99}-\frac{1}{100}}
$$ | 50 | 111 | 2 |
math | [ Mutual relations between sides and angles of triangles (other).]
In triangle $ABC$, the bisectors $AD$ and $BE$ are drawn. Find the measure of angle $C$, given that $AD \cdot BC = BE \cdot AC$ and $AC \neq BC$. | 60 | 60 | 2 |
math | Example 1. If the sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=a_{n}+3n+2, a_{1}=2$, find its general term formula. | a_{n}=\frac{1}{2}\left(3 n^{2}+n\right) | 48 | 23 |
math | $6.280 \sqrt{x}-\sqrt{x+1}-\sqrt{x+4}+\sqrt{x+9}=0$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$6.280 \sqrt{x}-\sqrt{x+1}-\sqrt{x+4}+\sqrt{x+9}=0$. | 0 | 86 | 1 |
math | $9 \cdot 67$ Let $n$ be a fixed integer, $n \geqslant 2$.
(a) Determine the smallest constant $c$, such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{1 \leqslant i \leqslant n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_... | \frac{1}{8} | 176 | 7 |
math | 24. Let $S$ be any nonempty set of $k$ integers. Find the smallest value of $k$ for which there always exist two distinct integers $x$ and $y$ in $S$ such that $x+y$ or $x-y$ is divisible by 2007 . | 1005 | 65 | 4 |
math | 403*. Solve the equation:
$$
\sqrt[4]{1-x^{2}}+\sqrt[4]{1-x}+\sqrt[4]{1+x}=3
$$ | 0 | 39 | 1 |
math | 30. (2007 Western China Mathematical Olympiad) Find all positive integers $n$, such that there exist non-zero integers $x_{1}$, $x_{2}, \cdots, x_{n}, y$, satisfying
$$
\left\{\begin{array}{l}
x_{1}+\cdots+x_{n}=0, \\
x_{1}^{2}+\cdots+x_{n}^{2}=n y^{2} .
\end{array}\right.
$$ | allpositiveintegersexcept13 | 108 | 7 |
math | Find all positive integers $n$ such that there exists the polynomial with degree $n$ satisfying $f(x^2+1)=f(x)^2+1$. | n \in \{2^k \mid k \in \mathbb{N} \cup \{0\}\} | 34 | 28 |
math | 12. Find the minimum value of the expression $x^{4}-2 x y+y^{4}$. | -\frac{1}{2} | 23 | 7 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.