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 | $$
\begin{array}{c}
\text { Example } 1 \text { Calculate } \frac{a}{a^{3}+a^{2} b+a b^{2}+b^{3}}+ \\
\frac{b}{a^{3}-a^{2} b+a b^{2}-b^{3}}+\frac{1}{a^{2}-b^{2}}-\frac{1}{b^{2}+a^{2}}-\frac{a^{2}+3 b^{2}}{a^{4}-b^{4}} .
\end{array}
$$
(1995, Tianjin City, Grade 7 "Mathematics Competition) | 0 | 146 | 1 |
math | 7. Let $m, n, p, l$ be natural numbers. If the following system:
$$
\left\{\begin{array}{l}
n p^{2}+m l^{2} \leq 23456 \\
m n+p^{7} l=1626 \\
m(l+1)^{p}+n p l=2197+m
\end{array}\right.
$$
has solutions, then present the one for which the sum $m+n+p+l$ is minimal. Otherwise, prove that there are no solutions. | nosolutions | 124 | 2 |
math | ## problem statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{\left(x^{3}-2 x-1\right)(x+1)}{x^{4}+4 x^{2}-5}$ | 0 | 53 | 1 |
math | 5. Let $f(x)$ be a polynomial of degree 2014,
such that $f(k)=\frac{1}{k}(k=1,2, \cdots, 2015)$. Then $f(2016)=$ $\qquad$ - . | \frac{1}{1008} | 64 | 10 |
math | 7・10 A fishing boat is fishing in the territorial sea of a foreign country without permission. Each time it casts a net, it causes the same value of loss in the fishing catch of the country. The probability of the boat being detained by the foreign coast patrol during each net casting is $1 / k$, where $k$ is a natural... | k-1 | 194 | 3 |
math | 3. Find all ordered triples ( $m, n, p$ ) of positive rational numbers such that the numbers
$$
m+\frac{1}{n p}, n+\frac{1}{p m}, p+\frac{1}{m n}
$$
are all integers.
(Ru.munija) | (1,1,1),(2,1,\frac{1}{2}),(4,\frac{1}{2},\frac{1}{2}) | 65 | 33 |
math | 10. Pay for an escalator moving upwards. A walks down from its top to its bottom, totaling 150 steps, B walks up from its bottom to its top, totaling 75 steps. Assuming A's speed (number of steps walked per unit time) is 3 times B's speed, how many steps of the escalator are visible at any given moment? (Assume this nu... | 120 | 88 | 3 |
math | ## Task 3 - 050723
Compare the sum of all three-digit natural numbers divisible by 4 with the sum of all three-digit even natural numbers not divisible by 4!
a) Which of the two sums is greater?
b) What is the difference between the two sums in absolute value? | 450 | 66 | 3 |
math | $4 \cdot 206$ There are two forces $f_{1} 、 f_{2}$ acting on the origin $O$ of the coordinate axes,
$$
\begin{array}{l}
\vec{f}_{1}=\overrightarrow{O A}=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right), \\
\overrightarrow{f_{2}}=\overrightarrow{O B}=2\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\ri... | 2.1 | 180 | 3 |
math |
Problem 3. Find the smallest positive integer $n$, for which there exist $n$ different positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the conditions:
a) the smallest common multiple of $a_{1}, a_{2}, \ldots, a_{n}$ is 1995 ;
b) for each $i, j \in\{1,2, \ldots, n\}$ the numbers $a_{i}$ and $a_{j}$ have a ... | 7 | 192 | 1 |
math | How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$? | 2002 | 56 | 4 |
math | 8.165. $2 \tan x - 2 \cot x = 3$. | -\frac{1}{2}\operatorname{arcctg}\frac{3}{4}+\frac{\pin}{2},n\inZ | 21 | 31 |
math | Example \ Let $a, b, c$ be 3 distinct real numbers, and the real-coefficient polynomial $p(x)$, when divided by $x-a$, $x-b$, and $x-c$, yields remainders $a$, $b$, and $c$ respectively. Find the remainder when the polynomial $p(x)$ is divided by $(x-a)(x-b)(x-c)$. | x | 83 | 1 |
math | 6. Find the set of values of the function $f(x)=4 \cos \left(\frac{\pi}{3} \sin \left(x^{2}+6 x+10-\sin x\right)\right)$.
# | [2;4] | 50 | 5 |
math | Find all pairs of primes $p, q<2023$ such that $p \mid q^2+8$ and $q \mid p^2+8$. | (2, 2), (17, 3), (11, 5) | 38 | 20 |
math | 42nd IMO 2001 shortlist Problem N2 Find the largest real k such that if a, b, c, d are positive integers such that a + b = c + d, 2ab = cd and a ≥ b, then a/b ≥ k. Solution | 3+2\sqrt{2} | 59 | 8 |
math | Problem 2. The distance between the roots of the quadratic trinomial $x^{2}+p x+q$ is 1. Find the coefficients $p$ and $q$, given that they are prime numbers.
Answer: $p=3, q=2$. | p=3,q=2 | 58 | 6 |
math | ## Problem 1
Let the sequence $\left(x_{n}\right)_{n \geq 2}$ be such that $x_{2}=1, x_{n+1}=\frac{n^{2}}{n-1} \cdot x_{n}, n \geq 2$.
If $S_{n}=\sum_{k=2}^{n}\left(1-\frac{1}{k}\right) \cdot \frac{1}{x_{k}}$, calculate $\lim _{n \rightarrow \infty} S_{n}$. | e-2 | 122 | 3 |
math | ## 1. Magic Drink
Petar and Tomo are playing a game that uses 3-crown and 5-crown coins. If they land on a golden field, they can buy a magic drink. Petar paid for the drink with 3-crown coins, while Tomo used 5-crown coins. What is the price of the magic drink if together they gave more than 60 but fewer than 70 coin... | 120 | 99 | 3 |
math | 9. Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is $1^{2}+2^{2}+4^{2}=21$ ). | 5035485 | 61 | 7 |
math | ## Task Condition
Find the angle between the planes:
$x-3 y-2 z-8=0$
$x+y-z+3=0$ | \frac{\pi}{2} | 31 | 7 |
math | Example 8 Find the maximum value of $n$ such that there exists an arithmetic sequence $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 3)$ satisfying
$$
\sum_{i=1}^{n}\left|a_{i}\right|=\sum_{i=1}^{n}\left|a_{i}+1\right|=\sum_{i=1}^{n}\left|a_{i}-2\right|=507 .
$$
(2005, China Southeast Mathematical Olympiad) | 26 | 125 | 2 |
math |
2.298. $\frac{\sqrt{21+8 \sqrt{5}}}{4+\sqrt{5}} \cdot \sqrt{9-4 \sqrt{5}}=\sqrt{5}-2$.
| \sqrt{5}-2 | 49 | 6 |
math | 7. Given a positive integer $n(n \geqslant 3)$. For any permutation $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ of $1,2, \cdots, n$, if $i<j<k$, then $x_{j}$ is said to be between $x_{i}$ and $x_{k}$ (for example, in the permutation $(1,3,2,4)$, 3 is between 1 and 4, and 4 is not between 1 and 2). Let the set $S=\left... | 2^{n-1} | 241 | 6 |
math | $$
14 \cdot 32 \text { Let } S=[\sqrt{1}]+[\sqrt{2}]+\cdots+[\sqrt{1988}] \text {. Find }[\sqrt{S}] \text {. }
$$
(Advanced Class for Science Experiment Mathematics Admission Test, 1988) | 241 | 71 | 3 |
math | 4. Solve in the set of real numbers the equation
$$
\frac{x-a_{1}}{a_{2}+\ldots+a_{n}}+\frac{x-a_{2}}{a_{1}+a_{3}+\ldots+a_{n}}+\ldots+\frac{x-a_{n}}{a_{1}+\ldots+a_{n-1}}=\frac{n x}{a_{1}+\ldots+a_{n}}
$$
where $n \geq 2$ and $a_{i}>0, i \in\{1,2, \cdots, n\}$. | a_{1}+a_{2}+\cdots+a_{n} | 133 | 16 |
math | 6. Given three points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)$ on the unit circle $x^{2}+y^{2}=1$ that satisfy
$$
x_{1}+x_{2}+x_{3}=y_{1}+y_{2}+y_{3}=0 \text {. }
$$
then $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=$ | \frac{3}{2} | 150 | 7 |
math | For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\in... | \frac{\pi}{2} | 152 | 8 |
math | B hits a 10-ring target with probabilities of $\frac{1}{5}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \frac{1}{10}$ for 10, 9, 8, 7, 6 points respectively. The probability of other (from 5 to 1) hits is $\frac{1}{12}$. $A$ pays $B$ for every hit of at least 6 points the amount in forints equal to the score. For any other ... | 96 | 154 | 2 |
math | Problem 6.8. A natural number $n$ is called good if 2020 gives a remainder of 22 when divided by $n$. How many good numbers exist? | 10 | 40 | 2 |
math | 6. If the sum of the areas of three square pieces of paper with integer side lengths is 2004, and the area of the largest square piece of paper is $S_{1}$, and the area of the smallest square piece of paper is $S_{2}$, then the maximum value of $\frac{S_{1}}{S_{2}}$ is $\qquad$ | 484 | 82 | 3 |
math | 3. Given $p$ is a prime number, the fractional part of $\sqrt{p}$ is $x$, and the fractional part of $\frac{1}{x}$ is $\frac{\sqrt{p}-31}{75}$. Find all prime numbers $p$ that satisfy the condition. | 2011 | 63 | 4 |
math | Example 4 (2005 Turkish Mathematical Olympiad) Find all functions $f:[0,+\infty) \rightarrow[0,+\infty)$, such that for all $x \in[0,+\infty)$, we have $4 f(x) \geqslant 3 x$, and $f[4 f(x)-3 x]=x$.
| f(x)=x | 81 | 4 |
math | Let the roots of the equation $\operatorname{Az} x^{2}-m x+n=0$ be $\tan \varphi$ and $\tan \phi$. Determine the angles $\varphi$ and $\phi$ without solving the equation. | \operatorname{tg}(\varphi+\phi)=\frac{}{1-n},\operatorname{tg}(\varphi-\phi)=\frac{\sqrt{^{2}-4n}}{1+n} | 51 | 46 |
math | 1. Solve the inequality:
$$
\log _{9 x} 3 x+\log _{3 x^{2}} 9 x^{2}<\frac{5}{2}
$$ | x\in(0,\frac{1}{27\sqrt{3}})\cup(\frac{1}{9},\frac{1}{\sqrt{3}})\cup(1,\infty) | 41 | 44 |
math | Example 5. Find the maximum area of a trapezoid inscribed in an ellipse with the major axis as one base. | \frac{3 \sqrt{3} a b}{4} | 27 | 14 |
math | 1. If the 5th term of the expansion of $\left(x \sqrt{x}-\frac{1}{x}\right)^{6}$ is $\frac{15}{2}$, then $\lim _{n \rightarrow \infty}\left(x^{-1}+x^{-2}+\cdots+x^{-a}\right)=$ | 1 | 73 | 1 |
math | 3. The function defined on the set of positive integers is
$$
f(x)=\left\{\begin{array}{ll}
3 x-1, & x \text { is odd, } \\
\frac{x}{2}, & x \text { is even. }
\end{array}\right.
$$
Let $x_{1}=12, x_{n+1}=f\left(x_{n}\right), n \in \mathbf{N}$, then the number of elements in the set $\left\{x \mid x=x_{n}\right.$, $n ... | 7 | 140 | 1 |
math | (50 points) Given a natural number $n \geqslant 5$. Try to find:
(1). In the $n$-element set $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, how many different numbers are produced by $a_{i}+a_{j}$ $(1<i<j \leqslant n)$ at least;
(2) Determine all $n$-element sets that achieve the above minimum value. | 2n-3 | 107 | 4 |
math | 18. Four different prime numbers $\mathrm{a}, \mathrm{~b}, \mathrm{c}, \mathrm{~d}$ satisfy the following properties:
(1) $a+b+c+d$ is also a prime number; (2) The sum of two of $\mathrm{a}, \mathrm{~b}, \mathrm{c}, \mathrm{~d}$ is also a prime number; (3) The sum of three of $\mathrm{a}, \mathrm{~b}, \mathrm{c}, \math... | 31 | 141 | 2 |
math | 7. (10 points) A and B are running a 10 km race. A completes the race in 50 minutes, at which point B is still 500 meters from the finish line. To give B a chance, they agree that in the second race, A will start 500 meters behind the starting line. Assuming both run at the same speed in both races, when the first pers... | 25 | 111 | 2 |
math | 1. (40 points) Let real numbers $a_{1}, a_{2}, \cdots, a_{n} \in[0,2]$, and define $a_{n+1}=a_{1}$, find the maximum value of $\frac{\sum_{i=1}^{n} a_{1}^{2} a_{i+1}+8 n}{\sum_{i=1}^{n} a_{i}^{2}}$. | 4 | 101 | 1 |
math | 193*. Find all integer values of $x$ and $y$ for which at least one of the numbers
$$
x^{2}-2 x y+2 y^{2}, \quad x^{2}+2 x y+2 y^{2}
$$
is divisible by 5. | allintegersxy,simultaneouslydivisibleornotdivisible5 | 64 | 15 |
math | The second problem of the second round of the Rákosi Mátyás competition read as follows: "On a railway line, a passenger train departs from location $A$ to location $C$. When the train passes through $B$, a freight train departs from $B$ towards $A$. When the freight train arrives at $A$, a fast train departs from $A$ ... | \begin{gathered}40\leqqc_{1}\leqq50\\40\geqqc_{2}\geqq33\frac{1}{3}\\85\frac{5}{7}\leqqc_{3}\leqq150\end{gathered} | 278 | 66 |
math | 1. Let $n$ be a natural number, $a, b$ be positive real numbers, and satisfy $a+b=2$, then the minimum value of $\frac{1}{1+a^{n}}+\frac{1}{1+b^{n}}$ is $\qquad$. | 1 | 60 | 1 |
math | 3.463 Find $\operatorname{ctg} \frac{x}{2}$, if it is known that $\sin x-\cos x=\frac{1+2 \sqrt{2}}{3}$. | \cot\frac{x}{2}=\frac{\sqrt{2}}{2},\cot\frac{x}{2}=3-2\sqrt{2} | 46 | 34 |
math | Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$. | n = 7 | 82 | 5 |
math | 4. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f\left(x^{2}+f(y)\right)=(x-y)^{2} \cdot f(x+y) \text {. }
$$
The second round of category A takes place
## on Tuesday, January 16, 2001
so that it starts in the morning and the contestants have 4 hours ... | f_1(x)=0f_2(x)=-x^2 | 156 | 15 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{\sqrt[3]{x^{2}-16}}$ | 0 | 42 | 1 |
math | 3. $\mathbf{C}$ is the set of complex numbers, let the set $A=\left\{z \mid z^{18}=1, z \in \mathbf{C}\right\}, B=\left\{w \mid w^{48}=1, w \in \mathbf{C}\right\}, D=\{z w \mid z \in A$ ， $w \in B\}$. Find the number of elements in $D$. | 144 | 103 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3 x}{\sqrt{2+x}-\sqrt{2}}
$$ | 6\sqrt{2} | 45 | 6 |
math | $11 \cdot 4$ Given that using a set of weights in groups of 6 can balance 63 weights of consecutive natural numbers, find this set of weights.
(21st All-Soviet Union Mathematical Olympiad, 1987) | x_{1}=1,x_{2}=2,x_{3}=4,x_{4}=8,x_{5}=16,x_{6}=32 | 56 | 32 |
math | Example 5 Let $a_{1}, a_{2}, \cdots, a_{n}$ be given non-zero real numbers. If the inequality
$$\begin{array}{l}
r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant \\
\sqrt[m]{x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}}-\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}
\en... | r_{i}=\left(\frac{a_{i}}{\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}}\right)^{m-1}(i=1,2, \cdots, n) | 244 | 62 |
math | 5. (10 points) A convoy of trucks is delivering supplies to a disaster victim resettlement point. Each truck has a carrying capacity of 10 tons. If each tent is allocated 1.5 tons of supplies, there will be less than one truck's worth of supplies left over. If each tent is allocated 1.6 tons of supplies, there will be ... | 213 | 104 | 3 |
math | A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \rig... | \lambda(\alpha) = \max\{\alpha, 1\} | 140 | 17 |
math | Example 6 Given three non-negative real numbers $a, b, c$ satisfying the conditions: $3a+2b+c=5, 2a+b-3c=1$. Let $s=$ $3a+b-7c$ have a maximum value of $M$ and a minimum value of $m$. Find $\mathrm{Mm}$.
(1989, Shanghai Junior High School Mathematics Competition) | \frac{5}{77} | 89 | 8 |
math | 12.167. The radius of the sector's arc is $R$, the central angle $A O B$ is $\alpha$. Through the midpoint $C$ of the radius $O A$, a line is drawn parallel to the radius $O B$ and intersects the arc $A B$ at point $D$. Find the area of triangle $O C D$. | \frac{R^{2}\sin\alpha}{8}(\sqrt{4-\sin^{2}\alpha}-\cos\alpha) | 78 | 29 |
math | 19.14*. Given a triangle $A B C$. Four circles of equal radius $\rho$ are constructed such that one of them touches the other three, and each of these three touches two sides of the triangle. Find $\rho$, if the radii of the inscribed and circumscribed circles of the triangle are $r$ and $R$ respectively. | \rho=\frac{rR}{2r+R} | 76 | 13 |
math | 12. $A, B, C, D$ are the 4 vertices of a regular tetrahedron, with each edge length being $1 \, \text{m}$. A gecko starts from point $A$ and crawls along the edges, following these rules: it does not change direction midway along any edge, and when it reaches each vertex, it has an equal probability of choosing any of ... | \frac{182}{729} | 126 | 11 |
math | Find the smallest constant $ C$ such that for all real $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds. | \frac{4}{3} | 59 | 7 |
math | Example 2 Given that $A$ and $B$ are two subsets of the set $\{1,2,3, \cdots, 100\}$, satisfying $|A|=|B|$, and $A \cap$ $B=\varnothing$. If $x \in A$ always implies $2 x+2 \in B$, then the maximum number of elements in the set $A \cup B$ is $\qquad$. | 66 | 96 | 2 |
math | 4. In quadrilateral $A B C D, \angle D A C=98^{\circ}, \angle D B C=82^{\circ}, \angle B C D=70^{\circ}$, and $B C=A D$. Find $\angle A C D$. | 28 | 62 | 2 |
math | 8.2. If $a, b, c, d$ are positive real numbers such that $a b=2$ and $c d=27$, find the minimum value of the expression $E=(a+1)(b+2)(c+3)(d+4)$. | 600 | 61 | 3 |
math | 1. Between the numbers $1,2,3,4,5,6,7,8,9$ insert arithmetic operation signs and parentheses so that the resulting expression has a value of 100. | 100 | 44 | 3 |
math | 12. Two people take turns rolling dice, each rolling two
at a time. The first one to get a sum greater than 6 on the two dice wins, otherwise the turn passes to the other person. The probability that the first person to roll wins is $\qquad$. | \frac{12}{17} | 58 | 9 |
math | 【Question 1】Calculate: $5 \times 13 \times 31 \times 73 \times 137=$ | 20152015 | 31 | 8 |
math | 2TIN asks, how many ways are there to divide the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots, 2^{2005}\right\}$ into two non-empty disjoint subsets $A$ and $B$, such that the equation $x^{2}-S(A) x+S(B)=0$ has integer roots, where $S(M)$ denotes the sum of all elements in the set $M$? | 1003 | 100 | 4 |
math | ## Task 21/63
A convex polyhedron with 53 vertices and 19 faces is modified by cutting off all its vertices with plane cuts, such that each cut exactly captures one vertex and no cut intersects or touches another.
How many edges, vertices, and faces does the resulting polyhedron have? | K=210,E=140,F=72 | 68 | 14 |
math | ## Task A-2.3.
Two quadratic functions $f_{1}(x)$ and $f_{2}(x)$ are given.
Function $f_{1}(x)$ achieves its minimum value at $x=-1$, and one of its roots is $x=3$. Function $f_{2}(x)$ achieves its maximum value at $x=3$, and one of its roots is $x=-1$.
Determine all values of $x$ for which the product $f_{1}(x) f_{... | x_{1}=1+2\sqrt{5},x_{2}=1-2\sqrt{5} | 118 | 24 |
math | Example 8 Let $n(n \geqslant 2)$ be a fixed integer. Determine the smallest constant $c$ such that
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$, and determine t... | \frac{1}{8} | 140 | 7 |
math | (7) Let $a, b, c$ be non-negative real numbers, then the minimum value of $\frac{c}{a}+\frac{a}{b+c}+\frac{b}{c}$ is | 2 | 45 | 1 |
math | 8th VMO 1969 Problem A2 Find all real x such that 0 < x < π and 8/(3 sin x - sin 3x) + 3 sin 2 x ≤ 5. | \frac{\pi}{2} | 48 | 7 |
math | 1.030. $\frac{\left(0.3275-\left(2 \frac{15}{88}+\frac{4}{33}\right): 12 \frac{2}{9}\right): 0.07}{(13-0.416): 6.05+1.92}$. | 0.5 | 80 | 3 |
math | 3. Given are positive numbers $a, b$ and $c$. Determine all triples of positive numbers $(x, y, z)$ such that
$$
x+y+z=a+b+c \quad \text { and } \quad 4 x y z-a^{2} x-b^{2} y-c^{2} z=a b c .
$$ | (x,y,z)=(\frac{b+}{2},\frac{+}{2},\frac{+b}{2}) | 73 | 27 |
math | $\sin (\alpha+\beta) \sin (\alpha-\beta)-(\sin \alpha+\sin \beta)(\sin \alpha-\sin \beta)=?$ | 0 | 33 | 1 |
math | Three. (20 points) Determine all real numbers $\theta$ such that the complex sequence $\{\cos n \theta+\mathrm{i} \sin n \theta\}$ forms an arithmetic sequence.
| \theta=2 k \pi, k \in \mathbf{Z} | 41 | 17 |
math | 3. In the tetrahedron $P-ABC$, $PC \perp$ plane $ABC$, $AB=8$, $BC=6$, $PC=9$, $\angle ABC=120^{\circ}$. Then the cosine value of the dihedral angle $B-AP-C$ is $\qquad$ | \frac{11 \sqrt{111}}{148} | 71 | 17 |
math | Example 17. Solve the equation
$$
(2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{101}{10(2-\sqrt{3})}
$$ | x_{1}=1+\sqrt{1+\log_{2+\sqrt{3}}10},\quadx_{2}=1-\sqrt{1+\log_{2+\sqrt{3}}10} | 62 | 44 |
math | 2. There is a class of four-digit numbers that, when divided by 5, leave a remainder of 1; when divided by 7, leave a remainder of 4; and when divided by 11, leave a remainder of 9. What is the smallest four-digit number in this class? | 1131 | 64 | 4 |
math | Example 1. Solve the equation
$$
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5}
$$ | x_{1}=\frac{7}{15},x_{2}=\frac{4}{5} | 36 | 23 |
math | 6. A factory produces sets of $n>2$ elephants of different sizes. According to the standard, the difference in mass between adjacent elephants within each set should be the same. The inspector checks the sets one by one using a balance scale without weights. For what smallest $n$ is this possible? | 5 | 62 | 1 |
math | There are three bins: one with 30 apples, one with 30 oranges, and one with 15 of each. Each is labeled "apples," "oranges," or "mixed." Given that all three labels are wrong, how many pieces of fruit must you look at to determine the correct labels? | 1 | 66 | 1 |
math | 272. Compute the approximate value: 1) $\sqrt[4]{17} ; 2$ ) $\operatorname{arc} \operatorname{tg} 0.98$ ; 3) $\sin 29^{\circ}$. | \sqrt[4]{17}\approx2.031,\operatorname{arctg}0.98\approx0.7754,\sin29\approx0.4848 | 57 | 46 |
math | 10. Let the function $f(x)=\frac{(x+1)^{2}+\ln \left(\sqrt{x^{2}+1}+x\right)}{x^{2}+1}$ have a maximum value of $M$ and a minimum value of $N$. Determine the value of $M+N$. | 2 | 70 | 1 |
math | 8. (3 points) The last digit of $3^{2003}+4^{2005} \times 5^{2007}$ is | 7 | 37 | 1 |
math | Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers. | \sum_{k=1}^{n} \frac{1}{k} | 81 | 17 |
math | ## Task B-3.5.
If $\operatorname{tg} x+\operatorname{ctg} x=-2, x \neq k \frac{\pi}{2}, k \in \mathbb{Z}$, what is $(\sin x-\cos x)^{10}$ ? | 32 | 64 | 2 |
math | ## Task 4 - 201234
Determine all integers $k$ for which the equation
$$
\frac{x}{k-4}+\frac{k}{2(k-4)}+\frac{k+4}{x}=0
$$
is solvable (i.e., has at least one solution $x$), where all solutions $x$ are integers. | k\in{-4,-3,-2,0,2,3} | 81 | 16 |
math | 11.007. Find the ratio of the volume of a cube to the volume of a regular tetrahedron, the edge of which is equal to the diagonal of the face of the cube. | 3 | 43 | 1 |
math | 7. A company invested in a project in 2009, with both cash inputs and cash revenues every year. It is known that
(1) In 2009, the company invested 10 million yuan, and the investment will decrease by $20\%$ each subsequent year;
(2) In 2009, the company earned 5 million yuan, and the revenue will increase by $25\%$ eac... | 2013 | 117 | 4 |
math | # Problem 4. (2 points)
How many positive numbers are there among the numbers of the form $\operatorname{ctg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019? | 1010 | 64 | 4 |
math | 9. (14 points) Let the function $f(x)$ be defined on the closed interval
$[0,1]$, satisfying $f(0)=0, f(1)=1$, and for any $x, y \in[0,1](x \leqslant y)$, we have
$$
f\left(\frac{x+y}{2}\right)=\left(1-a^{2}\right) f(x)+a^{2} f(y),
$$
where the constant $a$ satisfies $0<a<1$. Find the value of $a$. | a=\frac{\sqrt{2}}{2} | 124 | 11 |
math | 1. Captain Billy the pirate plundered 1010 gold doubloons and set sail on his ship to a deserted island to bury them as treasure. Every evening of the voyage, he paid each of his pirates one doubloon. On the eighth day of the voyage, the pirates plundered a Spanish caravel, and Billy's treasure doubled, while the numbe... | 30 | 136 | 2 |
math | Example 1.29. In the equations of the line $\frac{x}{2}=\frac{y}{-3}=\frac{z}{n}$, determine the parameter $n$ so that this line intersects with the line $\frac{x+1}{3}=\frac{y+5}{2}=\frac{z}{1}$, and find the point of their intersection. | M(2,-3,1) | 82 | 8 |
math | 11. Given the quadratic function $y=x^{2}+2 m x-3 m+1$, the independent variable $x$ and real numbers $p, q$ satisfy
$$
4 p^{2}+9 q^{2}=2, \frac{1}{2} x+3 p q=1,
$$
and the minimum value of $y$ is 1. Find the value of $m$. | -3or1 | 90 | 4 |
math | Auto: GToooanov A.C.
The sum of the digits in the decimal representation of a natural number $n$ is 100, and the sum of the digits of the number $44 n$ is 800. What is the sum of the digits of the number $3 n$? | 300 | 67 | 3 |
math | Example 1 Given a curve $C: x^{2}+y^{2}-2 x-2 y+1=0$ and a tangent line $l$ that intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. $O$ is the origin, $|O A|=a,|O B|=b, a>2$, $b>2$.
(1) Prove: $(a-2)(b-2)=2$;
(2) Find the equation of the locus of the midpoint of segment $A B$;
(3) Find the minimum... | 2\sqrt{2}+3 | 144 | 8 |
math | 2A. Calculate the value of the expression
$$
\frac{\sin 3 x}{\sin x}+\frac{\sin 6 x}{\sin 2 x}+\ldots+\frac{\sin 3 n x}{\sin n x}-\frac{\cos 3 x}{\cos x}-\frac{\cos 6 x}{\cos 2 x}-\ldots \frac{\cos 3 n x}{\cos n x}
$$ | 2n | 99 | 2 |
math | 8.176. $\sin ^{3} x(1+\operatorname{ctg} x)+\cos ^{3} x(1+\operatorname{tg} x)=2 \sqrt{\sin x \cos x}$. | \frac{\pi}{4}(8k+1),k\inZ | 52 | 16 |
math | 4. [4] Determine the remainder when
$$
2^{\frac{1 \cdot 2}{2}}+2^{\frac{2 \cdot 3}{2}}+\cdots+2^{\frac{2011 \cdot 2012}{2}}
$$
is divided by 7 . | 1 | 70 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.