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 | 3. Given two distinct points $A$ and $B$, find the locus of points $M$ in the plane that satisfy the condition $\overrightarrow{M A} \cdot \overrightarrow{M B}=k^{2}$ (where $k$ is a non-zero real constant). | |\overrightarrow{OM}|=\sqrt{k^{2}+^{2}} | 60 | 16 |
math | 8. Let $T$ be a triangle with side lengths 26,51 , and 73 . Let $S$ be the set of points inside $T$ which do not lie within a distance of 5 of any side of $T$. Find the area of $S$. | \frac{135}{28} | 61 | 10 |
math | For what real numbers $a, b, c$ is it true that if the sum of the first $n$ terms of a sequence is of the form $a n^{2}+b n+c$ for any positive integer $n$, then the sequence is an arithmetic sequence? | 0 | 58 | 1 |
math | [ Equilateral (regular) triangle ] [ Rotations by $\$ 60$ ^ \circ $\$$ and $\$ 120$ ^\circ\$
In rhombus $A B C D$, angle $A B C$ is $120^{\circ}$. Points $P$ and $Q$ are taken on sides $A B$ and $B C$, respectively, such that $A P=B Q$. Find the angles of triangle $P Q D$.
# | 60,60,60 | 105 | 8 |
math | 1. Given $a \in \mathbf{Z}$, and $x^{6}-33 x+20$ can be divided by $x^{2}-x+a$, then the value of $a$ is | 4 | 47 | 1 |
math | 2. Given an integer $n \geqslant 3$. Find the smallest positive integer $k$, such that there exists a $k$-element set $A$ and $n$ pairwise distinct real numbers $x_{1}, x_{2}, \cdots, x_{n}$, satisfying $x_{1}+x_{2}, x_{2}+x_{3}, \cdots$, $x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$. (Xiong Bin provided) | 3 | 121 | 1 |
math | We roll a die 10 times. What is the probability that any three consecutive rolls will be $1, 2, 3$? | 0.0367 | 30 | 6 |
math | 8,9 |
Given a trapezoid $A B C D$ with bases $A D=3$ and $B C=18$. Point $M$ is located on diagonal $A C$, such that $A M$ : $M C=1: 2$. A line passing through point $M$ parallel to the bases of the trapezoid intersects diagonal $B D$ at point $N$. Find $M N$. | 4 | 94 | 1 |
math | Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. | 453 | 86 | 3 |
math | Example 2 In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{2}=2, a_{n+2}=5 a_{n+1}-6 a_{n}, n=1,2$, $3, \cdots$, find the general formula for $\left\{a_{a}\right\}$. | a_{n}=2^{n-1} | 79 | 10 |
math | 7. [55] Consider sequences $a$ of the form $a=\left(a_{1}, a_{2}, \ldots, a_{20}\right)$ such that each term $a_{i}$ is either 0 or 1 . For each such sequence $a$, we can produce a sequence $b=\left(b_{1}, b_{2}, \ldots, b_{20}\right)$, where
$$
b_{i}=\left\{\begin{array}{ll}
a_{i}+a_{i+1} & i=1 \\
a_{i-1}+a_{i}+a_{i+1... | 64 | 198 | 2 |
math | I3.1 Suppose that $a=\cos ^{4} \theta-\sin ^{4} \theta-2 \cos ^{2} \theta$, find the value of $a$.
I3.2 If $x^{y}=3$ and $b=x^{3 y}+10 a$, find the value of $b$.
I3.3 If there is (are) $c$ positive integer(s) $n$ such that $\frac{n+b}{n-7}$ is also a positive integer, find the value of $c$.
I3.4 Suppose that $d=\log _... | -1,17,8,18 | 175 | 10 |
math | 4. Find the value of the expression $\frac{1}{a b}+\frac{1}{b c}+\frac{1}{a c}$, if it is known that $a, b, c$ are three different real numbers satisfying the conditions $a^{3}-2020 a^{2}+1010=0, b^{3}-2020 b^{2}+1010=0, \quad c^{3}-2020 c^{2}+1020=0$. | -2 | 116 | 2 |
math | 13. Determine the maximum positive integer $k$ such that $k^{2}$ divides $\frac{n !}{(n-6) !}$ for every $n>6$.
| 12 | 39 | 2 |
math | Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the le... | 2015029 | 138 | 7 |
math | Five. (15 points) If for any $n$ consecutive positive integers, there always exists a number whose sum of digits is a multiple of 8. Determine the minimum value of $n$. And explain the reason.
---
Translate the above text into English, please retain the original text's line breaks and format, and output the translati... | 15 | 72 | 2 |
math | Task 1. (5 points) Find $\frac{a^{12}-729}{27 a^{6}}$, if $\frac{a^{2}}{3}-\frac{3}{a^{2}}=4$.
# | 76 | 53 | 2 |
math | [Pythagorean Theorem (direct and inverse).]
The height of an isosceles triangle, dropped to the lateral side, divides it into segments of 2 and 1, counting from the vertex of the triangle. Find the base of the triangle. | \sqrt{6} | 54 | 5 |
math | 8.385. $3\left(\log _{2} \sin x\right)^{2}+\log _{2}(1-\cos 2 x)=2$. | (-1)^{k}\frac{\pi}{6}+\pik,k\inZ | 40 | 19 |
math | Given positive real numbers $a, b, c, d$ that satisfy equalities
$$
a^{2}+d^{2}-a d=b^{2}+c^{2}+b c \quad \text{and} \quad a^{2}+b^{2}=c^{2}+d^{2}
$$
find all possible values of the expression $\frac{a b+c d}{a d+b c}$.
Answer: $\frac{\sqrt{3}}{2}$. | \frac{\sqrt{3}}{2} | 106 | 10 |
math | 5.1. Let $S(n)$ be the sum of the digits in the decimal representation of the number $n$. Find $S\left(S\left(S\left(S\left(2017^{2017}\right)\right)\right)\right)$. Answer. 1. | 1 | 64 | 1 |
math | A group consisting of 5 girls and 7 boys plays "wedding": they select (in the following order) a couple, a registrar, two flower girls each with a gentleman, and two witnesses. How many ways can this "wedding party" be formed if 3 of the girls have 1 brother each present, and the couple cannot be siblings, and the regi... | 313200 | 99 | 6 |
math | 4. It is known that the number $\cos 6^{0}$ is a root of the equation $32 t^{5}-40 t^{3}+10 t-\sqrt{3}=0$. Find the other four roots of this equation. (Answers in the problem should be compact expressions, not containing summation signs, ellipses, and radicals.)
# | \cos66,\cos78,\cos138,\cos150 | 78 | 18 |
math | C/1 Find all real numbers $x$ that satisfy the equation
$$
\frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000},
$$
and simplify your answer(s) as much as possible. Justify your solution. | 2021 | 109 | 4 |
math | Solve the following system of equations:
$$
\begin{aligned}
\left(x^{3}+y^{3}\right)\left(x^{2}+y^{2}\right) & =64 \\
x+y & =2
\end{aligned}
$$ | {\begin{pmatrix}x_{1}=1+\sqrt{\frac{5}{3}},\\y_{1}=1-\sqrt{\frac{5}{3}};\end{pmatrix}\quad | 57 | 42 |
math | 12 (18 points) The inverse function of $f(x)$ is $y=\frac{x}{1+x}, g_{n}(x)+\frac{1}{f_{n}(x)}=0$, let $f_{1}(x)=f(x)$, and for $n>1, n \in \mathbf{N}^{*}$, $f_{n}(x)=f_{n-1}\left[f_{n-1}(x)\right]$. Find the analytical expression for $g_{n}(x)\left(n \in \mathbf{N}^{*}\right)$. | 2^{n-1}-\frac{1}{x} | 130 | 13 |
math | 10. (20 points) Given the parabola $C: y=\frac{1}{2} x^{2}$ and the circle $D: x^{2}+\left(y-\frac{1}{2}\right)^{2}=r^{2}(r>0)$ have no common points, a tangent line is drawn from a point $A$ on the parabola $C$ to the circle $D$, with the points of tangency being $E$ and $F$. When point $A$ moves along the parabola $C... | \left(0, \frac{\pi}{16}\right) | 148 | 15 |
math | In a quadrilateral $ABCD$, we have $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$ and $\angle BCD = 70^{\circ}$ . Let $ M, N$ be the mid-points of $AB$ and $CD$ respectively. Suppose $P$ is a point on the segment $M N$ such that $\frac{AM}{CN} = \frac{MP}{PN}$ and $AP = CP$ . Find $\angle AP C$. | 120^\circ | 113 | 5 |
math | 13. Given the sets $A=\left\{(x, y) \left\lvert\, \frac{y-3}{x-2}=a+1\right.\right\}, B=$ $\left\{(x, y) \mid\left(a^{2}-1\right) x+(a-1) y=15\right\}$. Then the possible values of $a$ that make $A \cap B=$ $\varnothing$ are . $\qquad$ | -4, -1, 1, \frac{5}{2} | 106 | 16 |
math | ## Task 2 - 190622
In a shelf of a HO retail store, there are six gift items priced at $15 \mathrm{M}, 16 \mathrm{M}, 18 \mathrm{M}, 19 \mathrm{M}, 20 \mathrm{M}$, and $31 \mathrm{M}$, with exactly one piece of each type.
One customer bought exactly two of these gifts, another exactly three. The second customer had t... | Thefirstbuyerboughtthegiftspricedat15\mathrm{M}18\mathrm{M},thebuyerconsequentlyboughtthegiftspricedat16\mathrm{M},19\mathrm{M},31\mathrm{M} | 136 | 58 |
math | 2. Solve the equation $\frac{\sin 8 x+\sin 4 x}{\cos 5 x+\cos x}=6|\sin 2 x|$. | -\arccos\frac{\sqrt{13}-3}{4}+2k\pi,-\arccos\frac{3-\sqrt{13}}{4}+2k\pi,k\pi,k\in\mathrm{Z} | 35 | 55 |
math | Let $a<b<c<d<e$ be real numbers. We calculate all possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of $e$.
## High School Statements | \frac{55}{2} | 72 | 8 |
math | 10. Let $[a]$ denote the greatest integer not exceeding the real number $a$. Then the largest positive integer solution to the equation
$$
\left[\frac{x}{7}\right]=\left[\frac{x}{8}\right]+1
$$
is $\qquad$ | 104 | 61 | 3 |
math | 4. A natural number $n$ when divided by 3 gives a remainder $a$, when divided by 6 gives a remainder $b$, and when divided by 9 gives a remainder $c$. It is known that $a+b+c=15$. Determine the remainder when the number $n$ is divided by 18. | 17 | 70 | 2 |
math | [ Numerical inequalities. Comparing numbers.]
What is greater: $(1.01)^{1000}$ or 1000? | (1.01)^{1000} | 32 | 12 |
math | 8.1. A bus with programmers left Novosibirsk for Pavlodar. When it had traveled 70 km, Pavel Viktorovich set off from Novosibirsk in a car along the same route, and caught up with the programmers in Karasuk. After that, Pavel drove another 40 km, while the bus traveled only 20 km in the same time. Find the distance fro... | 140 | 107 | 3 |
math | Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimu... | n^2 + n | 103 | 5 |
math | Problem 6.7. The Amur and Bengal tigers started running in a circle at 12:00, each at their own constant speed. By 14:00, the Amur tiger had run 6 more laps than the Bengal tiger. Then the Amur tiger increased its speed by 10 km/h, and by 15:00, it had run a total of 17 more laps than the Bengal tiger. How many meters ... | 1250 | 106 | 4 |
math | 17.1.9 ** $x$ is a real number, how many of the first 1000 positive integers can be expressed in the form $[2 x]+[4 x]+[6 x]+[8 x]$? | 600 | 51 | 3 |
math | For how many elements is it true that the number of repetitive 3-class variations is 225 greater than the number of variations without repetition? | 9 | 30 | 1 |
math | Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation. | p - 1 | 64 | 5 |
math | Half of the children in 5.A attend a dance club. All girls attend and one third of the 18 boys attend.
a) How many children are in 5.A?
b) How many girls are in 5.A? | 6 | 49 | 1 |
math | Putnam 1998 Problem A1 A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube? Solution | \frac{3\sqrt{2}}{3+\sqrt{2}} | 59 | 16 |
math | Evaluate the following expression: $$0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000$$ The terms with minus signs are exactly the powers of two.
| 1996906 | 51 | 7 |
math | 1. Solve the inequality $\sqrt{x^{2}-16} \cdot \sqrt{2 x-1} \leq x^{2}-16$. | x\in{4}\cup[5;+\infty) | 34 | 14 |
math | 2. When $x=\frac{\sqrt{29}-3}{2}$, the value of the algebraic expression
$$
x^{4}+5 x^{3}-3 x^{2}-8 x+9
$$
is $\qquad$. | 7 \sqrt{29}-32 | 55 | 9 |
math | $\left[\begin{array}{l}\text { Coordinate Method on the Plane }] \\ {[\quad \text { Circles (other). }}\end{array}\right]$
A circle with its center at point $M(3 ; 1)$ passes through the origin. Formulate the equation of the circle. | (x-3)^{2}+(y-1)^{2}=10 | 66 | 17 |
math | Task 3. Find all triples of real numbers $(x, y, z)$ that satisfy
$$
x^{2}+y^{2}+z^{2}+1=x y+y z+z x+|x-2 y+z| .
$$ | (y+1,y,y+1) | 54 | 8 |
math | 6.1. On a plane, 55 points are marked - the vertices of a certain regular 54-gon and its center. Petya wants to paint a triplet of the marked points in red so that the painted points are the vertices of some equilateral triangle. In how many ways can Petya do this? | 72 | 69 | 2 |
math | # Problem T-1
Given a pair $\left(a_{0}, b_{0}\right)$ of real numbers, we define two sequences $a_{0}, a_{1}, a_{2}, \ldots$ and $b_{0}, b_{1}, b_{2}, \ldots$ of real numbers by
$$
a_{n+1}=a_{n}+b_{n} \quad \text { and } \quad b_{n+1}=a_{n} \cdot b_{n}
$$
for all $n=0,1,2, \ldots$ Find all pairs $\left(a_{0}, b_{0... | b_{0}=0 | 207 | 5 |
math | Example 5 Given real numbers $x, y$ satisfy $x+y=3, \frac{1}{x+y^{2}}+\frac{1}{x^{2}+y}=\frac{1}{2}$.
Find the value of $x^{5}+y^{5}$. [3]
(2017, National Junior High School Mathematics League) | 123 | 79 | 3 |
math | The five numbers $17$, $98$, $39$, $54$, and $n$ have a mean equal to $n$. Find $n$. | 52 | 37 | 2 |
math | 3. Person A has a box, inside there are 4 balls in total, red and white; Person B has a box, inside there are 2 red balls, 1 white ball, and 1 yellow ball. Now, A randomly takes 2 balls from his box, B randomly takes 1 ball from his box. If the 3 balls drawn are all of different colors, then A wins. To ensure A has the... | 2 | 105 | 1 |
math | 4. The three sides of a triangle can be expressed as $m^{2}+m+1, 2m+1, m^{2}-1$, then the largest angle of this triangle is . $\qquad$ | \frac{2\pi}{3} | 47 | 9 |
math | 11.2. On the board, the expression $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}$ is written, where $a, b, c, d, e, f$ are natural numbers. If the number $a$ is increased by 1, then the value of this expression increases by 3. If in the original expression the number $c$ is increased by 1, then its value increases b... | 60 | 151 | 2 |
math | 4.6.43 $\stackrel{\star \star}{\star}$ Let $x_{1}, x_{2}, \cdots, x_{n}$ be non-negative real numbers, and satisfy
$$
\sum_{i=1}^{n} x_{i}^{2}+\sum_{1 \leqslant i<j \leqslant n}\left(x_{i} x_{j}\right)^{2}=\frac{n(n+1)}{2} \text {. }
$$
(1) Find the maximum value of $\sum_{i=1}^{n} x_{i}$;
(2) Find all positive integer... | 1,2,3 | 178 | 5 |
math | Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$. | \frac{3\sqrt{3}}{2} | 63 | 12 |
math | 5. $\frac{\sin 7^{\circ}+\sin 8^{\circ} \cos 15^{\circ}}{\cos 7^{\circ}-\sin 8^{\circ} \sin 15^{\circ}}=$ | 2-\sqrt{3} | 55 | 6 |
math | Question 52, in $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $\sin A \sin 2A=$ $(1-\cos A)(1-\cos 2A)$, if the area of $\triangle ABC$ is $S=\frac{\sqrt{3}}{12}\left(8 b^{2}-9 a^{2}\right)$, find the value of $\cos B$. | -\frac{\sqrt{7}}{14} | 101 | 11 |
math | 3. [5 points] Solve the system of equations
$$
\left\{\begin{array}{l}
\left(\frac{x^{4}}{y^{2}}\right)^{\lg y}=(-x)^{\lg (-x y)} \\
2 y^{2}-x y-x^{2}-4 x-8 y=0
\end{array}\right.
$$ | (-4;2),(-2;2),(\frac{\sqrt{17}-9}{2};\frac{\sqrt{17}-1}{2}) | 81 | 34 |
math | 8. Expand the binomial $\left(\frac{\sqrt{x+1}}{2 \sqrt[4]{x}}\right)^{n}$ in descending powers of $x$. If the coefficients of the first two terms form an arithmetic sequence, then the number of terms in the expansion where the power of $x$ is an integer is $\qquad$ . | 3 | 75 | 1 |
math | A sports competition lasting $n \quad(n>1)$ days distributed a total of $m$ medals. On the first day, 1 medal and $1 / 7$ of the remaining medals were awarded, on the second day, 2 medals and $1 / 7$ of the remaining medals, and so on. Finally, on the $n$-th, and last day, exactly $n$ medals were awarded, which were th... | n=6,=36 | 112 | 7 |
math | 2.12. Compute surface integrals of the first kind:
a) $\iint_{\sigma}|x| d S, \quad \sigma: x^{2}+y^{2}+z^{2}=1, \quad z \geqslant 0$
b) $\iint_{\sigma}\left(x^{2}+y^{2}\right) d S, \quad \sigma: x^{2}+y^{2}=2 z, \quad z=1$;
c) $\iint_{\sigma}\left(x^{2}+y^{2}+z^{2}\right) d S$, where $\sigma$ is the part of the cone... | \pi,\frac{4\pi}{15}(1+15\sqrt{3}),3\sqrt{2}\pi | 192 | 28 |
math | A quarter of the students in the class are non-swimmers. Half of the non-swimmers signed up for the swimming course. Four non-swimmers did not sign up for the course. How many students in the class can swim and how many students are there in total in the class? | 32 | 58 | 2 |
math | 3. Let $n \in \mathbf{Z}_{+}$. For positive integers $a, b, c$, we have $a+$ $b+c=5 n$. Find the maximum value of $(a, b)+(b, c)+(c, a)$. | G_{\max}={\begin{pmatrix}5n,&3\midn;\\4n,&3\nmidn0\end{pmatrix}.} | 57 | 36 |
math | 4.064. Three numbers, of which the third is 12, form a geometric progression. If 12 is replaced by 9, the three numbers form an arithmetic progression. Find these numbers. | 27,18,12;3,6,12 | 45 | 15 |
math | 10. There are several warriors, forming a rectangular formation that is exactly eight columns wide. If 120 more people are added to or 120 people are removed from the formation, a square formation can be formed in both cases. How many warriors are there in the original rectangular formation? | 904 | 62 | 3 |
math | 2. The kit's calf is twenty times heavier than the elephant's calf. The mother elephant is 50 times heavier than her calf and 30 times lighter than the mother whale, and she weighs 6 tons. How heavy is the elephant's calf, how heavy is the kit's calf, and how heavy is the mother whale? How many times heavier is the mot... | 75 | 82 | 2 |
math | 3.45 The recording of a six-digit number starts with the digit 2. If this digit is moved from the first position to the last, keeping the order of the other five digits, the newly obtained number will be three times the original number. Find the original number.
## Distance: path, speed, time | 285714 | 65 | 6 |
math | Determine the maximum value of $m^{2}+n^{2}$, where $m, n \in\{1,2, \cdots, 1981\}$, and $\left(n^{2}-\right.$ $\left.m n-m^{2}\right)^{2}=1$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 3524578 | 93 | 7 |
math | 4. Dwarves Glóin, Óin, and Thráin found 70 identical precious stones and want to divide them among themselves so that each of them gets no less than 10 stones. In how many ways can the dwarves do this? | 861 | 56 | 3 |
math | 4. 15 bags of potatoes and 12 bags of flour weigh 1 ton 710 kg, with the weight of one bag of potatoes being 30 kg less than the weight of one bag of flour. How much does one bag of potatoes and one bag of flour weigh? | one\bag\of\flour\weighs\80\,\\one\bag\of\potatoes\weighs\50\ | 62 | 32 |
math | Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$
Determine all integers $m$ such that every term of the sequence is a square. | m = 2 | 112 | 5 |
math | 2 boys and 3 girls went on a trip. Both the boys and the girls carried equally weighted backpacks, and the 5 backpacks together weighed 44 $\mathrm{kg}$. On the way - so that 1 girl could always rest - the boys carried the girls' backpacks, one each on their backs and one together, in their hands, while the girls took ... | 8 | 120 | 1 |
math | 4. determine all real solutions $(x, y, z)$ of the system
$$
\frac{4 x^{2}}{1+4 x^{2}}=y, \quad \frac{4 y^{2}}{1+4 y^{2}}=z, \quad \frac{4 z^{2}}{1+4 z^{2}}=x
$$
## Solution | (x,y,z)=(0,0,0)(x,y,z)=(\frac{1}{2},\frac{1}{2},\frac{1}{2}) | 84 | 35 |
math | \section*{Problem 21 - V01221}
The cross-section of a sewer pipe is to have the shape of a rectangle with a semicircle on top.
What height and width should it be given if the area of the cross-section is \(1 \mathrm{~m}^{2}\) and the manufacturing costs are to be as low as possible?
It should be taken into account t... | b\approx1.318\, | 105 | 10 |
math |
NT 1. Find all the integers pairs $(x, y)$ which satisfy the equation
$$
x^{5}-y^{5}=16 x y
$$
| (x,y)=(0,0)or(-2,2) | 37 | 13 |
math | (Bachet's weight problem) We have a two-pan balance with which we want to be able to weigh any object of an integer mass between $1 \text{and} 40 \text{ kg}$. What is the minimum number of integer-mass weights needed for this:
1. if these weights can only be placed on one pan of the balance?
2. if they can be placed o... | 1,3,9,27 | 87 | 8 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-3}{1}=\frac{y+2}{-1}=\frac{z-8}{0}$
$5 x+9 y+4 z-25=0$ | (4,-3,8) | 59 | 7 |
math | In Sweden, there is allegedly a very deep ground fissure or cave, into which if we drop a stone, we only hear the impact sound after $25 \mathrm{sec}$. How deep is the cave, if we also take into account the speed of sound? | 1867\mathrm{~} | 56 | 9 |
math | 2. (7p) Solve in $\mathbf{R}$ the equation:
$$
\{x\}+\{2 x\}+\{3 x\}=1
$$
where $\{a\}$ denotes the fractional part of the real number $a$.
Petru Vlad | {\frac{6p+1}{6},\frac{6p+2}{6},\frac{6p+3}{6},\frac{6p+4}{6}\midp\in{Z}} | 61 | 47 |
math | 11. Solve the equation $(\sin x)^{\arctan (\sin x+\cos x)}=(\csc x)^{\arctan (\sin 2 x)+\frac{\pi}{4}}$, where $2 n \pi<x<$ $(2 n+1) \pi, n$ is any integer. | x_{1}=2n\pi+\frac{\pi}{2},x_{2}=2n\pi+\frac{3\pi}{4} | 69 | 32 |
math | 3. Let $E$ be a given $n$-element set, and $A_{1}, A_{2}, \cdots, A_{k}$ be $k$ distinct non-empty subsets of $E$, satisfying: for any $1 \leqslant i<j \leqslant k$, either $A_{i} \cap A_{j}=\varnothing$ or one of $A_{i}$ and $A_{j}$ is a subset of the other. Find the maximum value of $k$.
(Cold Gangsong, problem contr... | 2n-1 | 120 | 4 |
math | 7. Given $1 \leqslant x^{2}+y^{2} \leqslant 4$, then the sum of the maximum and minimum values of $x^{2}-x y+y^{2}$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 6\frac{1}{2} | 75 | 8 |
math | 9. Mini-tournament (recommended from 7th grade, 1 point). Alyosha, Borya, and Vasya are holding a mini-tournament in tennis: each plays with each other once. The winner gets 1 point, the loser gets 0, and there are no draws in tennis. The absolute winner of the mini-tournament is the one who has a total of 2 points. It... | 0.24 | 143 | 4 |
math | 4. The brothers found a treasure of gold and silver. They divided it so that each received 100 kg. The eldest received the most gold - 25 kg - and one eighth of all the silver. How much gold was in the treasure? | 100 | 53 | 3 |
math | 11 Given $\alpha, \beta \in \mathbf{R}$, the intersection point of the lines $\frac{x}{\sin \alpha+\sin \beta}+\frac{y}{\sin \alpha+\cos \beta}=1$ and $\frac{x}{\cos \alpha+\sin \beta}+$ $\frac{y}{\cos \alpha+\cos \beta}=1$ lies on the line $y=-x$, then $\sin \alpha+\cos \alpha+\sin \beta+$ $\cos \beta=$ $\qquad$ | 0 | 115 | 1 |
math | Folkolo
Find the maximum value of the expression $ab + bc + ac + abc$, if $a + b + c = 12$ (where $a, b$, and $c$ are non-negative numbers). | 112 | 48 | 3 |
math | Example 1. The random variable $\xi$ is distributed according to the normal law with parameters $a$ and $\sigma^{2}$. From the sample $x_{1}, \ldots, x_{n}$ of values of $\xi$, the empirical moments $M_{1}^{*}=\bar{x}=2.3$ and $M_{2}^{*}=\overline{x^{2}}=8.7$ have been determined. Using these moments, find the paramete... | =2.3,\sigma^{2}=3.41 | 112 | 13 |
math | Question 11 Given $\alpha+\beta+\gamma=180^{\circ}(\alpha, \beta, \gamma \geqslant$ $0)$. Find the maximum value of $3 \cos \alpha+4 \cos \beta+5 \cos \gamma$. | \frac{769}{120} | 61 | 11 |
math | 1. The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal? | 6\sqrt{3} | 46 | 6 |
math | ## 9. Pools and Pipes
The hotel has three pools $A, B$, and $C$ that can be connected or separated by valves. Several identical pipes are used for filling. When all three pools are connected, filling them with water from three pipes takes 18 minutes. If only pools $A$ and $B$ are connected, filling them with two pipes... | 960 | 134 | 3 |
math | 6. Let $P(x)$ be a quadratic polynomial with real coefficients such that $P(11)=181$ and $x^{2}-2 x+2 \leq P(x) \leq 2 x^{2}-4 x+3$ for any real number $x$. Find $P(21)$.
(1 mark)
設 $P(x)$ 為二次多項式, 其系數皆為實數。已知 $P(11)=181$, 且對任意實數 $x$ 皆有 $x^{2}-2 x+2 \leq P(x) \leq 2 x^{2}-4 x+3$, 求 $P(21)$ 。
(1 分) | 721 | 165 | 3 |
math | 3. What are the sizes of the angles of a scalene triangle if the middle angle by size is the arithmetic mean of the remaining two angles and if the length of the longest side of the triangle is twice the length of the shortest side? | 30 | 49 | 2 |
math | Problem 4. Solve the equation:
$$
(x+1)^{2}+(x+3)^{2}+(x+5)^{2}+\ldots+(x+2021)^{2}=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-2020)^{2}
$$ | -0.5 | 82 | 4 |
math | Problem 10.1. Consider the inequality $\left|x^{2}-5 x+6\right| \leq x+a$, where $a$ is a real parameter.
a) Solve the inequality for $a=0$.
b) Find the values of $a$ for which the inequality has exactly three integer solutions.
Stoyan Atanassov | \in[-2,1) | 76 | 7 |
math | 11.102. A copper blank in the form of a rectangular parallelepiped with dimensions $80 \times 20 \times 5 \mathrm{~cm}$ is rolled into a sheet with a thickness of 1 mm. Determine the area of this sheet. | 8\mathrm{~}^{2} | 59 | 9 |
math | Example 1 Find the smallest positive period of the function $f(x)=\cos \sqrt{2} x+\sin \frac{3}{8} \sqrt{2} x$. | 8\sqrt{2}\pi | 39 | 7 |
math | Example 7 Real numbers $a$ and $b$ satisfy the equation $\frac{a^{2} b^{2}}{a^{4}-2 b^{4}}=$
1. Then $\frac{a^{2}-b^{2}}{19 a^{2}+96 b^{2}}=$ $\qquad$
(1996, Beijing Junior High School Mathematics Competition) | \frac{1}{134} | 84 | 9 |
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 |
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