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 | Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 647 | 69 | 3 |
math | 8. (10 points) A certain exam consists of 7 questions, each of which only concerns the answers to these 7 questions, and the answers can only be one of $1, 2, 3, 4$. It is known that the questions are as follows:
(1) How many questions have the answer 4?
(2) How many questions do not have the answer 2 or 3?
(3) What is... | 16 | 191 | 2 |
math | For what value of $\lambda$ does the following equation represent a pair of straight lines:
$$
\lambda x^{2}+4 x y+y^{2}-4 x-2 y-3=0
$$ | \lambda=4 | 45 | 4 |
math | Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). | 33 | 55 | 2 |
math | 4. Find all 4-digit numbers that are 7182 less than the number written with the same digits in reverse order.
ANSWER: 1909 | 1909 | 36 | 4 |
math | Example 9 For what value of $a$ does the inequality
$\log _{\frac{1}{a}}\left(\sqrt{x^{2}+a x+5}+1\right) \cdot \log _{5}\left(x^{2}+a x+6\right)+\log _{a} 3 \geqslant 0$ have exactly one solution. | 2 | 85 | 1 |
math | 8.7 For an integer $n>3$, we use $n$ ? to denote the product of all primes less than $n$ (called “$n$-question mark”). Solve the equation $n ?=2 n+16$.
| 7 | 53 | 1 |
math | (4) 8 singers participate in the arts festival, preparing to arrange $m$ performances, each time 4 of them will perform on stage, requiring that any two of the 8 singers perform together the same number of times. Please design a scheme so that the number of performances $m$ is the least. | 14 | 65 | 2 |
math | Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$. | 126 | 65 | 3 |
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 | 3. Let $A B C D$ be a rectangle with area 1 , and let $E$ lie on side $C D$. What is the area of the triangle formed by the centroids of triangles $A B E, B C E$, and $A D E$ ? | \frac{1}{9} | 58 | 7 |
math | 12. (6 points) A passenger liner has a navigation speed of 26 km/h in still water, traveling back and forth between ports $A$ and $B$. The river current speed is 6 km/h. If the liner makes 4 round trips in a total of 13 hours, then the distance between ports $A$ and $B$ is $\qquad$ kilometers. (Neglect the time for the... | 40 | 95 | 2 |
math | 8,9 One of the angles formed by the intersecting lines $a$ and $b$ is $15^{\circ}$. Line $a_{1}$ is symmetric to line $a$ with respect to line $b$, and line $b_{1}$ is symmetric to line $b$ with respect to $a$. Find the angles formed by the lines $a_{1}$ and $b_{1}$. | 45,45,135,135 | 88 | 13 |
math | 15. $z$ is a complex number, then the minimum value of $T=|z|+|z-2|+|z+\sqrt{3} i|$ is what?
---
The translation maintains the original format and line breaks as requested. | \sqrt{13} | 54 | 6 |
math | ## Task 6 - 210936
For a tetrahedron $A B C D$, let the edge lengths be $A B=10 \mathrm{~cm}, B C=6 \mathrm{~cm}, A C=8 \mathrm{~cm}, A D=13$ $\mathrm{cm}, B D=13 \mathrm{~cm}$, and the perpendicular from $D$ to the plane of triangle $A B C$ be $12 \mathrm{~cm}$ long.
Prove that these conditions uniquely determine th... | 13 | 136 | 2 |
math | $\square$ Example 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) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$.
Find the maximum value of $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$. (1997 C... | 189548 | 167 | 6 |
math | 4. In the arithmetic sequence $\left\{a_{n}\right\}$, it is known that $\left|a_{5}\right|=\left|a_{11}\right|, d>0$, the positive integer $n$ that makes the sum of the first $n$ terms $S_{n}$ take the minimum value is $\qquad$ . | 7or8 | 78 | 3 |
math | 1. The engines of a rocket launched vertically upward from the Earth's surface, providing the rocket with an acceleration of $20 \mathrm{~m} / \mathrm{c}^{2}$, suddenly stopped working 40 seconds after launch. To what maximum height will the rocket rise? Can this rocket pose a danger to an object located at an altitude... | 48 | 117 | 2 |
math | Find all sets $X$ consisting of at least two positive integers such that for every pair $m, n \in X$, where $n>m$, there exists $k \in X$ such that $n=m k^{2}$.
Answer: The sets $\left\{m, m^{3}\right\}$, where $m>1$. | {,^{3}} | 73 | 5 |
math | Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real. | g(x) = \sum_{k=3}^{n} \binom{n}{k} x^{n-k} | 91 | 27 |
math | Example. Find the sum of the series
$$
\sum_{n=1}^{\infty} \frac{\sin ^{n} x}{n}
$$
and specify the domain of convergence of the series to this sum. | S(x)=-\ln(1-\sinx),\quadx\neq\pi/2+2\pik | 50 | 27 |
math | João is playing a game where the only allowed operation is to replace the natural number $n$ with the natural number $a \cdot b$ if $a+b=n$, with $a$ and $b$ being natural numbers. For example, if the last number obtained was 15, he can replace it with $56=7 \cdot 8$, since $7+8=15$ and both are natural numbers.
a) St... | 2014 | 138 | 4 |
math | Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$ | 9 | 48 | 1 |
math | 12. Let $x \in \mathbf{R}$. Then the function
$$
f(x)=|2 x-1|+|3 x-2|+|4 x-3|+|5 x-4|
$$
has a minimum value of $\qquad$ | 1 | 62 | 1 |
math | Find the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ we have $f(x-f(x-y))+x=$ $f(x+y)$. | f(x)=x | 53 | 4 |
math | $370 \mathrm{~K}$ is to be divided among $A, B$, and $C$ such that $A$ receives as many times more than $B$ as $B$ receives more than $C$. How much does each receive if the share of $B$ and $C$ is $50 \mathrm{~K}$ more than the share of $A$? | A=160\mathrm{~K},B=120\mathrm{~K},C=90\mathrm{~K} | 83 | 32 |
math | Let the sum of the digits of a 1995-digit number divisible by 9 be $a$. We then add the digits of $a$, and let the result be $b$. What can the sum of the digits of $b$ be? | 9 | 53 | 1 |
math | N1) Find all integer values that the expression
$$
\frac{p q+p^{p}+q^{q}}{p+q}
$$
can take, where $p$ and $q$ are prime numbers. | 3 | 49 | 1 |
math | On the Island of Misfortune, only knights, who always tell the truth, and liars, who always lie, live. In the Parliament of the island, there are 101 deputies. To reduce the budget, it was decided to reduce the Parliament by one deputy. However, each deputy stated that if they were removed from the Parliament, the majo... | 51 | 95 | 2 |
math | 8.308. $\cos 3z - \cos^3 z + \frac{3}{4} \sin 2z = 0$. | z_{1}=\frac{\pi}{2}(2k+1);z_{2}=\pin;z_{3}=(-1)^{k}\frac{\pi}{6}+\pi, | 34 | 41 |
math | 77(1198). Find the coefficients $a, b$ and $c$ of the polynomial $a x^{2}+b x+c$, knowing that the equality
$$
2 x^{4}+x^{3}-41 x^{2}+83 x-45=\left(a x^{2}+b x+c\right)\left(x^{2}+4 x+9\right)
$$
is an identity. | =2,b=-7,=5 | 97 | 8 |
math | A three-digit number composed of different digits, 75% of which is made up of the same digits as the original, but none of them remain in their original position. Which number is this? | 216,324,432,540,648,756,864,972 | 41 | 31 |
math | Example 11 (25th IMO Shortlist) Find the last two digits of $\left[(\sqrt{29}+\sqrt{21})^{1984}\right]$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 71 | 67 | 2 |
math | I place $n$ pairs of socks (thus $2 n$ socks) in a line in such a way that the left sock is to the right of the right sock for each pair. How many different ways can I place my socks like this? | \frac{(2n)!}{2^n} | 51 | 10 |
math | Solve the systems:
a) $x+y+z=6, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}, x y+y z+x z=11$
b) $x(y+z)=2, y(z+x)=2, z(x+y)=3$
c) $x^{2}+y^{2}+x+y=32, 12(x+y)=7 x y$
d) $\frac{x^{2}}{y}+\frac{y^{2}}{x}=\frac{7}{2}, \frac{1}{y}+\frac{1}{x}=\frac{1}{2}$
e) $x+y+z=1, x y+x z+y z=-4, x^{3}+y^{3}+z^{3}=1$;
f) $x^... | ){1,2,3};b)(\\frac{\sqrt{2}}{2},\\frac{\sqrt{2}}{2},\\frac{3\sqrt{2}}{2});){3,4},{\frac{-16+8\sqrt{10}}{7},\frac{-16-8\sqrt{10}}} | 210 | 75 |
math | 1. (5 points) $111 \div 3+222 \div 6+333 \div 9=$ | 111 | 31 | 3 |
math | ## 255. Math Puzzle $8 / 86$
Felix rides his bicycle on a straight route between two towns that are ten kilometers apart, and the round trip takes him one hour.
The next day, a strong wind blows, so he rides 5 km/h faster on the way there and 5 km/h slower on the way back compared to his average speed from the previo... | 4 | 95 | 1 |
math | 697. Where are the points $(x, y)$ for which $y=2 x+1$? | 2x+1 | 24 | 4 |
math | Example 2 Given that the line $l$ passing through point $P(1,2)$ intersects the positive half-axes of the $x$-axis and $y$-axis at points $A, B$ respectively, find the minimum value of $|O A|+|O B|$. | 3+2\sqrt{2} | 64 | 8 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; y ; 0)$
$B(-2 ;-4 ; 6)$
$C(7 ; 2 ; 5)$ | A(0;1\frac{5}{6};0) | 60 | 14 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy:
$$
\forall x, y \in \mathbb{R}, \quad f(x+y)=2 f(x)+f(y)
$$ | f(x)=0 | 52 | 4 |
math | 4. Let $a, b \in \mathbf{R}$. Then the minimum value of $M=\frac{\left(a^{2}+a b+b^{2}\right)^{3}}{a^{2} b^{2}(a+b)^{2}}$ is $\qquad$ . | \frac{27}{4} | 65 | 8 |
math | Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the... | \frac{m^3}{3} | 169 | 9 |
math | ## Task A-2.2.
Determine the complex number $z$ such that
$$
\operatorname{Re} \frac{1}{1-z}=2 \quad \text { and } \quad \operatorname{Im} \frac{1}{1-z}=-1
$$ | \frac{3}{5}-\frac{1}{5}i | 63 | 15 |
math | Vladimir writes the numbers from 1 to 1000 on the board. As long as there is a number strictly greater than 9 written on the board, he chooses whichever he wants, and replaces it with the sum of its digits. How many times can he end up with the number 1? | 112 | 65 | 3 |
math | Third question: For all real numbers $x_{1}, x_{2}, \ldots, x_{60} \in[-1,1]$, find the maximum value of $\sum_{i=1}^{60} x_{i}^{2}\left(x_{i+1}-x_{i-1}\right)$, where $x_{0}=x_{60}, x_{61}=x_{1}$. | 40 | 93 | 2 |
math | Problem 3. Answer: $\frac{1}{64}$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The provided translation is already in the requested format and directly represents the translation of the given text. However, the not... | \frac{1}{64} | 102 | 8 |
math | Problem 6. Find all real numbers $x$ such that both numbers $x+\sqrt{3}$ and $x^{2}+\sqrt{3}$ are rational. | \frac{1}{2}-\sqrt{3} | 36 | 12 |
math | 14.3.27 ** Find the sum of all divisors $d=2^{a} \cdot 3^{b}(a, b>0)$ of $N=19^{88}-1$. | 744 | 47 | 3 |
math | 13.426 Two pumps were installed to fill a swimming pool with water. The first pump can fill the pool 8 hours faster than the second one. Initially, only the second pump was turned on for a time equal to twice the amount of time it would take to fill the pool if both pumps were working simultaneously. Then, the first pu... | 4 | 116 | 1 |
math | 6. Given $x, y, z \in \mathbf{R}_{+}$, and $\sqrt{x^{2}+y^{2}}+z=1$. Then the maximum value of $x y+2 x z$ is $\qquad$ | \frac{\sqrt{3}}{3} | 55 | 10 |
math | 3. The equation $x^{2}+a x+3=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}-\frac{39}{x_{2}}=x_{2}^{3}-\frac{39}{x_{1}}
$$
Find all possible values of $a$. | \4 | 84 | 2 |
math | 5. Given a real number $x$ between 0 and 1, consider its decimal representation $0, c_{1} c_{2} c_{3} \ldots$ We call $B(x)$ the set of different subsequences of six consecutive digits that appear in the sequence $c_{1} c_{2} c_{3} \ldots$
For example, $B(1 / 22)=\{045454,454545,545454\}$.
Determine the minimum numbe... | 7 | 160 | 1 |
math | 1. Find all integers $x, y$ that satisfy the equation
$$
x+y=x^{2}-x y+y^{2} \text {. }
$$ | (x,y)=(0,0),(0,1),(1,0),(1,2),(2,1),(2,2) | 34 | 27 |
math | 7. There are 5 power poles, 2 of which will leak electricity. There are 5 birds each randomly choosing one of the poles to rest (if a bird touches a leaking pole, it will be stunned by the electric shock and fall to the ground). The probability that only 2 of the poles have birds on them is $\qquad$ | \frac{342}{625} | 72 | 11 |
math | Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality. | S \leq \sqrt{abcd} | 54 | 10 |
math | \section*{Task 2 - 321022}
In a strength sports competition, Robert, Stefan, and Tilo are participating. Robert manages to do 20 pull-ups. Stefan aims to achieve at least \(80 \%\) of the combined performance of Robert and Tilo; Tilo wants to achieve at least \(60 \%\) of the combined performance of Robert and Stefan.... | 92 | 112 | 2 |
math | 10. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{n+1}=\frac{\sqrt{3} a_{n}-1}{a_{n}+\sqrt{3}}\left(n \in \mathbf{N}^{*}\right)$, then $a_{2004}=$ | 2+\sqrt{3} | 81 | 6 |
math | 13. (15 points) Divide the natural numbers from 1 to 30 into two groups, such that the product of all numbers in the first group \( A \) is divisible by the product of all numbers in the second group \( B \). What is the minimum value of \( \frac{A}{B} \)? | 1077205 | 70 | 7 |
math | ## Task 1 - 201221
For each $n=1,2,3, \ldots$ let
$$
a_{n}=\frac{1}{n^{2}} \sum_{k=1}^{n} k
$$
Furthermore, let $I_{1}, I_{2}, I_{3}$, and $I_{4}$ be the closed intervals
$$
I_{1}=[1 ; 2], \quad I_{2}=[0.53 ; 0.531], \quad I_{3}=[0.509 ; 0.51], \quad I_{4}=[0.4 ; 0.5]
$$
Investigate for each of these intervals whe... | a_{n}\inI_{1}exactlyforn=1,\,a_{n}\inI_{2}fornon,\,a_{n}\inI_{3}exactlyforn=50,51,52,53,54,55,\,a_{n}\inI_{4}fornon | 209 | 76 |
math | 5. The number of solutions to the equation $\cos \frac{x}{4}=\cos x$ in $(0,24 \pi)$ is | 20 | 31 | 2 |
math | Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard... | 1848 | 82 | 4 |
math | Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\] | 23 | 41 | 2 |
math | Question 4 Let $P$ be a polynomial of degree $3n$, such that
$$
\begin{array}{l}
P(0)=P(3)=\cdots=P(3 n)=2, \\
P(1)=P(4)=\cdots=P(3 n-2)=1, \\
P(2)=P(5)=\cdots=P(3 n-1)=0 .
\end{array}
$$
$$
\text { If } P(3 n+1)=730 \text {, find } n \text {. }
$$ | 4 | 124 | 1 |
math | What is the geometric place of the intersection points of the perpendicular tangents drawn to the circle $x^{2}+y^{2}=32$? | x^{2}+y^{2}=64 | 32 | 11 |
math | 13.285. On a 10 km stretch of highway, devoid of intersections, the bus stops only for passengers to get on and off. It makes a total of 6 intermediate stops, spending 1 minute at each, and always moves at the same speed. If the bus were to travel without stopping, it would cover the same distance at a speed exceeding ... | 24 | 105 | 2 |
math | Determine all real numbers $a$ such that there exists an infinite sequence of strictly positive real numbers $x_{0}, x_{1}, x_{2}, x_{3}, \ldots$ satisfying for all $n$ the equality
$$
x_{n+2}=\sqrt{a x_{n+1}-x_{n}} .
$$ | a > 1 | 74 | 4 |
math | 3. Given $\sqrt{x}+\sqrt{y}=35, \sqrt[3]{x}+\sqrt[3]{y}=13$. Then $x+y=$ $\qquad$ | 793 | 41 | 3 |
math | Example 4: Given 1 one-yuan note, 1 two-yuan note, 1 five-yuan note, 4 ten-yuan notes, and 2 fifty-yuan notes, how many different amounts of money can be paid using these notes?
(1986 Shanghai Competition Question) | 119 | 63 | 3 |
math | Points $A_{1}, B_{1}, C_{1}$ are the midpoints of the sides $B C, A C, A B$ of triangle $A B C$, respectively. It is known that $A_{1} A$ and $B_{1} B$ are the angle bisectors of triangle $A_{1} B_{1} C_{1}$. Find the angles of triangle $A B C$. | 60,60,60 | 90 | 8 |
math | In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of the... | 9 | 140 | 1 |
math | 1. Given that $a$, $b$, and $c$ are three distinct odd prime numbers, the equation $(b+c) x^{2}+(a+1) \sqrt{5} x+225=0$ has two equal real roots.
(1) Find the minimum value of $a$;
(2) When $a$ reaches its minimum value, solve this equation. | x=-\frac{3}{2} \sqrt{5} | 83 | 14 |
math | 149. Two equally matched opponents are playing chess. Find the most probable number of wins for any chess player if $2 N$ decisive (without draws) games will be played. | N | 38 | 1 |
math | $6 \cdot 148$ Find all functions $f(x)$ and $g(x) (x \in R)$ that satisfy
$$\sin x+\cos y \equiv f(x)+f(y)+g(x)-g(y), x, y \in R$$ | f(x)=\frac{\sin x+\cos x}{2}, g(x)=\frac{\sin x-\cos x}{2}+c | 58 | 30 |
math | \section*{Problem 2 - 071232}
It is the product
\[
\sin 5^{\circ} \sin 15^{\circ} \sin 25^{\circ} \sin 35^{\circ} \sin 45^{\circ} \sin 55^{\circ} \sin 65^{\circ} \sin 75^{\circ} \sin 85^{\circ}
\]
to be transformed into an expression that can be formed from natural numbers solely by applying the operations of additi... | \frac{1}{512}\sqrt{2} | 174 | 13 |
math | 3. Given a tetrahedron $S-ABC$, point $A_{1}$ is the centroid of $\triangle SBC$, and $G$ is on segment $AA_{1}$, satisfying $\frac{|AG|}{|GA_{1}|}=3$. Connecting $SG$ intersects the plane of $\triangle ABC$ at point $M$, then $\frac{|A_{1}M|}{|AS|}=$ $\qquad$ . | \frac{1}{3} | 94 | 7 |
math | 1. If the constant term in the expansion of $\left(\frac{1}{x}-x^{2}\right)^{n}$ is 15, then the coefficient of $x^{3}$ is | -20 | 43 | 3 |
math | Given the parabola $y^{2}=2 p x$ and the point $P(3 p, 0)$. Determine the point $Q$ on the parabola such that the distance $P Q$ is minimal. | Q_{1}(2p,2p)Q_{2}(2p,-2p) | 49 | 20 |
math | C1. The positive integer $N$ has six digits in increasing order. For example, 124689 is such a number.
However, unlike 124689 , three of the digits of $N$ are 3,4 and 5 , and $N$ is a multiple of 6 . How many possible six-digit integers $N$ are there? | 3 | 82 | 1 |
math | A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise? | 77 | 74 | 2 |
math | 6. (40 points) Let $f(x)$ be a polynomial of degree 2010 such that $f(k)=-\frac{2}{k}$ for $k=1,2, \cdots, 2011$.
Find $f(2012)$. | -\frac{1}{503} | 65 | 9 |
math |
4. Suppose 100 points in the plane are coloured using two colours, red and white, such that each red point is the centre of a circle passing through at least three white points. What is the least possible number of white points?
| 10 | 51 | 2 |
math | 1. Calculate: $25 \times 13 \times 2 + 15 \times 13 \times 7=$ | 2015 | 30 | 4 |
math | 289. Automobile Wheels. "You see, sir," said the car salesman, "the front wheel of the car you are buying makes 4 more revolutions than the rear wheel every 360 feet; but if you were to reduce the circumference of each wheel by 3 feet, the front wheel would make 6 more revolutions than the rear wheel over the same dist... | 15 | 116 | 2 |
math | ## Task 6 - 110936
Determine all ordered pairs $(x, y)$ of integers $x, y$ that are solutions to the following equation!
$$
2 x^{2}-2 x y-5 x-y+19=0
$$ | (x,y)\in{(-6,-11),(-1,-26),(0,19),(5,4)} | 59 | 26 |
math | Find all integers $n \geq 1$ such that $2^{n}-1$ has exactly $n$ positive integer divisors. | n\in{1,2,4,6,8,16,32} | 30 | 20 |
math | 4. Given that $a, b, c$ are the lengths of the three sides of $\triangle ABC$, and satisfy the conditions
$$
\frac{2 a^{2}}{1+a^{2}}=b, \frac{2 b^{2}}{1+b^{2}}=c, \frac{2 c^{2}}{1+c^{2}}=a \text {. }
$$
Then the area of $\triangle ABC$ is . $\qquad$ | \frac{\sqrt{3}}{4} | 100 | 10 |
math | 4. Let $\sin \alpha-\cos \alpha=\frac{1}{3}$. Then $\sin 3 \alpha+\cos 3 \alpha=$ $\qquad$ | -\frac{25}{27} | 37 | 9 |
math | Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ? | 18 | 54 | 2 |
math | 1. On the meadow, there are children and adults. The percentage of boys among all children is equal to the percentage of girls among all present people and also the number of all adults. How many boys, girls, and adults are on the meadow? | 32 | 53 | 2 |
math | 4. The product of the number 21 with some four-digit number $x$ is the cube of some integer $y$. Find the number $x$.
---
Note: The translation maintains the original text's structure and format, including the numbering and mathematical notation. | 3528 | 55 | 4 |
math | Example 3 The parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at points $A$ and $B$, and the $y$-axis at point $C$. If $\triangle A B C$ is a right triangle, then $a c=$ $\qquad$
(2003, National Junior High School Mathematics League) | a c=-1 | 81 | 4 |
math | Solve the following system of equations:
$$
\begin{aligned}
2 y+x-x^{2}-y^{2} & =0 \\
z-x+y-y(x+z) & =0 \\
-2 y+z-y^{2}-z^{2} & =0
\end{aligned}
$$ | x_{1}=0,y_{1}=0,z_{1}=0x_{2}=1,y_{2}=0,z_{2}=1 | 63 | 30 |
math | 1. A positive integer is called sparkly if it has exactly 9 digits, and for any $n$ between 1 and 9 (inclusive), the $n^{\text {th }}$ digit is a positive multiple of $n$. How many positive integers are sparkly? | 216 | 59 | 3 |
math | Problem 18. (4 points)
By producing and selling 4000 items at a price of 6250 rubles each, a budding businessman earned 2 million rubles in profit. Variable costs for one item amounted to 3750 rubles. By what percentage should the businessman reduce the production volume to make his revenue equal to the cost? (Provide... | 20 | 103 | 2 |
math | ## Task 2 - 250932
Determine all pairs $(a, b)$ of two-digit numbers $a$ and $b$ for which the following holds: If one forms a four-digit number $z$ by writing $a$ and $b$ consecutively in this order, then $z=(a+b)^{2}$. | (20,25),(30,25) | 76 | 13 |
math | 7. In the right triangle $\triangle ABC$, $\angle C=90^{\circ}, AB=c$. Along the direction of vector $\overrightarrow{AB}$, points $M_{1}, M_{2}, \cdots, M_{n-1}$ divide the segment $AB$ into $n$ equal parts. Let $A=M_{0}, B=M_{n}$. Then
$$
\begin{array}{l}
\lim _{n \rightarrow+\infty} \frac{1}{n}\left(\overrightarrow{... | \frac{c^{2}}{3} | 182 | 10 |
math | Problem 3. Borche's mother is three times older than Borche, and his father is four years older than Borche's mother. How old is each of them if together they are 88 years old? | Borche:12,Mother:36,Father:40 | 45 | 16 |
math | Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than or equal to $\frac12$ can be written as $\frac{m}{n}$, where $... | 171 | 116 | 3 |
math | 20. Each chocolate costs 1 dollar, each lioorice stick costs 50 cents and each lolly costs 40 cents. How many different combinations of these three items cost a total of 10 dollars? | 36 | 48 | 2 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.