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 | A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle. | 2 | 85 | 1 |
math | 2. Let $a, b, c, x, y, z$ all be positive numbers, and $a^{2}+b^{2}+c^{2}=25, x^{2}+y^{2}+z^{2}=36, a x+b y+c z=$ 30 , then $\frac{a+b+c}{x+y+z}=$ $\qquad$ . | \frac{5}{6} | 86 | 7 |
math | 3 [Arithmetic. Mental arithmetic, etc.]
Having walked $4 / 9$ of the length of the bridge, the pedestrian noticed that a car was catching up to him, which had not yet entered the bridge. Then he turned back and met it at the beginning of the bridge. If he had continued his movement, the car would have caught up with h... | 9 | 95 | 1 |
math | 4. Determine all natural numbers $n$ for which:
$$
[\sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+[\sqrt[3]{n}]=2 n
$$ | 33 | 45 | 2 |
math | For a given positive integer $m$, the series
$$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$
evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$. | 7 | 72 | 1 |
math | 12. In 2019, the Mathematics Olympiad underwent a trial reform: A city held 5 joint competitions in the second year of high school. A student who ranks in the top 20 of the city in 2 out of these 5 competitions can enter the provincial team training and does not need to participate in the remaining competitions. Each s... | \frac{269}{64} | 219 | 10 |
math | Example 6 Given the inequality
$$\sqrt{2}(2 a+3) \cos \left(\theta-\frac{\pi}{4}\right)+\frac{6}{\sin \theta+\cos \theta}-2 \sin 2 \theta<3 a+6$$
for $\theta \in\left[0, \frac{\pi}{2}\right]$ to always hold, find the range of values for $a$. | a > 3 | 93 | 4 |
math | Which is the largest positive integer that is 19 times larger than the sum of its digits? | 399 | 20 | 3 |
math | In a school, there are $m$ teachers and $n$ students. We assume that each teacher has exactly $k$ students, and that any two students always have exactly $\ell$ teachers in common. Determine a relation between $m, n, k, \ell$. | n(n-1)\ell=k(k-1) | 57 | 11 |
math | Task B-1.3. Determine the four-digit number which is 594 greater than the number obtained by swapping the two-digit beginning and the two-digit end (moving the first two digits to the end). The difference of the squares of the two-digit beginning and the two-digit end of the given number is 204. | 2014 | 69 | 4 |
math | Example 4 Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$, and for any $x \in \mathbf{R}$, $f(x+5) \geqslant f(x)+5$, $f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, find the value of $g(2002)$. | 1 | 98 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{2+4+\ldots+2 n}{n+3}-n\right)$ | -2 | 45 | 2 |
math | 4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $60^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio... | \sqrt{3}:1 | 108 | 6 |
math | Let $a, b, c, d \in \mathbb{R}$ such that $a+3 b+5 c+7 d=14$. Find the minimum possible value of $a^{2}+b^{2}+c^{2}+d^{2}$. | \frac{7}{3} | 61 | 7 |
math | 384. $\frac{1-\cos ^{2} x}{1-\sin ^{2} x}+\operatorname{tg} x \operatorname{ctg} x$.
384. $\frac{1-\cos ^{2} x}{1-\sin ^{2} x}+\tan x \cot x$. | \frac{1}{\cos^{2}x} | 74 | 12 |
math | Example 5. Find the derivative of the function $f(x)=\frac{x^{2}-2}{x^{2}+2}$. | \frac{8x}{(x^{2}+2)^{2}} | 30 | 17 |
math | ## Task B-3.3.
Given vectors $\vec{u}=\vec{m}+3 \vec{n}, \vec{v}=7 \vec{m}-5 \vec{n}, \vec{w}=\vec{m}-4 \vec{n} \text{ and } \vec{z}=7 \vec{m}-2 \vec{n}$, where $\vec{m}, \vec{n} \neq \overrightarrow{0}$. If vectors $\vec{u} \text{ and } \vec{v}$, and $\vec{w} \text{ and } \vec{z}$ are perpendicular, determine the ang... | 60 | 156 | 2 |
math | ## Task A-2.2.
Determine all real numbers $x$ for which
$$
\left\lfloor\frac{x^{2}+1}{x+2}\right\rfloor+\left\lfloor\frac{x-1}{2}\right\rfloor=\frac{x(3 x+1)}{2(x+2)}
$$
For a real number $t, \lfloor t\rfloor$ is the greatest integer not greater than $t$.
For example, if $t=3.14$, then $\lfloor t\rfloor=3$. | x\in{-1,-3,3,-7} | 126 | 12 |
math | (7) Given that $C$ and $F$ are two points on line segment $AB$, $AB=12$, $AC=6$, $D$ is any point on the circle with center $A$ and radius $AC$, the perpendicular bisector of line segment $FD$ intersects line $AD$ at point $P$. If the locus of point $P$ is a hyperbola, then the range of the eccentricity of this hyperbo... | (1,2] | 102 | 5 |
math | 3.28 Try to express $\sum_{k=0}^{n} \frac{(-1)^{k} C_{n}^{k}}{k^{3}+9 k^{2}+26 k+24}$ in the form $\frac{p(n)}{q(n)}$, where $p(n)$ and $q(n)$ are two polynomials with integer coefficients. | \frac{1}{2(n+3)(n+4)} | 84 | 14 |
math | Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$ | 2011 | 41 | 4 |
math | 4. Elisa makes so-called fake dice. On each side of a fake die, one of the numbers 1 through 6 is written, but not every number has to appear, and some numbers may appear more often. However, from all sides, it must look like a real die. This means: at each corner, three different numbers come together, and no two of t... | 19 | 192 | 2 |
math | Problem 2. Determine the perimeter of the scalene triangle $\mathrm{ABC}$ knowing that
$$
p^{2} \cdot \overrightarrow{G I}=(4 p-b-c) \cdot \overrightarrow{A I}+(4 p-c-a) \cdot \overrightarrow{B I}+(4 p-a-b) \cdot \overrightarrow{C I}
$$
where $\mathrm{G}$ is the centroid of $\triangle \mathrm{ABC}$, I is the incenter... | 12 | 132 | 2 |
math | 353. Solve the systems of equations:
a) $\left\{\begin{array}{l}2 x=(y+z)^{2}, \\ 2 y=(z+x)^{2}, \\ 2 z=(x+y)^{2}\end{array}\right.$,
b) $\left\{\begin{array}{l}x^{2}-x y-x z+z^{2}=0, \\ x^{2}-x z-y z+3 y^{2}=2, \\ y^{2}+x y+y z-z^{2}=2 .\end{array}\right.$ | )(0;0;0),(1/2;1/2;1/2),b)(1;1;1),(-1;-1;-1) | 126 | 34 |
math | C2. Charlotte writes a test consisting of 100 questions, where the answer to each question is either TRUE or FALSE. Charlotte's teacher announces that for every five consecutive questions on the test, the answers to exactly three of them are TRUE. Just before the test starts, the teacher whispers to Charlotte that the ... | 60 | 123 | 2 |
math | A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant... | 8 | 197 | 1 |
math | Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying
\[
\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
\] | 18 | 72 | 2 |
math | 1000. A certain natural number has only two prime divisors (in some powers), and its square has 35 different divisors. How many different divisors does the cube of this number have? | 70 | 44 | 2 |
math | 3. (2 points) A poor student wrote the following incorrect formulas for the sine and cosine of the difference: $\sin (\alpha-\beta)=$ $\sin \alpha-\sin \beta$ and $\cos (\alpha-\beta)=\cos \alpha-\cos \beta$. In his defense, he said that for some $\alpha$ and $\beta$ his formulas are still correct. Find all such pairs ... | \alpha=2\pin,n\in\mathbb{Z} | 91 | 15 |
math | Example 1 (24th All-Soviet Union Mathematical Olympiad, 1990) Determine for which natural numbers $n$, the number $3^{2 n+1}-2^{2 n+1}-6^{n}$ is composite.
Determine for which natural numbers $n$, the number $3^{2 n+1}-2^{2 n+1}-6^{n}$ is composite. | n\geqslant2 | 87 | 7 |
math | 3.160. $\frac{5}{6+7 \sin 2 \alpha}$, if $\operatorname{tg} \alpha=0.2$. | \frac{65}{113} | 36 | 10 |
math | 8. Given $a b=1$, and $\frac{1}{1-2^{x} a}+\frac{1}{1-2^{y+1} b}=1$, then the value of $x+y$ is $\qquad$. | -1 | 53 | 2 |
math | 4. A sequence of integers $a_{1}, a_{2}, a_{3}, \ldots$ is defined by
$$
\begin{array}{c}
a_{1}=k, \\
a_{n+1}=a_{n}+8 n \text { for all integers } n \geq 1 .
\end{array}
$$
Find all values of $k$ such that every term in the sequence is a square. | 1 | 95 | 1 |
math | 18. The number $\frac{20!\times 22!}{16!\times 11!}$ has $N$ prime factors, which are not necessarily distinct.
What is the value of $N(N-2)$ ? | 960 | 51 | 3 |
math | 10.008. In an isosceles triangle with a lateral side equal to 4 cm, a median of the lateral side is drawn. Find the base of the triangle if the median is 3 cm. | \sqrt{10} | 47 | 6 |
math | Example 6. Find $\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$. | 0 | 35 | 1 |
math | 5. Let $\pi \leqslant x<2 \pi$. Define
$$
\begin{aligned}
P= & \frac{1}{2} \cos x-\frac{1}{4} \sin 2 x-\frac{1}{8} \cos 3 x+ \\
& \frac{1}{16} \sin 4 x+\frac{1}{32} \cos 5 x-\frac{1}{64} \sin 6 x- \\
& \frac{1}{128} \cos 7 x+\cdots, \\
Q= & 1-\frac{1}{2} \sin x-\frac{1}{4} \cos 2 x+\frac{1}{8} \sin 3 x+ \\
& \frac{1}{1... | -\frac{17}{19} | 265 | 9 |
math | ## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=p+3q$
$b=p-2q$
$|p|=2$
$|q|=3$
$(\widehat{p, q})=\frac{\pi}{3}$ | 15\sqrt{3} | 63 | 7 |
math | Solve the following equation over the set of integer pairs:
$$
(x+2)^{4}-x^{4}=y^{3} \text {. }
$$ | -1,0 | 34 | 4 |
math | Example 3 Find all integer pairs $(m, n)(m, n \geqslant 2)$, such that for any integer $x$ we have
$$
x^{n} \equiv x(\bmod m)
$$ | m=p_{1} p_{2} \cdots p_{k}, n=1+u\left[p_{1}-1, p_{2}-1, \cdots, p_{k}-1\right] | 50 | 47 |
math | A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$. | 2^m - 1 | 95 | 6 |
math | 6. (20 points) A one-kilogram model of a sports car body was made from aluminum at a scale of 1:10. What is the mass of the actual body if it is also entirely made of aluminum? | 1000 | 49 | 4 |
math | 2.264. $\frac{x^{3}+5 x^{2}+3 x-9}{x^{3}+x^{2}-5 x+3}$.
2.264. $\frac{x^{3}+5 x^{2}+3 x-9}{x^{3}+x^{2}-5 x+3}$. | \frac{x+3}{x-1} | 78 | 10 |
math | Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a strai... | (m-1)(n-1) = 0 | 135 | 13 |
math | ## Task 3 - 030713
How can one determine without performing the specified arithmetic operations whether the number
$$
\frac{378 \cdot 436-56}{378+436 \cdot 377}
$$
is greater or less than 1? | \frac{377\cdot436+380}{377\cdot436+378}>1 | 69 | 30 |
math | Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits. | \lim_{n \to \infty} x_n = 0 | 127 | 16 |
math | For non-negative integers $x$, the function $f(x)$ is defined as follows:
$$
f(0)=0, f(x)=f\left(\left[\frac{x}{10}\right]\right)+\left[\lg \frac{10}{x-10\left[\frac{x-1}{10}\right]}\right] .
$$
What is the value of $x$ when $f(x)$ reaches its maximum in the range $0 \leqslant x \leqslant 2006$? | 1111 | 117 | 4 |
math | ## Task $1 / 86$
We are looking for the smallest natural number $n$, which is the product of 3 prime factors $p_{1} ; p_{2} ; p_{3}$, and it holds that: $p_{3}=55 \cdot p_{1} \cdot p_{2}+1$ and $p_{3}>p_{2}>p_{1}$. | 1986 | 87 | 4 |
math | Example 8 Find the minimum value of $\frac{a}{\sin \theta}+\frac{b}{\cos \theta}\left(a, b>0, \theta \in\left(0, \frac{\pi}{2}\right)\right)$. | (\sqrt[3]{^{2}}+\sqrt[3]{b^{2}})^{4} | 56 | 21 |
math | 8. Let the area of $\triangle A B C$ be $1, D$ be a point on side $B C$, and $\frac{B D}{D C}=\frac{1}{2}$. If a point $E$ is taken on side $A C$ such that the area of quadrilateral $A B D E$ is $\frac{4}{5}$, then the value of $\frac{A E}{E C}$ is $\qquad$. | \frac{7}{3} | 99 | 7 |
math | 4. For the set $\{00,01, \cdots, 98,99\}$, a subset $X$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain? | 55 | 67 | 2 |
math | 13. Let $x_{1}, x_{2}, x_{3}$ be the roots of the equation $x^{3}-17 x-18=0$, with $-4<x_{1}<-3$, and $4<x_{3}<5$.
(1) Find the integer part of $x_{2}$;
(2) Find the value of $\arctan x_{1}+\arctan x_{2}+\arctan x_{3}$. | -\frac{\pi}{4} | 104 | 7 |
math | 4- 47 Let $a, b$ be real numbers, and $x^{4}+a x^{3}+b x^{2}+a x+1=0$ has at least one real root. Try to find the minimum value that $a^{2}+b^{2}$ can take. | \frac{4}{5} | 68 | 7 |
math | 4. If the complex coefficient equation $(4+3 \mathrm{i}) x^{2}+m x+$ $4-3 \mathrm{i}=0$ has real roots, then the minimum value of the modulus of the complex number $m$ is $\qquad$ | 8 | 56 | 1 |
math | ## Task Condition
Write the canonical equations of the line.
$x-3 y+2 z+2=0$
$x+3 y+z+14=0$ | \frac{x+8}{-9}=\frac{y+2}{1}=\frac{z}{6} | 35 | 25 |
math | 12. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, $F_{1} 、 F_{2}$ are its left and right foci, $Q$ is any point on the ellipse $C$, and the centroid and incenter of $\triangle F_{1} Q F_{2}$ are $G 、 I$, respectively. The line $I G$ is parallel to the $x$-axis. Then the eccentricity of the el... | \frac{1}{2} | 125 | 7 |
math | 3. Given $x_{1}, x_{2}, \cdots, x_{100}$ are non-negative real numbers, and for $i=1,2, \cdots, 100$, we have $x_{i}+x_{i+1}+x_{i+2} \leqslant 1$, where $x_{101}=x_{1}, x_{102}=x_{2}$. Find the maximum value of the sum $S=\sum_{i=1}^{100} x_{i} x_{i+2}$. | \frac{25}{2} | 130 | 8 |
math | Task B-1.2. If $\left[\left(\frac{a}{b}-\frac{b}{a}\right):\left((a+b)\left(\frac{1}{b}-\frac{1}{a}\right)\right)-\frac{a}{b}\right]:\left(\frac{a}{b}+1\right)=\frac{1}{2014}$, what is $\frac{a}{b}$? | \frac{2013}{2015} | 96 | 13 |
math | Task 1. Find all pairs $(x, y)$ of integers that satisfy
$$
x^{2}+y^{2}+3^{3}=456 \sqrt{x-y} .
$$ | (30,21),(-21,-30) | 43 | 14 |
math | Problem 2. The probability of hitting the ring with a ball in one throw is 0.7. Find the probability that in five throws there will be hits in the ring: a) no more than three; b) no less than four. | 0.5282 | 51 | 6 |
math | One, (40 points) Find the smallest real number $\lambda$, such that there exists a sequence $\left\{a_{n}\right\}$ with all terms greater than 1, for which $\prod_{i=1}^{n+1} a_{i}<a_{n}^{\lambda}$ holds for any positive integer $n$. | 4 | 74 | 1 |
math | 3B. Find all values of the real number $k$ for which the expression
$$
\sin ^{6} x+\cos ^{6} x+k\left(\sin ^{4} x+\cos ^{4} x\right)
$$
does not depend on $x$. | -\frac{3}{2} | 63 | 7 |
math | 39.2. Solve the system of equations in integers:
$$
\left\{\begin{array}{l}
x+y+z=6 \\
x+y z=7
\end{array}\right.
$$
$$
\text { (8-10 grades) }
$$ | \begin{aligned}&x_{1}=7,\quady_{1}=0,\quadz_{1}=-1;\\&x_{2}=7,\quady_{2}=-1,\quadz_{2}=0\text{;}\\&x_{3}=1,\quady_{3}=3,\quadz_{3}=2\\&x_{4}=1,\quady_{4}=2, | 60 | 88 |
math | 7th Irish 1994 Problem B2 p, q, r are distinct reals such that q = p(4-p), r = q(4-q), p = r(4-r). Find all possible values of p+q+r. | 6,7 | 53 | 3 |
math | Consider a set with $n$ elements.
How many subsets of odd cardinality exist? | 2^{n-1} | 18 | 6 |
math | 5. If the arithmetic sequence $\left\{a_{n}\right\}$ and the positive integer $m(m \geqslant 3)$ satisfy: $a_{1}=1, a_{m}=2$, and $\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{m-1} a_{m}}=3$, then $a_{1}+a_{2}+\cdots+a_{m}=$ $\qquad$ . | \frac{21}{2} | 119 | 8 |
math | 44. Solve the inequality $4 \log _{16} \cos 2 x+2 \log _{4} \sin x+\log _{2} \cos x+$ $+3<0$. | 0<x<\frac{\pi}{24} | 47 | 11 |
math | One, (40 points) Find the smallest integer $c$, such that there exists a sequence of positive integers $\left\{a_{n}\right\}(n \geqslant 1)$ satisfying:
$$
a_{1}+a_{2}+\cdots+a_{n+1}<c a_{n}
$$
for all $n \geqslant 1$. | 4 | 84 | 1 |
math | $$\begin{array}{ll}
8 \cdot 3 & \text { Let the sequence } x_{1}, x_{2}, x_{3}, \cdots \text { satisfy } \\
& 3 x_{n}-x_{n-1}=n, n=2,3, \cdots \\
\text { and } & \left|x_{1}\right|<1971 .
\end{array}$$
Find \( x_{1971} \), accurate to 0.000001. | 985.250000 | 117 | 10 |
math | Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size he... | 13 | 134 | 4 |
math | 7 Find the coefficient of $x^{2}$ in the expansion of $(1+x)(1+2 x)(1+4 x) \cdots\left(1+2^{n-1} \cdot x\right)$. | \frac{1}{3}(2^{n}-1)(2^{n}-2) | 49 | 19 |
math | 6. An old clock is 12 seconds behind in 14 days. What time will it show on April 9, 2019, at 10:00 AM if it was set to the correct time on January 1, 2019, at 10:00 AM? (2019 is not a leap year.) | 9:58:36 | 80 | 7 |
math | 143*. Find the condition under which the difference of two irreducible fractions is equal to their product. | \frac{}{b},\frac{}{+b} | 22 | 13 |
math | (IMO SL 2020 A3) Let $a, b, c, d$ be strictly positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Determine the smallest value that
$$
\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}
$$
can take. | 8 | 80 | 1 |
math | 1. Given the sets $A=\{x, x y, x+y\}, B=\{0,|x|, y\}$ and $A=B$, then $x^{2018}+y^{2018}=$ | 2 | 53 | 1 |
math | Determine all integers $a, b, c$ satisfying the identities:
$$
\begin{gathered}
a+b+c=15 \\
(a-3)^{3}+(b-5)^{3}+(c-7)^{3}=540
\end{gathered}
$$ | (a-3)(b-5)(c-7)=180 | 64 | 15 |
math | Example 3. Calculate the mass of the surface $z=x y$, located inside the cylinder $x^{2}+\frac{y^{2}}{4}=1$, if the density is $\rho=\frac{|z|}{\sqrt{1+x^{2}+y^{2}}}$. | 2 | 62 | 1 |
math | Tom is chasing Jerry on the coordinate plane. Tom starts at $(x, y)$ and Jerry starts at $(0, 0)$. Jerry moves to the right at $1$ unit per second. At each positive integer time $t$, if Tom is within $1$ unit of Jerry, he hops to Jerry’s location and catches him. Otherwise, Tom hops to the midpoint of his and Jerry’s l... | \frac{\sqrt{3}}{3} | 190 | 11 |
math | Let $f\colon \mathbb R ^2 \rightarrow \mathbb R$ be given by $f(x,y)=(x^2-y^2)e^{-x^2-y^2}$.
a) Prove that $f$ attains its minimum and its maximum.
b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$ and determine for which of them $f$ has global or loca... | (1,0) | 114 | 6 |
math | 6.200. $\left\{\begin{array}{l}(x+y)^{2}+2 x=35-2 y, \\ (x-y)^{2}-2 y=3-2 x\end{array}\right.$ | (-5,-2),(3,2),(-3,-4),(1,4) | 54 | 18 |
math | Let's calculate the value of the expression under a) and simplify the expressions under b) and c):
a) $0.027^{-\frac{1}{3}}-\left(-\frac{1}{6}\right)^{-2}+256^{0.75}+0.25^{0}+(-0.5)^{-5}-3^{-1}$.
b) $\left[\frac{\left(a^{\frac{3}{4}}-b^{\frac{3}{4}}\right)\left(a^{\frac{3}{4}}+b^{\frac{3}{4}}\right)}{a^{\frac{1}{2}}-b^... | 0 | 312 | 1 |
math | ## Task Condition
Are the vectors $a, b$ and $c$ coplanar?
$a=\{1 ;-1 ; 4\}$
$b=\{1 ; 0 ; 3\}$
$c=\{1 ;-3 ; 8\}$ | 2 | 56 | 1 |
math | Example 2: In a certain year, the total coal production of a coal mine, apart from a certain amount of coal used for civilian, export, and other non-industrial purposes each year, the rest is reserved for industrial use. According to the standard of industrial coal consumption of a certain industrial city in that year,... | 10 | 137 | 2 |
math | 1. Question: How many real roots does the equation $x^{2}|x|-5 x|x|+2 x=0$ have (where $|x|$ represents the absolute value of $x$)? | 4 | 44 | 1 |
math | Problem 1. A barrel is filled with water to $\frac{5}{6}$ of its volume. If another 10 liters of water are added to the barrel, it will be filled to $\frac{7}{8}$ of its volume. How many liters of water does the barrel hold? | 240 | 62 | 3 |
math | 5. Let $G$ be the centroid of $\triangle A B C$. If $B G \perp C G$, $B C=\sqrt{2}$, then the maximum value of $A B+A C$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2\sqrt{5} | 76 | 6 |
math | Find all $n>1$ and integers $a_1,a_2,\dots,a_n$ satisfying the following three conditions:
(i) $2<a_1\le a_2\le \cdots\le a_n$
(ii) $a_1,a_2,\dots,a_n$ are divisors of $15^{25}+1$.
(iii) $2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)$ | (4, 4, 15^{25} + 1) | 144 | 18 |
math | 5. Solve the system of equations:
$$
\left\{\begin{array}{l}
x y-2 y=x+106 \\
y z+3 y=z+39 \\
z x+3 x=2 z+438
\end{array}\right.
$$ | (38,4,9)(-34,-2,-15) | 62 | 17 |
math | 8. Given the function $f(x)=\mathrm{e}^{x}\left(x-a \mathrm{e}^{x}\right)$ has exactly two critical points $x_{1} 、 x_{2}\left(x_{1}<x_{2}\right)$. Then the range of values for $a$ is $\qquad$ | \left(0, \frac{1}{2}\right) | 72 | 14 |
math | 8. Let the inequality $x^{4}+(a-2) x^{2}+a \geqslant 0$ hold for all real numbers $x$, the range of real number $a$ is $\qquad$ . | \geqslant4-2\sqrt{3} | 51 | 13 |
math | 32 Let $\mathbf{R}^{+}$ be the set of all real numbers excluding 0, find all functions $f: \mathbf{R}^{*} \rightarrow \mathbf{R}^{*}$ such that $f(x)+f(y)=f(x y f(x+y))$, where $x$, $y \in \mathbf{R}^{*}$ and $x+y \neq 0$. | f(x)=\frac{1}{x} | 93 | 10 |
math | 7. If the distance between the two directrices of an ellipse is twice the distance between the two foci, then its eccentricity $e=$ $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{\sqrt{2}}{2} | 59 | 10 |
math | 8.158. $\operatorname{tg} x \operatorname{tg} 20^{\circ}+\operatorname{tg} 20^{\circ} \operatorname{tg} 40^{\circ}+\operatorname{tg} 40^{\circ} \operatorname{tg} x=1$. | 30+180k,\quadk\in\mathbb{Z} | 75 | 18 |
math | Four, (50 points) Find all positive integers $n$, such that $2^{n}+2n$ is a factorial of a positive integer.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Note: The provided text is already in English, so no translation is n... | 4 | 126 | 1 |
math | Test: Given that $a$ is a natural number, a quadratic trinomial with integer coefficients and $a$ as the leading coefficient has two distinct positive roots less than 1. Find the minimum value of $a$.
---
The above text translated into English, preserving the original text's line breaks and format. | 5 | 65 | 1 |
math | Ten points are given in the plane and no three are collinear. Four distinct segments connecting pairs of these points are chosen at random, but all with the same probability. What is the probability that three of the chosen segments form a triangle? | \frac{16}{473} | 48 | 10 |
math | 3. Real numbers $x, y, z$ satisfy $x^{2}+y^{2}-x+y=1$. Then the range of the function
$$
f(x, y, z)=(x+1) \sin z+(y-1) \cos z
$$
is . $\qquad$ | \left[-\frac{3 \sqrt{2}+\sqrt{6}}{2}, \frac{3 \sqrt{2}+\sqrt{6}}{2}\right] | 67 | 39 |
math | B2. Let $A B C D$ be a square with side length 1 . Points $X$ and $Y$ are on sides $B C$ and $C D$ respectively such that the areas of triangles $A B X, X C Y$, and $Y D A$ are equal. Find the ratio of the area of $\triangle A X Y$ to the area of $\triangle X C Y$. | \sqrt{5} | 87 | 5 |
math | ## Task B-4.1.
Given is the function
$$
f: \mathbf{R} \rightarrow \mathbf{R}, \quad f(x)=\frac{1}{2}\left(a^{x}+a^{-x}\right)
$$
where $a$ is a positive real number different from 1. What is $f(p+t)+f(p-t)$ if $f(t)=20$ and $f(p)=25 ?$ | 1000 | 99 | 4 |
math | Example 2.42. Compute the integral $I=\int_{0}^{1} \frac{x^{m}-x^{n}}{\ln x} d x$, $(m>0, n>0)$. | \ln|\frac{+1}{n+1}| | 47 | 12 |
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