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 | 6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from twenty consecutive natural numbers to some number $a$ is 360, and the sum of the distances from these same twenty numbers to the number $a^{2}$ is 345. Find all possible values of $a$. | =-\frac{1}{2},=\frac{3}{2} | 75 | 15 |
math | 33 Find the largest real number $\lambda$, such that for a real-coefficient polynomial $f(x)=x^{3}+a x^{2}+c$ with all roots being non-negative real numbers, if $x \geqslant 0$, then $f(x) \geqslant \lambda(x-a)^{3}$, and find the condition for equality. | -\frac{1}{27} | 81 | 8 |
math | 23.7. Using the result of problem 23.6, calculate the following sums and products:
a) $\operatorname{ctg}^{2} \frac{\pi}{2 n+1}+\operatorname{ctg}^{2} \frac{2 \pi}{2 n+1}+\ldots+\operatorname{ctg}^{2} \frac{n \pi}{2 n+1}$;
b) $\frac{1}{\sin ^{2} \frac{\pi}{2 n+1}}+\frac{1}{\sin ^{2} \frac{2 \pi}{2 n+1}}+\ldots+\frac{... | \frac{\sqrt{2n+1}}{2^{n}} | 316 | 15 |
math | $10 \cdot 40$ Find all such three-digit numbers, if the number itself is increased by 3, then the sum of its digits decreases to $\frac{1}{3}$ of the original.
(Kyiv Mathematical Olympiad, 1963) | 117,207,108 | 58 | 11 |
math | 1. Given the function
$$
f(x)=\left|8 x^{3}-12 x-a\right|+a
$$
has a maximum value of 0 on the interval $[0,1]$. Then the maximum value of the real number $a$ is $\qquad$ | -2 \sqrt{2} | 64 | 7 |
math | A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers... | 61 | 97 | 2 |
math | 2. In the set of integers, solve the equation
$$
x^{2}+x y+y^{2}=x^{2} y^{2}
$$ | (0,0),(1,-1),(-1,1) | 34 | 14 |
math | Sam dumps tea for $6$ hours at a constant rate of $60$ tea crates per hour. Eddie takes $4$ hours to dump the same
amount of tea at a different constant rate. How many tea crates does Eddie dump per hour?
[i]Proposed by Samuel Tsui[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{90}$
Sam dumps a total of $6 \cdot 60 = 3... | 90 | 145 | 2 |
math | Example 5.17. Find the interval of convergence of the power series
$$
1-\frac{x}{2 \cdot 2}+\frac{x^{2}}{3 \cdot 2^{2}}-\frac{x^{3}}{4 \cdot 2^{3}}+\ldots+(-1)^{n} \frac{x^{n}}{(n+1) 2^{n}}+\ldots
$$ | -2<x\leq2 | 91 | 7 |
math | Example 5 Determine all polynomials $p(x)$ that satisfy the following conditions:
$$
p\left(x^{2}+1\right)=[p(x)]^{2}+1, p(0)=0.
$$
(32nd Putnam $A-2$) | p(x)=x | 61 | 4 |
math | Example 1 Let the set $A=\left\{a^{2}, a+1,-3\right\}, B=\left\{a-3,2 a-1, a^{2}+1\right\}$, and $A \cap$ $B=\{-3\}$, find the value of the real number $a$. | -1 | 74 | 2 |
math | Problem 6. Calculate $2 \operatorname{arctg} 4+\arcsin \frac{8}{17}$. | \pi | 30 | 2 |
math | Example 4 Try to find the unit digit of the integer part of $(\sqrt{2}+\sqrt{3})^{2012}$.
[2] | 7 | 35 | 1 |
math | Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value. | \frac{3}{2}e^{-\frac{1}{3}} | 46 | 16 |
math | 5. Now arrange for seven students to participate in five sports events, requiring that students A and B cannot participate in the same event, each event must have participants, and each person only participates in one event. Then the number of different schemes that meet the above requirements is $\qquad$ | 15000 | 58 | 5 |
math | 7. A die is rolled twice in succession, and the numbers obtained are $a$ and $b$ respectively. Then the probability $p=$ $\qquad$ that the cubic equation $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$ has three distinct real roots is $\qquad$ .(Answer with a number). | \frac{3}{4} | 84 | 7 |
math | Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy
$$f(\sin{x})+f(\cos{x})=2020$$
for any real number $x.$ | f(x) = g(1 - 2x^2) + 1010 | 49 | 21 |
math | Problem 11.6. The quadratic trinomial $P(x)$ is such that $P(P(x))=x^{4}-2 x^{3}+4 x^{2}-3 x+4$. What can $P(8)$ be? List all possible options. | 58 | 59 | 2 |
math | 13.346. The volume of substance A is half the sum of the volumes of substances B and C, and the volume of substance B is 1/5 of the sum of the volumes of substances A and C. Find the ratio of the volume of substance C to the sum of the volumes of substances A and B. | 1 | 69 | 1 |
math | 2. Problem: A cube with sides $1 \mathrm{~m}$ in length is filled with water, and has a tiny hole through which the water drains into a cylinder of radius $1 \mathrm{~m}$. If the water level in the cube is falling at a rate of $1 \mathrm{~cm} / \mathrm{s}$, at what rate is the water level in the cylinder rising? | \frac{1}{\pi}\mathrm{}/\mathrm{} | 86 | 14 |
math | Example 2. Find the general integral of the homogeneous equation
$$
\left(x^{2}-y^{2}\right) d y-2 y x d x=0
$$ | x^{2}+y^{2}=Cy | 39 | 10 |
math | $4 \cdot 37$ Try to point out, if the equation
$$x^{3}+a x^{2}+b x+c=0$$
has three real roots in arithmetic progression, what necessary and sufficient conditions should the real numbers $a, b, c$ satisfy? | 2 a^{3}-9 a b+27 c=0 \text{ and } a^{2}-3 b \geqslant 0 | 62 | 32 |
math | I4.1 Let $a$ be a real number.
If $a$ satisfies the equation $\log _{2}\left(4^{x}+4\right)=x+\log _{2}\left(2^{x+1}-3\right)$, find the value of $a$ | 2 | 64 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-2 x}}{x+\sin x^{2}}$ | 3 | 42 | 1 |
math | 936. What is the greatest rate at which the function $u(M)=\frac{10}{x^{2}+y^{2}+z^{2}+1}$ can increase as the point $M(x, y, z)$ passes through the point $M_{0}(-1 ; 2 ;-2)$? In what direction should the point $M$ move as it passes through the point $M_{1}(2 ; 0 ; 1)$, so that the function $u(M)$ decreases at the grea... | \frac{3}{5} | 115 | 7 |
math | 3.276. $\cos ^{2}\left(45^{\circ}+\alpha\right)-\cos ^{2}\left(30^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)$. | -\sin2\alpha | 71 | 5 |
math | What is the smallest eight-digit positive integer that has exactly four digits which are 4 ? | 10004444 | 18 | 8 |
math | Example 8 Given non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the inequality $x_{1}+x_{2}+\cdots+x_{n} \leqslant \frac{1}{2}$, find the minimum value of $f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\prod_{i=1}^{n}\left(1-x_{i}\right)$. | \frac{1}{2} | 105 | 7 |
math | 3. Given the function $f(x)=\sin \omega x+\sin 2 x$, where $\omega \in \mathbf{N}_{+}, \omega \leqslant 2023$. If $f(x)<2$ always holds, then the number of constants $\omega$ that satisfy the condition is $\qquad$ | 1770 | 73 | 4 |
math | 1. Given $x \in[0,2 \pi]$, and $\sin x=\sin \left[\arcsin \frac{2}{3}-\arcsin \left(-\frac{1}{3}\right)\right], x=$ | \arcsin\frac{4\sqrt{2}+\sqrt{5}}{9} | 54 | 21 |
math | 7. Given $x, y, z \in \mathbf{R}$, then $\sum \frac{x^{2}}{(3 x-2 y-z)^{2}}$ has the minimum value of $\qquad$ ("sum" indicates cyclic sum).
| \frac{5}{49} | 55 | 8 |
math | $7.4 \quad 2.5^{\frac{4+\sqrt{9-x}}{\sqrt{9-x}}} \cdot 0.4^{1-\sqrt{9-x}}=5^{10} \cdot 0.1^{5}$. | x_{1}=8,x_{2}=-7 | 57 | 11 |
math | 14. (12 points) Li Gang reads a book. On the first day, he read $\frac{1}{5}$ of the book. On the second day, he read 24 pages. On the third day, he read 150% of the total number of pages he read in the first two days. At this point, there is still $\frac{1}{4}$ of the book left to read. How many pages does the book ha... | 240 | 101 | 3 |
math | 5. The function $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfies for all $x, y, z \in \mathbf{R}$
$$
f(x+y)+f(y+z)+f(z+x) \geqslant 3 f(x+2 y+z) .
$$
Then $f(1)-f(0)=$ $\qquad$ | 0 | 85 | 1 |
math | 5. Find all real numbers $x$ and $y$ that satisfy the equation
$$
(x+y)^{2}=(x+3)(y-3)
$$
Solve the problem independently. You have 150 minutes for solving.
The use of notes, literature, or a pocket calculator is not allowed.
Mathematical Competition for High School Students in Slovenia
## Invitational Competition... | -3,3 | 102 | 4 |
math | 8. Find the value of $\frac{1}{3^{2}+1}+\frac{1}{4^{2}+2}+\frac{1}{5^{2}+3}+\cdots$. | \frac{13}{36} | 46 | 9 |
math | 2. On a line, points $A_{1}, A_{2}, \ldots, A_{20}$ are taken in this order, such that $A_{1} A_{2}=6 \text{~cm}$, $A_{2} A_{3}=12 \text{~cm}$, $A_{3} A_{4}=18 \text{~cm}$, and so on.
a) What is the length of the segment $\left[A_{1} A_{20}\right]$? What about the segment $\left[A_{15} A_{20}\right]$?
b) Determine $i... | 14 | 191 | 2 |
math | Aurick throws $2$ fair $6$-sided dice labeled with the integers from $1$ through $6$. What is the probability that the sum of the rolls is a multiple of $3$? | \frac{1}{3} | 45 | 7 |
math | 7. Draw a line $l$ through the point $P(1,1)$ such that the midpoint of the chord intercepted by the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ is exactly $P$. Then the equation of the line $l$ is $\qquad$ . | 4 x+9 y=13 | 70 | 8 |
math | Problem 7. Aся, Borya, Vasilina, and Grisha bought tickets to the cinema for one row. It is known that:
- There are a total of 9 seats in the row, numbered from 1 to 9.
- Borya did not sit in seat 4 or 6.
- Aся sat next to Vasilina and Grisha, and no one sat next to Borya.
- There were no more than two seats between A... | 5 | 125 | 1 |
math | Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that:
$f(1997)=1998$
$\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$ | t = -1 | 85 | 5 |
math | 2. In the underwater kingdom, there live octopuses with seven and eight legs. Those with 7 legs always lie, while those with 8 legs always tell the truth. One day, three octopuses had the following conversation.
Green octopus: “We have 24 legs together.”
Blue octopus: “You are right!”
Red octopus: “Nonsense, Green i... | TheGreenOctopushas7legs,theBlueOctopushas7legs,theRedOctopushas8legs | 99 | 24 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}-3 n+2}-n\right)
$$ | -\frac{3}{2} | 42 | 7 |
math | Example 19 (2000 Hebei Province Competition Question) Given $x^{2}+y^{2}-2 x-2 y+1=0(x, y \in \mathbf{R})$, then the minimum value of $F(x, y)=\frac{x+1}{y}$ is $\qquad$ | \frac{3}{4} | 71 | 7 |
math | 2. The general solution of the equation $\cos \frac{x}{4}-\cos x$ is ( ), within $(0,24 \pi)$, there are ( ) distinct solutions. | 20 | 40 | 2 |
math | [Inscribed, circumscribed, and exscribed circles; their radii] Pythagorean Theorem (direct and inverse).
Find the radius of the circumscribed circle of a right triangle if the radius of its inscribed circle is 3, and one of the legs is 10. | \frac{29}{4} | 62 | 8 |
math | 6. The product of all of the positive integer divisors of $6^{16}$ equals $6^{k}$ for some integer $k$. Determine the value of $k$. | 2312 | 38 | 4 |
math | Determine a value of $n$ for which the number $2^{8}+2^{11}+2^{n}$ is a perfect square. | 12 | 33 | 2 |
math | 4. If $12 x=4 y+2$, determine the value of the expression $6 y-18 x+7$. | 4 | 29 | 1 |
math | 2. 22 Let $S=\{1,2, \cdots, 1990\}$. If the sum of the elements of a 31-element subset of $S$ is divisible by 5, it is called a good subset of $S$. Find the number of good subsets of $S$. | \frac{1}{5}C_{1990}^{31} | 69 | 18 |
math | 11.2. A trinomial of degree $p$ is a function of the form $f(x)=x^{p}+a x^{q}+1$, where $p, q-$ are natural numbers, $q<p$, and $a$ is an arbitrary real number (possibly equal to 0). Find all pairs of trinomials that give a trinomial of degree 15 when multiplied.
ANSWER: $\left(1+x^{5}\right)\left(1-x^{5}+x^{10}\right... | (1+x^{5})(1-x^{5}+x^{10}),(1-x^{3}+x^{9})(1+x^{3}+x^{6}),(1-x^{6}+x^{9})(1+x^{3}+x^{6}) | 517 | 59 |
math | ## Task Condition
Find the $n$-th order derivative.
$y=\lg (2 x+7)$ | y^{(n)}=(-1)^{n-1}\cdot\frac{2^{n}\cdot(n-1)!}{\ln10}\cdot(2x+7)^{-n} | 24 | 42 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\frac{\cos x-\cos 3 x}{x}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 4 | 71 | 1 |
math | Task 9. Find the largest negative root $x_{0}$ of the equation $\frac{\sin x}{1+\cos x}=2-\operatorname{ctg} x$. In the answer, write $x_{0} \cdot \frac{3}{\pi}$. | -3.5 | 59 | 4 |
math | Let $ABC$ be a triangle whose angles measure $A$, $B$, $C$, respectively. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$, find the number of possible integer values for $\tan B$. (The values of $\tan A$ and $\tan C$ need not be integers.)
[i] Prop... | 11 | 105 | 2 |
math | \section*{Aufgabe 3 - 091033}
Geben Sie
a) eine notwendige und hinreichende,
b) eine notwendige und nicht hinreichende sowie
c) eine hinreichende und nicht notwendige
Bedingung dafür an, da \(\sqrt{1-\left|\log _{2}\right| 5-x||}>0\) gilt!
Die anzugebenden Bedingungen sind dabei so zu formulieren, dass sie in der... | x\in(3;\frac{9}{2})\quad\text{oder}\quadx\in(\frac{11}{2};7) | 144 | 33 |
math | 【Example 2】How many six-digit numbers can be formed using three odd numbers, two 2s, and one 8? | 7500 | 28 | 4 |
math | ## Task 1 - 300731
In a textbook from the year 1525, the following problem is stated in essence:
A dog is hunting a fox. In the time it takes the fox to make 9 jumps, the dog makes 6 jumps, but with 3 jumps, the dog covers the same distance as the fox does with 7 jumps.
How many of its jumps does the dog need to cat... | 72 | 136 | 2 |
math | Clara leaves home by bike at 1:00 p.m. for a meeting scheduled with Quinn later that afternoon. If Clara travels at an average of $20 \mathrm{~km} / \mathrm{h}$, she would arrive half an hour before their scheduled meeting time. If Clara travels at an average of $12 \mathrm{~km} / \mathrm{h}$, she would arrive half an ... | 15 | 124 | 2 |
math | 5. Toss an unfair coin 5 times, if the probability of getting exactly 1 head is equal to the probability of getting exactly 2 heads and is not zero, then the probability of getting exactly 3 heads is $\qquad$ .
| \frac{40}{243} | 51 | 10 |
math | II. (50 points) Find all prime pairs $(p, q)$ such that $p q \mid 5^{p}+5^{q}$.
untranslated text remains unchanged. | (2,3),(3,2),(2,5),(5,2),(5,5),(5,313),(313,5) | 39 | 33 |
math | Alice is given a rational number $r>1$ and a line with two points $B \neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is p... | r = \frac{q+1}{q} \text{ for some integer } q \text{ with } 1 \leqslant q \leqslant 1010 | 178 | 42 |
math | Example 13 (1992 Nordic Competition Problem) Find real numbers $x, y, z$ greater than 1 that satisfy the equation
$$
x+y+z+\frac{3}{x-1}+\frac{3}{y-1}+\frac{3}{z-1}=2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2}) .
$$ | \frac{1}{2}(3+\sqrt{13}) | 87 | 14 |
math | Solve the equation
$$
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5} .
$$ | x=\frac{7}{15} \text{ or } \frac{4}{5} | 34 | 21 |
math | 5. The maximum value of the function $y=2 x-5+\sqrt{11-3 x}$ is $\qquad$ . | \frac{65}{24} | 30 | 9 |
math | Exercise 3. Let $x, y, z$ be non-zero real numbers such that $x+y+z=0$. Suppose that
$$
\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+1
$$
Determine the value of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$. | -1 | 102 | 2 |
math | 12.64 Two individuals, A and B, simultaneously solve the radical equation $\sqrt{x+a}+\sqrt{x+b}=7$. When copying the problem, A mistakenly copies it as: $\sqrt{x-a}+\sqrt{x+b}=7$, and as a result, finds one of the roots to be 12. B mistakenly copies it as $\sqrt{x+a}+\sqrt{x+d}=7$, and as a result, finds one of the ro... | (,b)=(3,4) | 174 | 8 |
math | Example 5. Find the integral $\int \sin ^{4} x \cos ^{2} x d x$. | \frac{1}{16}x-\frac{1}{64}\sin4x-\frac{1}{48}\sin^{3}2x+C | 26 | 35 |
math | 4.3. Form the equation of the plane passing through the points $P_{0}(2,-1,2), P_{1}(4,3,0), P_{2}(5,2,1)$. | x-2y-3z+2=0 | 46 | 11 |
math | Example 2.5.3 Rolling a die once, the probabilities of the outcomes being $1,2, \cdots, 6$ are all $\frac{1}{6}$. If the die is rolled 10 times consecutively, what is the probability that the sum of the outcomes is 30? | \frac{2930455}{6^{10}}\approx0.0485 | 68 | 24 |
math | 3.172. Find $\operatorname{ctg} 2 \alpha$, if it is known that $\sin \left(\alpha-90^{\circ}\right)=-\frac{2}{3}$ and $270^{\circ}<\alpha<360^{\circ}$. | \frac{\sqrt{5}}{20} | 67 | 11 |
math | 2. Given $a, b \in \mathbf{R}$, the circle $C_{1}: x^{2}+y^{2}-2 x+4 y-b^{2}+5=0$ intersects with $C_{2}: x^{2}+y^{2}-2(a-6) x-2 a y$ $+2 a^{2}-12 a+27=0$ at two distinct points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$, and $\frac{y_{1}+y_{2}}{x_{1}+x_{2}}+\frac{x_{1}-x_{2}}{y_{1}-y_{2}}$ $=0$, then $a=... | 4 | 176 | 1 |
math | 3. The function
$$
f(x)=\sqrt{x^{2}+\left(\frac{x^{2}}{4}-2\right)^{2}}+\sqrt{x^{2}+\left(\frac{x^{2}}{4}-1\right)^{2}}
$$
has a minimum value of . $\qquad$ | 3 | 70 | 1 |
math | 11. What is the remainder when $2006 \times 2005 \times 2004 \times 2003$ is divided by 7 ? | 3 | 41 | 1 |
math | 3. For each of the following sets of integers, express their greatest common divisor as a linear combination of these integers
a) $6,10,15$
b) $70,98,105$
c) $280,330,405,490$. | 7=0 \cdot 70+(-1) \cdot 98+1 \cdot 105 | 67 | 25 |
math | Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$ | \left[ \frac{7}{18}, \frac{5}{6} \right] | 82 | 22 |
math | 7. Let $[x]$ denote the greatest integer not exceeding $x$, then $S=[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\cdots+[\sqrt{99}]$, find the value of $[\sqrt{S}]$.
Let's translate the problem and solution step by step:
1. **Understanding the Notation:**
- $[x]$ represents the greatest integer less than or equal to $x$.
2. ... | 24 | 840 | 2 |
math | 45. 18 $k \star$ Find the smallest real number $\lambda$ such that the inequality
$$
5(a b c+a b d+a c d+b c d) \leqslant \lambda a b c d+12
$$
holds for any positive real numbers $a, b, c, d$ satisfying $a+b+c+d=4$. | 8 | 81 | 1 |
math | Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds. | 4 | 154 | 1 |
math | 1. For a right-angled triangle with a hypotenuse of 2009, if the two legs are also integers, then its area is $\qquad$ . | 432180 | 37 | 6 |
math | 32nd Putnam 1971 Problem B2 Let X be the set of all reals except 0 and 1. Find all real valued functions f(x) on X which satisfy f(x) + f(1 - 1/x) = 1 + x for all x in X. | f(x)=\frac{x^3-x^2-1}{2x^2-2x} | 64 | 22 |
math | The equations of the sides of a quadrilateral are:
$$
y=-x+7, \quad y=\frac{x}{2}+1, \quad y=-\frac{3}{2} x+21, \quad y=\frac{7}{4} x+\frac{3}{2}
$$
Determine the coordinates of the vertices of the quadrilateral and its area. | 35 | 82 | 2 |
math | Example 5: Through the right focus $F$ of the hyperbola $x^{2}-y^{2}=1$, a chord $M N$ is drawn intersecting the right branch, and $P$ is the midpoint of the chord $M N$.
(1) Find the equation of the locus of point $P$;
(2) Draw $P Q \perp M N$ intersecting the $x$-axis at point $Q$, prove that $\frac{|M N|}{|F Q|}=\sq... | \sqrt{2} | 117 | 5 |
math | 1. For which values of the parameter $a$ does the system
$$
\begin{aligned}
& x^{3}-a y^{3}=\frac{1}{2}(a+1)^{2} \\
& x^{3}+a x^{2} y+x y^{2}=1
\end{aligned}
$$
have at least one solution $(x, y)$ that satisfies the condition $x+y=0$? | 0,-1,1 | 94 | 5 |
math | 2. Solve the equation $\left(\frac{x}{243}\right)^{\log _{2}\left(\frac{9 x}{4}\right)}=\frac{729}{x^{4}}$. | \frac{243}{4},\frac{1}{9} | 46 | 16 |
math | 18.4.3 $\star \star$ A positive integer $n$ is not divisible by $2$ or $3$, and there do not exist non-negative integers $a$, $b$ such that $\left|2^{a}-3^{b}\right|=n$. Find the minimum value of $n$. | 35 | 67 | 2 |
math | In the following equalities, each letter represents a digit, and different letters represent different digits. Under the line, in each column, the result of the operation indicated vertically is given. Determine the value of the letters so that both the horizontal and vertical operations appear with the correct result.... | 98-24=74 | 142 | 8 |
math | ## Task B-3.2.
Let $A=1202^{2}+2^{2021}$. Determine the units digit of the number $A^{2021}$. | 6 | 44 | 1 |
math | Find the equation of the circumcircle of triangle $A_{1} A_{2} A_{3}$ in barycentric coordinates.
# | \sum_{i<j}x_{\mathrm{i}}x_{\mathrm{j}}a_{\mathrm{ij}}{}^{2}=0 | 30 | 30 |
math | Find the maximum value of the area of a triangle having side lengths \(a, b, c\) with
\[
a^{2}+b^{2}+c^{2}=a^{3}+b^{3}+c^{3}
\] | \frac{\sqrt{3}}{4} | 54 | 10 |
math | 14. A cube with an edge length of $n$ was cut into cubes, each with an edge length of 1. The total volume of the resulting cubes will obviously remain the same, but the surface area will undoubtedly increase. By what factor? | n | 52 | 1 |
math | A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn do... | 350 | 140 | 3 |
math | 5. Among the $n$ positive integers from 1 to $n$, those with the most positive divisors are called the "prosperous numbers" among these $n$ positive integers. For example, among the positive integers from 1 to 20, the numbers with the most positive divisors are $12, 18, 20$, so $12, 18, 20$ are all prosperous numbers a... | 10080 | 130 | 5 |
math | [ Decimal numeral system]
A 1992-digit number is written. Each two-digit number formed by adjacent digits is divisible by 17 or 23. The last digit of the number is 1. What is the first?
# | 2 | 51 | 1 |
math | Task B-2.1. Determine the absolute value of the complex number $z$ for which
$$
\frac{\bar{z}}{1+2 i}-\frac{2 z}{1-2 i}=5
$$ | 5\sqrt{5} | 50 | 6 |
math | 2B. Solve the equation:
$$
\log _{\frac{1}{8}}(2 x)-4 \log _{\frac{1}{4}} x \cdot \log _{8} x=0
$$ | x_{1}=\frac{1}{\sqrt{2}},x_{2}=2 | 48 | 19 |
math | Four consecutive odd numbers, when their sum is added to their product, as well as all possible two-factor and three-factor products formed from these numbers without repetition, result in 26,879. Which four numbers are these? | 9,11,13,15-17,-15,-13,-11 | 48 | 22 |
math | Let's determine the sum of the following sequence:
$$
S_{n}=2\binom{n}{2}+6\binom{n}{3}+\cdots+(n-2)(n-1)\binom{n}{n-1}+(n-1) n\binom{n}{n}
$$ | s_{n}=n(n-1)2^{n-2} | 67 | 15 |
math | Determine all distinct positive integers $x$ and $y$ such that
$$
\frac{1}{x}+\frac{1}{y}=\frac{2}{7}
$$ | \frac{1}{28}+\frac{1}{4}=\frac{2}{7} | 40 | 22 |
math | M1. Consider the sequence $5,55,555,5555,55555, \ldots$
Are any of the numbers in this sequence divisible by 495 ; if so, what is the smallest such number? | 555555555555555555 | 56 | 18 |
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