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 | Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\log_{10}x)(\log_{10} yz) + (\log_{10}y) (\log_{10}z) = 468$. Find $\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}$. | 75 | 100 | 2 |
math | 20. (6 points) If the ten-digit number $\mathrm{a} 2016 \mathrm{~b} 2017$ is divisible by 33, then, how many such ten-digit numbers are there? | 3 | 52 | 1 |
math | 11. On a plane, a parabola divides the plane into two parts, two parabolas can divide the plane into at most seven parts, then how many parts can $n$ parabolas divide the plane into at most? | 2n^{2}-n+1 | 50 | 8 |
math | Determine all integers $a$ and $b$ such that
\[(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}\]
is a perfect square. | (a, b) = (0, 0) | 55 | 13 |
math | We say "$s$ grows to $r$" if there exists some integer $n>0$ such that $s^n = r.$ Call a real number $r$ "sparcs" if there are only finitely many real numbers $s$ that grow to $r.$ Find all real numbers that are sparse. | -1, 0, 1 | 66 | 8 |
math | I1.4 Suppose $d$ is the number of positive factors of $500+c$, find the value of $d$. | 24 | 28 | 2 |
math | Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square. | n = 1 | 42 | 5 |
math | Find all integer solutions of the indefinite system of equations
$$\left\{\begin{array}{l}
x+y+z=3 \\
x^{3}+y^{3}+z^{3}=3
\end{array}\right.$$ | (x, y, z)=(1,1,1),(-5,4,4),(4,-5,4),(4,4,-5) | 51 | 31 |
math | Example 2. Given $\mathrm{x}+\mathrm{y}=1$, find the maximum value of $\sqrt{2 \mathrm{x}+1}$ $+\sqrt{2 y+1}$. | 2 \sqrt{2} | 42 | 6 |
math | Question 233: Let the difference between any two adjacent interior angles of a convex $\mathrm{n}$-sided polygon be $20^{\circ}$, find the maximum possible value of $\mathrm{n}$. | 34 | 46 | 2 |
math | 7. Given $a, b \in \mathbf{R}^{+}$ and $\frac{\sin ^{4} x}{a}+\frac{\cos ^{4} x}{b}=\frac{1}{a+b}$, then $\frac{\sin ^{8} x}{a^{3}}+\frac{\cos ^{8} x}{b^{3}}=$ $\qquad$ | \frac{1}{(+b)^{3}} | 86 | 11 |
math | (14) (13 points) Let the set $I=\{1,2,3, \cdots, n\}\left(n \in \mathbf{N}^{*}\right)$, choose two non-empty subsets $A$ and $B$ of $I$, such that the smallest number in $B$ is greater than the largest number in $A$, denote the number of different selection methods as $a_{n}$, obviously $a_{1}=0, a_{2}=\mathrm{C}_{2}^{... | (n-3)\cdot2^{n}+n+3 | 170 | 13 |
math | 1. [20] Let $A B C$ be an equilateral triangle with side length 2 that is inscribed in a circle $\omega$. A chord of $\omega$ passes through the midpoints of sides $A B$ and $A C$. Compute the length of this chord. | \sqrt{5} | 61 | 5 |
math | Ha az előbbi feladatban azt is tudni akarjuk, hogy a hamis pénz könnyebb-e, vagy nehezebb a többinél, hány pénzdarab a legtöbb, amennyi közül még három mérés elegendő ennek a kérdésnek az eldöntéséhez?
If in the previous task we also want to know whether the counterfeit coin is lighter or heavier than the others, how ... | 12 | 129 | 2 |
math | 4. Starting from 1, alternately add 4 and 3, to get the following sequence of numbers $1,5,8,12,15,19,22 \ldots \ldots$ The number in this sequence that is closest to 2013 is $\qquad$ . | 2014 | 68 | 4 |
math | 8. [25] Suppose $x$ is a fifth root of unity. Find, in radical form, all possible values of
$$
2 x+\frac{1}{1+x}+\frac{x}{1+x^{2}}+\frac{x^{2}}{1+x^{3}}+\frac{x^{3}}{1+x^{4}} .
$$ | 4 | 75 | 1 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-10.5,10.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 462 | 59 | 3 |
math | 4. For a quadruple of points $A, B, C, D$ in the plane, no three of which are collinear, let $f(A, B, C, D)$ denote the measure of the largest angle formed by these points (out of a total of 12 such angles). Determine $\min f(A, B, C, D)$, where the minimum is taken over all such quadruples of points. | 90 | 89 | 2 |
math | 36*. A polyhedron is called regular if all its faces are regular and congruent polygons to each other and all its polyhedral angles are also regular and congruent to each other ${ }^{2}$. Investigate the possibility of constructing simple regular polyhedra with $n$-sided faces and $m$-sided angles and determine the pos... | 5 | 269 | 1 |
math | Problem 8. For what values of the parameter $a$ does the equation $x^{3}+6 x^{2}+a x+8=0$ have no more than two solutions? | [-15;+\infty) | 42 | 8 |
math | Example 2 Find the integer solutions of the equation $\frac{1}{x^{2}}+\frac{1}{x y}+\frac{1}{y^{2}}=1$.
untranslated text remains the same as requested. | (1,-1),(-1,1) | 48 | 10 |
math | Consider an $n\times{n}$ grid formed by $n^2$ unit squares. We define the centre of a unit square as the intersection of its diagonals.
Find the smallest integer $m$ such that, choosing any $m$ unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram. | 2n | 74 | 2 |
math | 4. Given that $x$ is a
four-digit number, the sum of its digits is $y$. If the value of $\frac{x}{y}$ is minimized, then $x=$ $\qquad$ | 1099 | 44 | 4 |
math | ## Problem Statement
Find the angle between the planes:
$2 x+2 y+z-1=0$
$x+z-1=0$ | \frac{\pi}{4} | 30 | 7 |
math | Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$? | 1024 | 88 | 4 |
math | Example 1. Find the value of $\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$ | -\frac{1}{2} | 41 | 7 |
math | In an acute triangle $ABC$, the bisector $AL$, the altitude $BH$, and the perpendicular bisector of the side $AB$ intersect at one point. Find the value of the angle $BAC$. | 60^\circ | 44 | 4 |
math | 9. Given $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $\Gamma$:
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$
with $\left|F_{1} F_{2}\right|=2$, and $A$ is the right endpoint of $\Gamma$. The line $l$ passes through point $A$ and is perpendicular to the $x$-axis, and $P$ is a moving point on... | (\sqrt{2},1)or(\sqrt{2},-1) | 167 | 16 |
math | The triangle $ABC$ lies on the coordinate plane. The midpoint of $\overline{AB}$ has coordinates $(-16, -63)$, the midpoint of $\overline{AC}$ has coordinates $(13, 50)$, and the midpoint of $\overline{BC}$ has coordinates $(6, -85)$. What are the coordinates of point $A$? | (-9, 72) | 84 | 7 |
math | 6. Let $x, y, z \in (0,1)$, satisfy
$$
\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2 \text {. }
$$
Find the maximum value of $x y z$.
(Tang Lihua, provided) | \frac{27}{64} | 81 | 9 |
math | 5. The terms of the sequence $\left\{a_{n}\right\}$ are all positive, and the sum of the first $n$ terms $S_{n}$ satisfies
$$
S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right) .
$$
Then $a_{n}=$ | a_{n}=\sqrt{n}-\sqrt{n-1} | 79 | 14 |
math | Lake 1. Given the arithmetic sequence with the first term $a_{1}=3, a_{n+1}=a_{n}+5$, find up to the 5th term. | 23 | 41 | 2 |
math | 13.356. The distance between points $A$ and $B$ is 308 m. A point moves from point $A$ towards $B$, covering 15 m in the first second and 1 m less in each subsequent second. Another point moves from point $B$ in the opposite direction, covering 20 m in the first second and 3 m more in each subsequent second. At what di... | 105 | 126 | 3 |
math | 3.22. Calculate
$$
\int_{L} \frac{\sin 3 z}{(6 z-\pi)^{3}} d z
$$
where $L$ is the unit circle centered at the origin. | -\frac{i\pi}{24} | 49 | 9 |
math | 9. (16 points) Let $n$ be a positive odd number, and the complex number $z$ satisfies $z^{2^{n}-1}=1$. Find $Z_{n}=\prod_{k=0}^{n-1}\left(z^{2^{k}}+\frac{1}{z^{2^{k}}}-1\right)$.
| 1 | 78 | 1 |
math | 10. If the function $f(x)=-\frac{1}{2} x^{2}+\frac{13}{2}$ has a minimum value of $2a$ and a maximum value of $2b$ on the interval $[a, b]$, find $[a, b]$. | [1,3]or[-2-\sqrt{17},\frac{13}{4}] | 66 | 22 |
math | Determine the number of pairs of real numbers, $(x, y)$, with $0 \leq x \leq \frac{\pi}{8}$ and $0 \leq y \leq \frac{\pi}{8}$ and $\cos ^{6}(1000 x)-\sin ^{6}(1000 y)=1$. | 15876 | 77 | 5 |
math | Example 7. Find the integral $\int \frac{d x}{\sqrt{x^{2}-6 x+3}}$. | \ln|x-3+\sqrt{x^{2}-6x+3}|+C | 27 | 18 |
math | 1. The last digit in the notation of a natural number is 2016 times smaller than the number itself. Find all such numbers. Answer: $4032,8064,12096,16128$. | 4032,8064,12096,16128 | 55 | 21 |
math | Example 22 (Mathematical Problem 1433 from "Mathematical Bulletin") Solve the system of equations in the real numbers:
$$
\left\{\begin{array}{l}
y=x^{3}(3-2 x), \\
z=y^{3}(3-2 y), \\
x=z^{3}(3-2 z) .
\end{array}\right.
$$ | (0,0,0),(1,1,1),(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}) | 84 | 35 |
math | 12. There is a well, at the bottom of which there is a spring that continuously gushes out water, with the same amount of water gushing out every minute. If 4 water pumps are used to pump out the water, it takes 40 minutes to finish; if 5 water pumps are used, it takes 30 minutes to finish. Now, it is required to pump ... | 6 | 104 | 1 |
math | 8.4. We will call a number remarkable if it can be decomposed into the sum of 2023 addends (not necessarily distinct), each of which is a natural composite number. Find the largest integer that is not remarkable. | 8095 | 50 | 4 |
math | Problem 4. The numbers
$$
x=a-2b+2010, y=2b-3c+2010 \text{ and } z=3c-a+2010
$$
are consecutive integers. Determine $x, y$ and $z$. | 2009,2010,2011 | 64 | 14 |
math | ## Task 2 - 030522
In a state-owned enterprise, a certain machine part was produced daily in a quantity of 12 pieces until the end of June. Through the socialist competition, it became possible to produce 2 more pieces daily.
a) How many machine parts of this type are now produced monthly - 26 working days?
b) How m... | 312 | 96 | 3 |
math | Let's determine all functions $f$ defined on the set of positive real numbers that satisfy the equation $f(x)+2 f\left(\frac{1}{x}\right)=3 x+6$ for all $x$ in their domain.
---
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
... | f(x)=\frac{2}{x}-x+2 | 129 | 13 |
math | 14. If the system of equations $\left\{\begin{array}{l}\sin x=m \sin ^{3} y, \\ \cos x=m \cos ^{3} y\end{array}\right.$ has real solutions for $x, y$, then the range of positive real values for $m$ is | [1,2] | 69 | 5 |
math | 10.52 A circle contains 6000 points, no three of which are collinear.
（1）Is it possible to divide this circle into 2000 regions, each containing exactly three points? How can this be done?
(2) If the three points in each region satisfy: the distance between any two points is an integer and does not exceed 9, then using... | 22 | 108 | 2 |
math | 374. Minimum of a function. Find the minimum value of the function
$$
f(x)=(x+a+b)(x+a-b)(x-a+b)(x-a-b)
$$
without using differential calculus.[^22] | -4a^2b^2 | 49 | 8 |
math | 30.9. Find the largest three-digit number such that the number minus the sum of its digits is a perfect square. | 919 | 26 | 3 |
math | Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resultin... | 945 | 209 | 3 |
math | 9. (5 points) A non-zero natural number $a$ satisfies the following two conditions:
(1) $0.2 a=m \times m$;
(2) $0.5 a=n \times n \times n$.
where $m, n$ are natural numbers, then the minimum value of $a$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and o... | 2000 | 99 | 4 |
math | 4. The radii of the excircles of a certain triangle are 2, 3, and 6 cm. Find the radius of the circle inscribed in this triangle. | 1 | 37 | 1 |
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 | 5. In $\triangle A B C$, $A B=6, B C=4$, and the length of the median on side $A C$ is $\sqrt{10}$. Then the value of $\sin ^{6} \frac{A}{2}+\cos ^{6} \frac{A}{2}$ is | \frac{211}{256} | 71 | 11 |
math | Find the $x, y$ integers such that $x^{3}(y+1)+y^{3}(x+1)=19$. | (2,1),(1,2),(-20,-1),(-1,-20) | 30 | 21 |
math | 【Question 3】
The store has a safe, the password is $3854 \square 942$, and the digit in the fifth position from left to right is forgotten, but it is remembered that the password is the product of $5678 \times 6789$. So, what should be filled in $\square$? | 7 | 76 | 1 |
math | 2. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is known that $A B=4, A A_{1}=A D=2$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}, B C$ respectively. Then the volume of the tetrahedron $B_{1}-E F G$ is $\qquad$ | 3 | 104 | 1 |
math | 2.31. The side of the base of a regular triangular pyramid is equal to $a$. A plane is drawn through one of the sides of the base, perpendicular to the opposite lateral edge and dividing this edge in the ratio $\boldsymbol{m}: \boldsymbol{n}$, counting from the vertex of the base. Determine the total surface area of th... | \frac{^{2}\sqrt{3}}{4}(1+\sqrt{\frac{3(+2n)}{}}) | 76 | 27 |
math | If an urn is divided into two parts and one part contains $v_{1}$ red and $f_{1}$ white balls, and the other part contains $v_{2}$ red and $f_{2}$ white balls, what is the probability of drawing a red ball blindly? What would the probability be if there were no partition in the urn, and under what conditions would the ... | v_{1}+f_{1}=v_{2}+f_{2} | 83 | 18 |
math | 378. Two bodies simultaneously leave one point: one with a speed of $v_{1}=5 t \mathrm{~m} / \mathrm{c}$, the other with a speed of $v_{2}=3 t^{2} \mathrm{~m} / \mathrm{c}$. At what distance from each other will they be after $20 \mathrm{c}$, if they move in a straight line in the same direction? | 7000(\mathrm{}) | 96 | 7 |
math | 3.3. ** Given a finite sequence $\left\{a_{n}\right\}$ with $2 k$ terms (integer $k \geqslant 2$), the first term $a_{1}=2$. Let the sum of the first $n$ terms be $S_{n}$, and $a_{n+1}=(a-1) S_{n}+2(n=1,2, \cdots, 2 k-1)$, where the constant $a>1$.
(1) Prove that the sequence $\left\{a_{n}\right\}$ is a geometric seque... | 2,3,4,5,6,7 | 353 | 11 |
math | 7. Let the three sides of $\triangle A B C$ be $a, b, c$, and $b>\max \{a, c\}$. There are three non-zero real numbers $x_{0}, y_{0}, z_{0}$ such that the line $a x+b y+c=0$ passes through the point $\left(\frac{z_{0}}{x_{0}}, \frac{2 y_{0}}{x_{0}}\right)$, and the point $\left(\frac{z_{0}}{y_{0}}, \frac{x_{0}}{y_{0}}\... | \frac{5}{3} | 174 | 7 |
math | Three, a football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. According to the competition rules, each pair of teams plays at most one match. Additionally, the two teams from the same city do not play against each other. After a period of matches, it was found that, excep... | 15 | 100 | 2 |
math | 9. Given $z \in \mathbf{C}$. If the equation in terms of $x$
$$
4 x^{2}-8 z x+4 \mathrm{i}+3=0
$$
has real roots. Then the minimum value of $|z|$ is $\qquad$ | 1 | 64 | 1 |
math | 6.42 Given the parabola $y=a x^{2}+b x+c$ has a line of symmetry at $x=-2$, it is tangent to a certain line at one point, this line has a slope of 2, and a y-intercept of 1, and the parabola intersects the $y=0$ at two points, the distance between which is $2 \sqrt{2}$. Try to find the equation of this parabola. | y=x^{2}+4 x+2 \text{ or } y=\frac{1}{2} x^{2}+2 x+1 | 101 | 32 |
math | XXVII OM - I - Zadanie 4
Samolot leci bez zatrzymywania się po najkrótszej drodze z Oslo do miasta $ X $ leżącego na równiku w Ameryce Południowej. Z Oslo startuje dokładnie w kierunku zachodnim. Wiedząc, że współrzędne geograficzne Oslo są : $ 59^{\circ}55 szerokości północnej i $ 10^{\circ}43 długości wschodniej, ob... | Quito,10000 | 212 | 8 |
math | Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n!)=f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$ for all distinct positive integers $m, n$. | f\equiv1,f\equiv2,f(n)=n | 63 | 12 |
math | Example. Expand the function $y=x+1$ into a trigonometric Fourier series in complex form on the interval $(-2,2)$. | x+1=1+\frac{2i}{\pi}\sum_{\substack{n=-\infty\\(n\neq0)}}^{+\infty}\frac{(-1)^{n}}{n}e^{in\pix/2},x\in(-2,2) | 31 | 64 |
math | Let's find such numbers that when multiplied by 12345679, we get numbers consisting of 9 identical digits. | 9,18,27,36,45,54,63,72,81 | 29 | 25 |
math | Example 6 (2008 National High School Mathematics Joint Competition, First Test) Let $f(x)$ be a function defined on $\mathbf{R}$. If $f(0)=2008$, and for any $x \in \mathbf{R}$, it satisfies $f(x+2)-f(x) \leqslant 3 \cdot 2^{x}, f(x+6)-f(x) \geqslant 63 \cdot 2^{x}$, then $f(2008)=$ $\qquad$ . | 2^{2008}+2007 | 124 | 12 |
math | 3. (7p) Consider the sequence of real numbers $\left(x_{n}\right)_{n \circledast 1}$ with $x_{1}=1, x_{2}=2, x_{3}=3$ and $x_{n+3}=x_{n}, \forall n \in \mathbb{N}^{*}$. Calculate $\lim _{n \rightarrow \infty} \frac{x_{1}+x_{2}+\ldots+x_{n}}{n}$. | 2 | 110 | 1 |
math | 1. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E$, $F$, and $G$ are the midpoints of edges $B C$, $C C_{1}$, and $C D$ respectively. Then the angle formed by line $A_{1} G$ and plane $D E F$ is $\qquad$ | 90^{\circ} | 86 | 6 |
math | It is known that $x+2 y+3 z=1$. What is the minimum value that the expression $x^{2}+y^{2}+z^{2}$ can take? | \frac{1}{14} | 41 | 8 |
math | Example 2 Given the function $f(x)=\frac{m-2 \sin x}{\cos x}$ is monotonically decreasing on the interval $\left(0, \frac{\pi}{2}\right)$, try to find the range of the real number $m$. | (-\infty,2] | 59 | 7 |
math | Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now? | 45 | 59 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(6-\frac{5}{\cos x}\right)^{\operatorname{ctg}^{2} x}$ | e^{-\frac{5}{2}} | 47 | 9 |
math | 7. It is known that the four-digit number $\overline{A B C D}$ satisfies the following property: $\overline{A B} 、 \overline{B C} 、 \overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, for example, $4=2^{2}, 81=9^{2}$, so we call $4 、 81$ perfect squar... | 13462 | 121 | 5 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 3} \frac{\log _{3} x-1}{\tan \pi x}$ | \frac{1}{3\pi\ln3} | 39 | 12 |
math | [ Pairing and grouping; bijections ] [ Examples and counterexamples. Constructions ] Tolya laid out 101 coins of 1, 2, and 3 kopecks in a row. It turned out that between any two 1-kopeck coins there is at least one coin, between any two 2-kopeck coins there are at least two coins, and between any two 3-kopeck coins th... | 25or26 | 113 | 5 |
math | 3. 16 Evaluate
(1) $\cos \frac{2 \pi}{n}+\cos \frac{4 \pi}{n}+\cdots+\cos \frac{2(n-1) \pi}{n}$;
(2) $\sin \frac{2 \pi}{n}+\sin \frac{4 \pi}{n}+\cdots+\sin \frac{2(n-1) \pi}{n}$. | \cos\frac{2\pi}{n}+\cos\frac{4\pi}{n}+\cdots+\cos\frac{2(n-1)\pi}{n}=-1,\sin\frac{2\pi}{n}+\sin\frac{4\pi}{n}+\cdots+\sin\frac{2(n-1)\pi}{n}=0 | 95 | 81 |
math | 41st Putnam 1980 Problem A6 Let R be the reals and C the set of all functions f : [0, 1] → R with a continuous derivative and satisfying f(0) = 0, f(1) = 1. Find inf C ∫ 0 1 | f '(x) - f(x) | dx. | \frac{1}{e} | 79 | 7 |
math | Let's find a point that is at distances $m, n, p$ from three planes. How many solutions does this problem have? | 8 | 28 | 1 |
math | For some positive integer $n$, a coin will be flipped $n$ times to obtain a sequence of $n$ heads and tails. For each flip of the coin, there is probability $p$ of obtaining a head and probability $1-p$ of obtaining a tail, where $0<p<1$ is a rational number.
Kim writes all $2^{n}$ possible sequences of $n$ heads and ... | (n,\frac{1}{2}) | 146 | 8 |
math | Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$. | 20 | 28 | 2 |
math | Example 4 Find all positive integers $n$ that satisfy the following condition: if $a \in \mathbf{Z}, (a, n)=1$, then $n \mid \left(a^{2}-1\right)$.
| 1,2,3,4,6,8,12,24 | 50 | 17 |
math | Problem 2. The teacher wrote two numbers on the board. Petya divided the first number by the second. Vasya added both numbers and divided the resulting sum by the doubled first number. It turned out that Petya and Vasya got the same result, which is not equal to 1. What is this result? | -\frac{1}{2} | 70 | 7 |
math | Solve the following equations:
$$
\begin{aligned}
& \frac{1.618+\frac{1}{x}}{1.618-\frac{1}{x}}=1.618+\frac{1}{1.618} \\
& \frac{0.618+\frac{1}{x}}{0.618-\frac{1}{x}}=0.618+\frac{1}{0.618}
\end{aligned}
$$
For 2nd graders, substitute the values 1.618 and 0.618 with $(\sqrt{5}+1) / 2$ and $(\sqrt{5}-1) / 2$, respectiv... | x_{1}=\frac{\sqrt{5}+1}{2},\quadx_{2}=\sqrt{5}+2 | 165 | 29 |
math | 1.4.4 * Given the function $f(x)=\log \frac{1}{2}\left(\frac{\sqrt{a^{2}+1}-a}{a}\right)^{x}$ is a decreasing function on $\mathbf{R}$. Find the range of real number $a$. | 0<<\frac{\sqrt{3}}{3} | 65 | 12 |
math | (solved by Mathieu Aria, Jeanne Nguyen and Thomas Williams). Let $n \geqslant 3$ and $x_{1}, \ldots, x_{n-1}$ be non-negative integers. We assume:
$$
\begin{aligned}
x_{1}+x_{2}+\cdots+x_{n-1} & =n \\
x_{1}+2 x_{2}+\cdots+(n-1) x_{n-1} & =2 n-2
\end{aligned}
$$
Calculate the minimum value of:
$$
\sum_{k=1}^{n-1} k(2... | 3n^{2}-3n | 148 | 7 |
math | Determine all trios of integers $(x, y, z)$ which are solution of system of equations
$\begin{cases} x - yz = 1 \\ xz + y = 2 \end{cases}$
| (x, y, z) \in \{(1, 0, 2), (1, 2, 0)\} | 48 | 28 |
math | Example 6 Given that for all real numbers $x$, the inequality $\left[\left(\log _{3} m\right)^{2}-\log _{3}\left(27 m^{2}\right)\right] x^{2}-\left(\log _{3} m-3\right) x-1<0$ always holds, find the range of real numbers $m$. | (3^{-\frac{1}{5}},27] | 85 | 13 |
math | Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$
Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$-digit integer $ELMO?$ | 1880 | 67 | 4 |
math | 11. In an isosceles right $\triangle A B C$, it is known that $\angle A B C=90^{\circ}$, and the coordinates of points $A, B$ are $A(1,0), B(3,1)$, then the coordinates of vertex $C$ are $\qquad$ . | (2,3)or(4,-1) | 72 | 11 |
math | # 2. Option 1
There are candies in three boxes. It is known that there are 2 times fewer candies in the first box than in the second. It is also known that there are a total of 24 candies in the first and third boxes, and a total of 34 candies in the second and third boxes. How many candies are there in total in the b... | 44 | 82 | 2 |
math | ## problem statement
Based on the definition of the derivative, find $f^{\prime}(0)$ :
$$
f(x)=\left\{\begin{array}{c}
\sqrt[3]{1-2 x^{3} \sin \frac{5}{x}}-1+x, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 1 | 83 | 1 |
math | 1. Find the sum of all three-digit numbers that can be formed from the digits $1,2,3$ and 4. | 17760 | 28 | 5 |
math | ## Task 10/65
A gentleman cashes a check at the bank and does not check the amount paid out. In a store, he pays a bill of 26.66 MDN from this money. To his surprise, the remaining amount is twice as large as the amount the check was issued for. The gentleman therefore goes back to the bank, where it turns out that th... | 26.80 | 101 | 5 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{7 n^{2}+18 n-15}{7 n^{2}+11 n+15}\right)^{n+2}$ | e | 60 | 1 |
math | 8.1. Experienced sawyer Garik knows how to make cuts. In one day of continuous work, he can saw 600 nine-meter logs into identical three-meter logs (they differ from the original only in length). How much time will the experienced sawyer Garik need to saw 400 twelve-meter logs (they differ from the nine-meter logs only... | 1 | 86 | 1 |
math | 3. Let $\mathbb{R}_{+}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{+}^{3} \rightarrow \mathbb{R}_{+}$ such that for all positive real numbers $x, y, z$ and $k$ the following three conditions hold:
(i) $x f(x, y, z)=z f(z, y, x)$;
(ii) $f\left(x, k y, k^{2} z\right)=k f(x, y, z)$;
(iii) $... | \frac{y+\sqrt{y^{2}+4xz}}{2x} | 141 | 19 |
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