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 | Let $ ABC$ be a right angled triangle of area 1. Let $ A'B'C'$ be the points obtained by reflecting $ A,B,C$ respectively, in their opposite sides. Find the area of $ \triangle A'B'C'.$ | 3 | 49 | 1 |
math | Given the equation $x^{2}+p x+q=0$, we know that $p$ and $q$ are integers, their greatest common divisor is $r(>1)$, and the roots are real. Consider the sum of the $n$-th powers of the roots, where $n$ is a natural number. For which exponents $k$ can we state that the sum is divisible by $r^{k}$? | e_{n}=[\frac{n+1}{2}] | 93 | 13 |
math | The operation $*$ is defined by $a*b=a+b+ab$, where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\]
[i]2017 CCA Math Bonanza Team Round #3[/i] | 5 | 107 | 1 |
math | 12. Given that the center of ellipse $C$ is at the origin, the foci are on the $x$-axis, the eccentricity is $\frac{\sqrt{3}}{2}$, and the area of the triangle formed by any three vertices of ellipse $C$ is $\frac{1}{2}$.
(1) Find the equation of ellipse $C$;
(2) If a line $l$ passing through $P(\lambda, 0)$ intersects... | \lambda\in(-1,-\frac{1}{3})\cup(\frac{1}{3},1) | 139 | 25 |
math | How many integers $a$ divide $2^{4} \times 3^{2} \times 5$? | 30 | 25 | 2 |
math | 3. A real number $x$ is called interesting if, by erasing one of the digits in its decimal representation, one can obtain the number $2x$. Find the largest interesting number. | 0.375 | 40 | 5 |
math | Subject 3. In the equilateral triangle $\mathrm{ABC}$ with $\mathrm{AB}=6 \mathrm{~cm}$, perpendiculars $\mathrm{A}^{\prime} \mathrm{A} \perp(\mathrm{ABC}), \mathrm{B}^{\prime} \mathrm{B} \perp(\mathrm{ABC})$ are raised from the same side of the plane (ABC), such that $\mathrm{AA}^{\prime}=\mathrm{BB}^{\prime}=6 \sqrt{... | \frac{\sqrt{39}}{8} | 150 | 11 |
math | Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions. | (1, 2) \text{ and } (2.5, -1) | 71 | 19 |
math | 3. For an integer $n \geq 3$, we say that $A=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ is an $n$-list if every $a_{k}$ is an integer in the range $1 \leq a_{k} \leq n$. For each $k=1, \ldots, n-1$, let $M_{k}$ be the minimal possible non-zero value of $\left|\frac{a_{1}+\ldots+a_{k+1}}{k+1}-\frac{a_{1}+\ldots+a_{k}}{k}... | 4(n-1) | 245 | 5 |
math | Example 3. Solve the equation: $\sqrt[5]{171-x}+\sqrt[5]{104+x}=5$.
| x_1=139, x_2=-72 | 31 | 14 |
math | 3. Determine all prime numbers $p$ for which $\frac{p^{2}-p-2}{2}$ is a cube of a natural number. | 127 | 32 | 3 |
math | 7. The solution set of the inequality $|x|^{3}-2 x^{2}-4|x|+3<0$ is | (-3,\frac{1-\sqrt{5}}{2})\cup(\frac{-1+\sqrt{5}}{2},3) | 29 | 30 |
math | $6 \cdot 82$ Find the smallest real number $A$, such that for every quadratic polynomial $f(x)$ satisfying the condition
$$
|f(x)| \leqslant 1 \quad(0 \leqslant x \leqslant 1)
$$
the inequality $f^{\prime}(0) \leqslant A$ holds. | 8 | 82 | 1 |
math | Determine all numbers congruent to 1 modulo 27 and to 6 modulo 37. | 487\pmod{999} | 22 | 11 |
math | 4. In triangle $K L M$, medians $L D$ and $M E$ intersect at point $G$. The circle constructed on segment $L G$ as a diameter passes through vertex $M$ and is tangent to line $D E$. It is known that $G M=6$. Find the height $K T$ of triangle $K L M$, the angle $L G M$, and the area of triangle $K L M$. | KT=18,\angleLGM=60,S_{KLM}=54\sqrt{3} | 94 | 23 |
math | ## Task B-1.5.
Determine the last two digits of the number whose square ends with 44. | 12,62,38,88 | 25 | 11 |
math | Matekváros and Fizikaváros are located in different time zones. A flight departs from Fizikaváros at 8 AM local time and arrives in Matekváros the same day at noon local time. The flight departs again two hours after arrival and arrives back in Fizikaváros at 8 PM local time. The travel time is the same in both directi... | 1PM | 107 | 2 |
math | Question 228, Set $S=\{1,2, \ldots, 10\}$ has several five-element subsets satisfying: any two elements in $S$ appear together in at most two five-element subsets. Ask: What is the maximum number of five-element subsets? | 8 | 60 | 1 |
math | 7. Given $\sin ^{2}\left(x+\frac{\pi}{8}\right)-\cos ^{2}\left(x-\frac{\pi}{8}\right)=$ $\frac{1}{4}$, and $x \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value of $\tan x$ is $\qquad$ . | \frac{2 \sqrt{14}+\sqrt{7}}{7} | 84 | 18 |
math | Anton, Artem, and Vera decided to solve 100 math problems together. Each of them solved 60 problems. We will call a problem difficult if it was solved by only one person, and easy if it was solved by all three. How much does the number of difficult problems differ from the number of easy ones?
# | 20 | 69 | 2 |
math | Example 10 Let $x>y>0, xy=1$, find the minimum value of $\frac{3x^3+125y^3}{x-y}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 10 Let $x>y>0, xy=1$, find the minimum value of $\frac{3x^3+125y^3}{x-y}$. | 25 | 104 | 2 |
math | A five-member committee, whose members are $A, B, C, D, E$, decides every issue by voting, and resolutions are adopted by a simple majority. The smartest member of the committee is $A$, who makes a wrong decision only 5% of the time. (For simplicity, we assume that every decision is either correct or incorrect.) $B, C$... | 0.0145 | 164 | 6 |
math | One. (20 points) Given that $a$, $b$, and $c$ are three distinct real numbers, and the three quadratic equations
\[
\begin{array}{l}
x^{2} + a x + b = 0, \\
x^{2} + b x + c = 0, \\
x^{2} + c x + a = 0
\end{array}
\]
each pair of which has exactly one common root. Find the value of $a^{2} + b^{2} + c^{2}$. | 6 | 118 | 1 |
math | 10.5. Find all pairs $(x ; y)$ of real numbers that satisfy the conditions: $x^{3}+y^{3}=1$ and $x^{4}+y^{4}=1$. | (0;1);(1;0) | 46 | 10 |
math | ## Task B-3.2.
Ana, Bruno, Cvita, Dino, and Ema are trying to arrange themselves in five seats in a row. In how many ways can they do this if Ana does not want to sit next to either Bruno or Cvita, and Dino does not want to sit next to Ema? | 28 | 69 | 2 |
math | 3. Let $\frac{\sin ^{4} x}{3}+\frac{\cos ^{4} x}{7}=\frac{1}{10}$. Then for a given positive integer $n, \frac{\sin ^{2 n} x}{3^{n-1}}+\frac{\cos ^{2 n} x}{7^{n-1}}=$ $\qquad$ | \frac{1}{10^{n-1}} | 84 | 12 |
math | 5. Let the sequence of natural numbers from $1 \sim 8$ be $a_{1}, a_{2}$, $\cdots, a_{8}$. Then
$$
\begin{array}{l}
\left|a_{1}-a_{2}\right|+\left|a_{2}-a_{3}\right|+\left|a_{3}-a_{4}\right|+\left|a_{4}-a_{5}\right|^{\prime}+ \\
\left|a_{5}-a_{6}\right|+\left|a_{6}-a_{7}\right|+\left|a_{7}-a_{8}\right|+\left|a_{8}-a_{1... | 32 | 174 | 2 |
math | 3. Xiaoming goes to the supermarket to buy milk. If he buys fresh milk at 6 yuan per box, the money he brings is just enough; if he buys yogurt at 9 yuan per box, the money is also just enough, but he buys 6 fewer boxes than fresh milk. Xiaoming brings a total of yuan. | 108 | 69 | 3 |
math | 34.14. Find all differentiable functions $f$ for which $f(x) f^{\prime}(x)=0$ for all $x$.
## 34.5. Functional equations for polynomials | f(x)= | 47 | 3 |
math | 11. How many integers between 1 and 2005 (inclusive) have an odd number of even digits? | 1002 | 26 | 4 |
math | 377. The density function $f(x)$ of a random variable $X$, whose possible values are contained in the interval $(a, b)$, is given. Find the density function of the random variable $Y=3X$. | (y)=\frac{1}{3}f(y/3) | 49 | 14 |
math | ## Task 32/63
The service life of a motorcycle tire type was determined experimentally. When mounted on the rear wheel, an average of $15000 \mathrm{~km}$ driving distance was achieved until complete wear, whereas when mounted on the front wheel, $25000 \mathrm{~km}$ was achieved.
a) After which driving distance shou... | 9375 | 137 | 4 |
math | 1st Irish 1988 Problem 8 The sequence of nonzero reals x 1 , x 2 , x 3 , ... satisfies x n = x n-2 x n-1 /(2x n-2 - x n-1 ) for all n > 2. For which (x 1 , x 2 ) does the sequence contain infinitely many integral terms? | x_1=x_2= | 81 | 7 |
math | Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$ | 1 | 107 | 1 |
math | 1. One car covers a distance of 120 km 18 minutes faster than the other. If the first car reduced its speed by 12 km/h, and the second car increased its speed by $10 \%$, they would spend the same amount of time on the same distance. Find the speeds of the cars. | 100\mathrm{}/\mathrm{},80\mathrm{}/\mathrm{} | 69 | 19 |
math | 6. (3 points) In space, there is a cube $1000 \times 1000 \times 1000$ with a vertex at the origin and faces parallel to the coordinate planes. Vectors are drawn from the origin to all integer points inside and on the boundary of this cube. Find the remainder when the sum of the squares of the lengths of these vectors ... | 0 | 90 | 1 |
math | 10. (7 points) In the multiplication problem below, each box must be filled with a digit; each Chinese character represents a digit, different characters represent different digits, and the same character represents the same digit. Therefore, the final product of this multiplication problem is . $\qquad$
将上面的文本翻译成英文，请... | 39672 | 81 | 5 |
math | 7.070. $4^{\log _{9} x^{2}}+\log _{\sqrt{3}} 3=0.2\left(4^{2+\log _{9} x}-4^{\log _{9} x}\right)$. | 1;3 | 60 | 3 |
math | ## A5.
Let $a, b, c$ and $d$ be real numbers such that $a+b+c+d=2$ and $a b+b c+c d+d a+a c+b d=0$.
Find the minimum value and the maximum value of the product $a b c d$.
| \()=-1 | 65 | 4 |
math | Example 4 Let $x_{i}>0(i=1,2, \cdots, n), \mu>0$, $\lambda-\mu>0$, and $\sum_{i=1}^{n} x_{i}=1$. For a fixed $n\left(n \in \mathbf{N}_{+}\right)$, find the minimum value of $f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\sum_{i=1}^{n} \frac{x_{i}}{\lambda-\mu x_{i}}$. | \frac{n}{n \lambda-\mu} | 124 | 10 |
math | 4. In the record of three two-digit numbers, there are no zeros, and in each of them, both digits are different. Their sum is 41. What could be their sum if the digits in them are swapped? | 113 | 47 | 3 |
math | Example 1.5 Let $N$ denote the number of 5-digit numbers where the ten-thousands place is not 5 and all digits are distinct, find $N$.
| 24192 | 38 | 5 |
math | ## Task 1 - V00601
Before the merger of individual agricultural enterprises in a village into an LPG, a tractor brigade had to frequently change work locations due to the scattered fields. As a result, each tractor had 2.3 hours of idle time per day (8 hours).
After the merger, each tractor could work continuously in... | 3 | 121 | 1 |
math | 3. Given $A=\left\{x \mid x^{2}-m x+m^{2}-19=0\right\}, B=\left\{x \mid \log _{2}\left(x^{2}-5 x+\right.\right.$ $8)=1\}, C=\left\{x \mid x^{2}+2 x-8=0\right\}$, and $A \cap B \neq \varnothing, A \cap C=\varnothing$, find the value of $m$. | -2 | 114 | 2 |
math | Task 1. Find all prime numbers $p$ for which the natural number
$$
3^{p}+4^{p}+5^{p}+9^{p}-98
$$
has at most 6 positive divisors.
Note. You may use that 9049 is a prime number. | 2,3,5 | 68 | 5 |
math | 3. On the board is written a three-digit number **8. A trio of students guessed the properties of this number.
Zoran: All its digits are even and it has an even number of different prime divisors.
Daniel: It is divisible by 9 and is the square of some natural number.
Nikola: It is less than 400 and 13 times the squa... | 108or468 | 112 | 7 |
math | We call a positive integer $n$ $\textit{sixish}$ if $n=p(p+6)$, where $p$ and $p+6$ are prime numbers. For example, $187=11\cdot17$ is sixish, but $475=19\cdot25$ is not sixish. Define a function $f$ on the positive integers such that $f(n)$ is the sum of the squares of the positive divisors of $n$. For example, $f(10)... | g(x) = x^2 + 2x + 37 | 266 | 15 |
math | Example 2: Let $1 \leqslant k<n$, consider all positive integer sequences of length $n$, and find the total number of terms $T(n, k)$ in these sequences that are equal to $k$. (29th IMO Shortlist) | (n-k+3) \cdot 2^{n-k-2} | 56 | 15 |
math | Three. (50 points) Let $M=\{1,2, \cdots, 65\}, A \subseteq M$ be a subset. If $|A|=33$, and there exist $x, y \in A, x<y$, $x \mid y$, then $A$ is called a "good set". Find the largest $a \in M$ such that any 33-element subset containing $a$ is a good set. | 21 | 99 | 2 |
math | 8.2.1. (12 points) The first, second, and third terms of a geometric progression are pairwise distinct and are equal to the second, fourth, and seventh terms of a certain arithmetic progression, respectively, and the product of these three numbers is 64. Find the first term of the geometric progression. | \frac{8}{3} | 67 | 7 |
math | ## Task A-1.2.
Gargamel has caught $N$ Smurfs and distributed them into three bags. When Papa Smurf was moved from the first bag to the second, Grouchy from the second to the third, and Smurfette from the third to the first, the average height of the Smurfs in the first bag decreased by 8 millimeters, while the averag... | 21 | 125 | 2 |
math | A $7\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\times 1$ board in which all three colors are used at leas... | 106 | 201 | 3 |
math | 14th Chinese 1999 Problem B3 How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? Solution | 576 | 84 | 3 |
math | Example 9. At what angles do the curves $y=x^{2}$, $y^{2}=x$ intersect? | 90,\operatorname{arctg}\frac{3}{4} | 26 | 16 |
math | 4. Given an isosceles trapezoid \(ABCD (AD \parallel BC, AD > BC)\). A circle \(\Omega\) is inscribed in angle \(BAD\), touches segment \(BC\) at point \(C\), and intersects \(CD\) again at point \(E\) such that \(CE = 7\), \(ED = 9\). Find the radius of the circle \(\Omega\) and the area of trapezoid \(ABCD\). | R=2\sqrt{7},S_{ABCD}=56\sqrt{7} | 102 | 20 |
math | Solve the following system of equations:
$$
\frac{x}{a}=\frac{y}{b}=\frac{z}{c}, x+y+z=d
$$ | \frac{}{+b+},\quadb\frac{}{+b+},\quad\frac{}{+b+} | 36 | 29 |
math | 10. The base of the tetrahedron $S-ABC$ is an equilateral triangle with side length 4. It is known that $AS=BS=\sqrt{19}, CS=3$. Find the surface area of the circumscribed sphere of the tetrahedron $S-ABC$. | \frac{268}{11}\pi | 66 | 11 |
math | [ Midline of the trapezoid ] $[\quad$ Area of the trapezoid $\quad]$
The diagonals of the trapezoid are 6 and 8, and the midline is 5. Find the area of the trapezoid. | 24 | 59 | 2 |
math | 22. Find the number of triangles whose sides are formed by the sides and the diagonals of a regular heptagon (7-sided polygon). (Note: The vertices of triangles need not be the vertices of the heptagon.) | 287 | 48 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{\cos \left(\frac{\pi x}{2}\right)}{1-\sqrt{x}}$ | \pi | 44 | 2 |
math | 2. On the shores of a circular island (viewed from above), there are cities $A, B, C$, and $D$. The straight asphalt road $A C$ divides the island into two equal halves. The straight asphalt road $B D$ is shorter than road $A C$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also h... | 450 | 157 | 3 |
math | 2. In the positive term sequence $\left\{a_{n}\right\}$, $a_{1}=10, a_{n+1}=10 \sqrt{a_{n}}$, find the general formula for this sequence. | a_{n}=10^{2-(\frac{1}{2})^{n-1}} | 51 | 21 |
math | Example 1 The elements of set $A$ are all integers, the smallest of which is 1, and the largest is 100. Except for 1, each element is equal to the sum of two numbers (which can be the same) in set $A$. Find the minimum number of elements in set $A$.
| 9 | 70 | 1 |
math | Question 23 Let $f(n)=\left\{\begin{array}{ll}n-12 & n>2000 ; \\ f[f(n+14)] & n \leqslant 2000\end{array}(n \in \mathbf{N})\right.$. Try to find all the fixed points of $f(n)$. | 1989or1990 | 81 | 9 |
math | ## Problem Statement
Find the derivative.
$$
y=\sqrt{(4+x)(1+x)}+3 \ln (\sqrt{4+x}+\sqrt{1+x})
$$ | \sqrt{\frac{4+x}{1+x}} | 37 | 11 |
math | Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition:
for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds. | f(x) = c^{\frac{1}{\ln x}} | 79 | 16 |
math | 81. Determine the sum of the coefficients of the polynomial that results from expanding and combining like terms in the expression $\left(1+x-3 x^{2}\right)^{1965}$. | -1 | 43 | 2 |
math | 10. (5 points) The sum of three numbers, A, B, and C, is 2017. A is 3 less than twice B, and B is 20 more than three times C. Then A is $\qquad$ . | 1213 | 56 | 4 |
math | Example 4 The inequality about the real number $x$ is $\left|x-\frac{(a+1)^{2}}{2}\right| \leqslant \frac{(a-1)^{2}}{2}$ and $x^{2}-3(a+1) x+$ $2(3 a+1) \leqslant 0$ (where $a \in \mathbf{R})$. The solution sets of these inequalities are denoted as $A$ and $B$ respectively. Find the range of values for $a$ that makes $... | 1\leqslant\leqslant3or=-1 | 127 | 15 |
math | 3.167. Find the number $\alpha \in\left(\frac{\pi}{2}, \pi\right)$, if it is known that $\operatorname{tg} 2 \alpha=-\frac{12}{5}$. | \pi-\arctan\frac{2}{3} | 53 | 13 |
math | 1. (7 points) The numerator and denominator of the fraction are positive numbers. The numerator was increased by 1, and the denominator by 100. Can the resulting fraction be greater than the original? | \frac{1}{200}<\frac{2}{300} | 44 | 18 |
math | Example 4 Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{n+1}=a_{n}^{3}-3 a_{n}$, and $a_{1}=\frac{3}{2}$. Find the general term formula of $\left\{a_{n}\right\}$. | a_{n}=2 \cos \left(3^{n-1} \arccos \frac{3}{4}\right) | 70 | 29 |
math | 26 Determine all non-empty subsets $A, B, C$ of $\mathbf{N}^{*}$ such that:
(1) $A \cap B=B \cap C=C \cap A=\varnothing$;
(2) $A \cup B \cup C=\mathbf{N}^{*}$;
(3) For all $a \in A, b \in B, c \in C$, we have $a+c \in A, b+c \in B, a+b \in C$ | ({3k-2\midk\in{Z}^{+}},{3k-1\midk\in{Z}^{+}},{3k\midk\in{Z}^{+}}) | 111 | 47 |
math | 9. If $P(x, y)$ is a point on the hyperbola $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$, then the minimum value of $|x-y|$ is $\qquad$ . | 2 | 54 | 1 |
math | 4.063. Given two geometric progressions consisting of the same number of terms. The first term and the common ratio of the first progression are 20 and $3 / 4$, respectively, while the first term and the common ratio of the second progression are 4 and $2 / 3$, respectively. If the terms of these progressions with the ... | 7 | 105 | 1 |
math | Example 9. Find the mathematical expectation of a random variable $X$,
uniformly distributed on the interval $[2,8]$. | 5 | 29 | 1 |
math | Example 10 (1998 National High School Mathematics League Question) Let the function $f(x)=a x^{2}+8 x+3(a<0)$. For a given negative number $a$, there is a largest positive number $L(a)$, such that the inequality $|f(x)| \leqslant 5$ holds for the entire interval $[0, L(a)]$. For what value of $a$ is $L(a)$ the largest?... | \frac{\sqrt{5}+1}{2} | 225 | 12 |
math | 10.1. The area of the quadrilateral formed by the midpoints of the bases and diagonals of a trapezoid is four times smaller than the area of the trapezoid itself. Find the ratio of the lengths of the bases of the trapezoid. | 3:1 | 59 | 3 |
math | Example 6 Let $a, b, c$ be positive real numbers, and $abc + a + c = b$. Try to determine the maximum value of $P=\frac{2}{a^{2}+1}-\frac{2}{b^{2}+1}+\frac{3}{c^{2}+1}$.
(1999, Vietnam Mathematical Olympiad) | \frac{10}{3} | 83 | 8 |
math | 7. In the expansion of $\left(x+\frac{4}{x}-4\right)^{5}$, the coefficient of $x^{3}$ is $\qquad$ .(answer with a specific number) | 180 | 45 | 3 |
math | 6. From a class of 20 people, a team of three students is formed to participate in mathematics, Russian language, and informatics olympiads. In this class, all students excel academically. How many ways are there to form the team of olympiad participants, if each member of the team participates in one olympiad? | 6840 | 73 | 4 |
math | A (192 digit) $123456789101112 \ldots 979899100$ number, delete 100 digits from it so that the remaining digits read together form the largest possible number. | 9999978596061\ldots99100 | 60 | 21 |
math | 11.2. On a coordinate plane, a grasshopper jumps starting from the origin. The first jump, one cm long, is directed along the OX axis, each subsequent jump is 1 cm longer than the previous one, and is directed perpendicular to the previous one in either of two directions at its choice. Can the grasshopper end up at the... | No | 82 | 1 |
math | Let $x$ and $y$ be distinct real numbers such that
\[ \sqrt{x^2+1}+\sqrt{y^2+1}=2021x+2021y. \]
Find, with proof, the value of
\[ \left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right). \] | \frac{1011}{1010} | 85 | 13 |
math | side 28 find $\sigma_{3}(62)=$ ? | 268128 | 15 | 6 |
math | Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied:
1- $b>a$ and $b-a$ is a prime number
2- The last digit of the number $a+b$ is $3$
3- The number $ab$ is a square of an integer. | (4, 9) | 69 | 7 |
math | 72. In the triangular pyramid $A B C D$, the faces $A B C$ and $A B D$ have areas $p$ and $q$ and form an angle $\alpha$ between them. Find the area of the section of the pyramid passing through the edge $A B$ and the center of the sphere inscribed in the pyramid. | \frac{2pq\cos\frac{\alpha}{2}}{p+q} | 74 | 19 |
math | A 25-meter long coiled cable is cut into pieces of 2 and 3 meters. In how many ways can this be done if the order of the different sized pieces also matters? | 465 | 40 | 3 |
math | 3. Suppose $f$ is a function that assigns to each real number $x$ a value $f(x)$, and suppose the equation
$$
f\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)=f\left(x_{1}\right)+f\left(x_{2}\right)+f\left(x_{3}\right)+f\left(x_{4}\right)+f\left(x_{5}\right)-8
$$
holds for all real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}... | 2 | 145 | 1 |
math | 【Question 3】
In a math competition, the average score of boys in a class is 73 points, and the average score of girls is 77 points, with the overall average score of the class being 74 points. It is also known that there are 22 more boys than girls. Therefore, the total number of students in the class is $\qquad$ peopl... | 44 | 83 | 2 |
math | For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$. | \{1, 2, 3, 4, 20, 21, 22, 23, 24, 100, 101, 102, 103, 104, 120, 121, 122, 123, 124\} | 65 | 85 |
math | $1 \frac{2 \cos 10^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}}=$ $\qquad$ | \sqrt{3} | 38 | 5 |
math | 6.1. (16 points) In a hut, several inhabitants of the island gathered, some from the Ah tribe, and the rest from the Ukh tribe. The inhabitants of the Ah tribe always tell the truth, while the inhabitants of the Ukh tribe always lie. One of the inhabitants said: "There are no more than 16 of us in the hut," and then ad... | 15 | 173 | 2 |
math | Problem 10.3. Find all pairs of positive integers $(m, n), m>n$, such that
$$
\left[m^{2}+m n, m n-n^{2}\right]+[m-n, m n]=2^{2005}
$$
where $[a, b]$ denotes the least common multiple of $a$ and $b$.
Ivan Landjev | =2^{1002},n=2^{1001} | 86 | 17 |
math | 3. The equation concerning $x, y$
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{x y}=\frac{1}{2011}
$$
has $\qquad$ groups of positive integer solutions $(x, y)$. | 12 | 62 | 2 |
math | Example 14 A permutation $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ of the set $\{1,2, \cdots, n\}$ is called a derangement (or wrong permutation) if $a_{i} \neq i$ $(i=1,2, \cdots, n)$. Find the number of derangements $D_{n}$. | D_{n}=n!(1-\frac{1}{1!}+\frac{1}{2!}-\cdots+(-1)^{n}\frac{1}{n!}) | 93 | 40 |
math | Example 2 Given $x, y, z \in \mathbf{R}$, and satisfying $x+y+z=2$, find the minimum value of $T=x^{2}-2 x+2 y^{2}+3 z^{2}$. | -\frac{5}{11} | 54 | 8 |
math | Example 2 Given $a+b+c=1$,
$$
b^{2}+c^{2}-4 a c+6 c+1=0 \text{. }
$$
Find the value of $a b c$. | 0 | 49 | 1 |
math | 3. (3 points) Equilateral triangles $A B C$ and $A_{1} B_{1} C_{1}$ with side length 10 are inscribed in the same circle such that point $A_{1}$ lies on the arc $B C$, and point $B_{1}$ lies on the arc $A C$. Find $A A_{1}^{2}+B C_{1}^{2}+C B_{1}^{2}$. | 200 | 101 | 3 |
math | ## Task A-2.3.
Determine all triples $(x, y, z)$ of positive real numbers that satisfy the system of equations
\[
\begin{aligned}
3\lfloor x\rfloor - \{y\} + \{z\} & = 20.3 \\
3\lfloor y\rfloor + 5\lfloor z\rfloor - \{x\} & = 15.1 \\
\{y\} + \{z\} & = 0.9
\end{aligned}
\]
For a real number $t$, $\lfloor t\rfloor$ de... | 7.9,2.8,2.1 | 214 | 11 |
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