task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | 9. Let $\alpha, \beta$ satisfy the equations $\alpha^{3}-3 \alpha^{2}+5 \alpha-4=0, \beta^{3}-3 \beta^{2}+5 \beta-2=0$, then $\alpha+\beta=$ | \alpha+\beta=2 | 58 | 6 |
math | 5. Calculate the maximum product of all natural numbers whose sum is 2019. | 3^{673} | 19 | 6 |
math | 6. Let $x, y, z \geqslant 0$, and $x y+y z+z x=1$, (1) find the maximum value of $f(x, y, z)=x(1-$ $\left.y^{2}\right)\left(1-z^{2}\right)+y\left(1-z^{2}\right)\left(1-x^{2}\right)+z\left(1-x^{2}\right)\left(1-y^{2}\right)$; (2) find the minimum value of $\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}$. | \frac{4\sqrt{3}}{9} | 142 | 12 |
math | 11.2. Find all quadruples of real numbers $(a, b, c, d)$ such that $a(b+c)=b(c+d)=c(d+a)=d(a+b)$. | 8\inftyiniteseriesofsolutions,obtainedmultiplyingeachofthe(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,1,1,1),(1,-1,1,-1),(1,-1\\sqrt{2},-1,1\\sqrt{2})anyreal | 41 | 86 |
math | 10.2.1. (12 points) Find the greatest integer value of $a$ for which the equation
$$
\sqrt[3]{x^{2}-(a+7) x+7 a}+\sqrt[3]{3}=0
$$
has at least one integer root. | 11 | 65 | 2 |
math | 2. Let $x_{0}, x_{1}, x_{2}, \ldots, x_{2002}$ be consecutive integers, for which
$$
-x_{0}+x_{1}-x_{2}+\ldots-x_{2000}+x_{2001}-x_{2002}=2003
$$
Calculate the number $x_{2002}$. | x_{2002}=-1002 | 93 | 12 |
math | (Legendre's Formula)
We recall the factorial notation: for an integer $n, n! = n \times (n-1) \times \ldots \times 2 \times 1$.
For a prime number $p$, calculate $v_{p}(n!)$. | v_{p}(n!)=\Sigma_{k=1}^{+\infty}\lfloor\frac{n}{p^{k}}\rfloor | 60 | 32 |
math | 287. How many solutions does the system of equations generally have
$$
\begin{gathered}
a x^{2}+b x y+c y^{2}=d \\
a_{1} x^{2}+b_{1} x y+c_{1} y^{2}=d_{1} ?
\end{gathered}
$$
In particular, how many solutions does the system of question 280 have? | 4 | 92 | 1 |
math | 8. Suppose the equation $z^{2009}+z^{2008}+1=0$ has roots of modulus 1, then the sum of all roots of modulus 1 is | -1 | 44 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{2^{3 x}-3^{5 x}}{\sin 7 x-2 x}$ | \frac{1}{5}\ln\frac{2^{3}}{3^{5}} | 43 | 20 |
math | 11. Let the sequence $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $a_{1}=a_{2}=1, a_{3}=2$, and for any natural number $n$, $a_{n} a_{n+1} a_{n+2} \neq 1$, and $a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3}$, then the value of $a_{1}+a_{2}+\cdots+a_{100}$ is $\qquad$ . | 200 | 155 | 3 |
math | 66. a) $(x+y)\left(x^{2}-x y+y^{2}\right)$; b) $(x+3)\left(x^{2}-3 x+9\right)$; c) $(x-1)\left(x^{2}+x+1\right)$; d) $(2 x-3)\left(4 x^{2}+6 x+9\right)$ | )x^{3}+y^{3};b)x^{3}+27;)x^{3}-1;)8x^{3}-27 | 86 | 31 |
math | 4A. Find all real numbers $a$ and $b, b>0$, such that the roots of the given equations $x^{2}+a x+a=b$ and $x^{2}+a x+a=-b$ are four consecutive integers. | =-1,b=1or=5,b=1 | 55 | 11 |
math | Example 1. Solve the equation $y^{\prime \prime}+y^{\prime}-2 y=x^{3}-2$. | y_{\text{on}}=C_{1}e^{-2x}+C_{2}e^{x}-\frac{x^{3}}{2}-\frac{3x^{2}}{4}-\frac{9x}{4}-\frac{7}{8} | 29 | 61 |
math | $\left[\begin{array}{l}{[\text { Equilateral (equilateral) triangle ] }} \\ {[\quad \text { Law of Cosines }}\end{array}\right]$
An equilateral triangle $ABC$ with side 3 is inscribed in a circle. Point $D$ lies on the circle, and the chord $AD$ equals $\sqrt{3}$. Find the chords $BD$ and $CD$. | \sqrt{3},2\sqrt{3} | 91 | 11 |
math | Example 8 Solve the equation $3 x^{3}-[x]=3$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | \sqrt[3]{\frac{4}{3}} | 43 | 12 |
math | 1. Find the positive integer $x(0 \leqslant x \leqslant 9)$, such that the tens digit of the product of $\overline{2 x 7}$ and 39 is 9. | 8 | 51 | 1 |
math | Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
[i]Proposed by Lewis Chen[/i] | 499499 | 66 | 6 |
math | Compute $\sqrt{(31)(30)(29)(28)+1}$. | 869 | 19 | 3 |
math | 514. Calculate approximately $3.002^{4}$. | 81.216 | 16 | 6 |
math | 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 $\frac{\sum_{i=1}^{n} a_{i}^{2} a_{i+1}+8 n}{\sum_{i=1}^{n} a_{i}^{2}}$. | 4 | 94 | 1 |
math | 【Question 18】
If $a, b, c, d, e$ are consecutive positive integers, and $a<b<c<d<e, b+c+d$ is a perfect square, $a+b+c+d+e$ is a perfect cube, what is the minimum value of $c$? | 675 | 65 | 3 |
math | 11. (2007 Slovakia-Slovakia-Poland Mathematics Competition) Given $n \in\{3900,3901, \cdots, 3909\}$. Find the value of $n$ such that the set $\{1,2, \cdots, n\}$ can be partitioned into several triples, and in each triple, one number is equal to the sum of the other two. | 3900or3903 | 95 | 9 |
math | 10. $A$ is the center of a semicircle, with radius $A D$ lying on the base. $B$ lies on the base between $A$ and $D$, and $E$ is on the circular portion of the semicircle such that $E B A$ is a right angle. Extend $E A$ through $A$ to $C$, and put $F$ on line $C D$ such that $E B F$ is a line. Now $E A=1, A C=\sqrt{2},... | \sqrt{2-\sqrt{2}} | 179 | 9 |
math | $2 \cdot 40$ Find all natural numbers $n$ such that $n \cdot 2^{n}+1$ is divisible by 3.
(45th Moscow Mathematical Olympiad, 1982) | 6k+1,6k+2 | 50 | 9 |
math | 5. In Pascal's Triangle, each number is the sum of the two numbers directly above it. The first few rows of this triangle are as follows:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline Row 0 & \multicolumn{6}{|c|}{1} \\
\hline Row 1 & & & 1 & & 1 & \\
\hline Row 2 & & 1 & & 2 & & 1 \\
\hline Row 3 & & & 3 & & 3 & 1 \\
\hline ... | 62 | 192 | 2 |
math | 3. (CAN) Find the minimum value of $$ \max (a+b+c, b+c+d, c+d+e, d+e+f, e+f+g) $$ subject to the constraints (i) $a, b, c, d, e, f, g \geq 0$, (ii) $a+b+c+d+e+f+g=1$. | \frac{1}{3} | 83 | 7 |
math | For what values of the parameter $a$ is the sum of the squares of the roots of the equation $x^{2}+2 a x+2 a^{2}+4 a+3=0$ the greatest? What is this sum? (The roots are considered with their multiplicity.)
# | 18 | 63 | 2 |
math | [b]p1.[/b] Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly
a) Three other points?
b) Four other points?
[b]p2.[/b] Martian bank notes can have denomination of $1, 3, 5, 25$ marts. Is it possible to change a note of $25$ mar... | 1376 | 227 | 6 |
math | Let $a, b, c$, and $\alpha, \beta, \gamma$ denote the sides and angles of a triangle, respectively. Simplify the following expression:
$$
\frac{\alpha \cdot \cos \alpha+b \cdot \cos \beta-c \cdot \cos \gamma}{\alpha \cdot \cos \alpha-b \cdot \cos \beta+c \cdot \cos \gamma}
$$ | \frac{\operatorname{tg}\gamma}{\operatorname{tg}\beta} | 86 | 18 |
math | Example 4 Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta, \alpha \beta$. Find the sum of the coefficients of this polynomial.
(2005 China Western Mathematical Olympiad) | 1 | 63 | 1 |
math | 9. (12 points) Five students, Jia, Yi, Bing, Ding, and Wu, ranked in the top 5 (no ties) in a math competition and stood in a row for a photo. They each said the following:
Jia said: The two students next to me have ranks that are both behind mine;
Yi said: The two students next to me have ranks that are adjacent to m... | 23514 | 208 | 5 |
math | Problem 10.3. Petya and Daniil are playing the following game. Petya has 36 candies. He lays out these candies in the cells of a $3 \times 3$ square (some cells may remain empty). After this, Daniil chooses four cells forming a $2 \times 2$ square and takes all the candies from there. What is the maximum number of cand... | 9 | 94 | 1 |
math | 3. [4 points] Solve the inequality $\left(\sqrt{x^{3}+x-90}+7\right)\left|x^{3}-10 x^{2}+31 x-28\right| \leqslant 0$. | 3+\sqrt{2} | 57 | 6 |
math | 2. Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b x+c$ have three real roots $x_{1}, x_{2}, x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$. | \frac{3\sqrt{3}}{2} | 131 | 12 |
math | 18. Among all tetrahedra with edge lengths 2, 3, 3, 4, 5, 5, what is the maximum volume? Prove your conclusion.
(1983 National Competition Problem) | \frac{8\sqrt{2}}{3} | 50 | 12 |
math | 1. How many 4-digit numbers exist where the digit in the thousands place is greater than the digit in the hundreds place? | 4500 | 26 | 4 |
math | 9. Find all values of $a$ such that the roots $x_{1}, x_{2}, x_{3}$ of the polynomial $x^{3}-6 x^{2}+a x+a$ satisfy $\left(x_{1}-3\right)^{3}+\left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0$. | -9 | 86 | 2 |
math | G3.2 Let $n$ be the integral part of $\frac{1}{\frac{1}{1980}+\frac{1}{1981}+\cdots+\frac{1}{2009}}$. Find the value of $n$. | 66 | 59 | 2 |
math | 4. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy the condition:
$$
x + \sqrt{x} - f(x) = \sqrt{f(f(x))}, \quad \forall x \in \mathbb{N}
$$
Gabriel Daniilescu, Brăila | f(n)=n,\foralln\in\mathbb{N} | 73 | 15 |
math | Example 4 (2002 China Western Mathematical Olympiad) Consider a square on the complex plane, whose 4 vertices correspond to the 4 roots of a certain monic quartic equation with integer coefficients $x^{4}+p x^{3}+q x^{2}+r x+s=0$. Find the minimum value of the area of such a square.
| 2 | 79 | 1 |
math | Three. (20 points) Given the hyperbola $\Gamma: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $(a>0, b>0)$. If for any $\triangle A B C$ with vertices all on the hyperbola $\Gamma$, the orthocenter of $\triangle A B C$ is also on the hyperbola $\Gamma$. Explore the condition that the hyperbola $\Gamma$ should satisfy. | a=b | 106 | 2 |
math | 8. Let the function
$$
f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n},
$$
where $a_{0}, a_{1}, \cdots, a_{n}$ are non-negative integers. It is known that $f(1)=4$, $f(5)=152$. Then $f(6)=$ $\qquad$ | 254 | 96 | 3 |
math | 10.55 Find the digital sum of the product (as a function of $n$)
$$
9 \cdot 99 \cdot 9999 \cdot \cdots \cdot\left(10^{2^{n}}-1\right) \text {. }
$$
where the number of digits of each factor is twice the number of digits of the preceding factor.
(21st United States of America Mathematical Olympiad, 1992) | 9\cdot2^{n} | 101 | 7 |
math | 313. Find the scalar product of the vectors
$$
\bar{p}=\bar{i}-3 \bar{j}+\bar{k}, \bar{q}=\bar{i}+\bar{j}-4 \bar{k}
$$ | -6 | 51 | 2 |
math | 19. [11] Integers $0 \leq a, b, c, d \leq 9$ satisfy
$$
\begin{array}{c}
6 a+9 b+3 c+d=88 \\
a-b+c-d=-6 \\
a-9 b+3 c-d=-46
\end{array}
$$
Find $1000 a+100 b+10 c+d$. | 6507 | 95 | 4 |
math | Teacher Ana went to buy cheese bread to honor the students who won in the OBMEP, given that
- every 100 grams of cheese bread cost $R \$ 3.20$ and correspond to 10 cheese breads; and
- each person eats, on average, 5 cheese breads.
In addition to the teacher, 16 students, one monitor, and 5 parents will be present at... | 1200 | 164 | 4 |
math | Problem 6. Calculate $2 \operatorname{arctg} 5+\arcsin \frac{5}{13}$. | \pi | 30 | 2 |
math | Example 10 Find the least common multiple of 513, 135, and 3114. | 887490 | 27 | 6 |
math | 7.050. $\log _{2} \frac{x-5}{x+5}+\log _{2}\left(x^{2}-25\right)=0$. | 6 | 40 | 1 |
math | 11.3. Find the domain and range of the function
$$
y=\sqrt{x^{2}+3 x+2}+\frac{1}{\sqrt{2 x^{2}+6 x+4}}
$$ | y\geq\sqrt[4]{8} | 49 | 11 |
math | Let $I_{m}=\textstyle\int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x .$ For which integers $m, 1 \leq m \leq 10$ is $I_{m} \neq 0 ?$ | m = 3, 4, 7, 8 | 74 | 13 |
math | 4. A sphere is inscribed in a regular triangular prism, touching all three sides and both bases of the prism. Find the ratio of the surface areas of the sphere and the prism. | \frac{2\pi}{9\sqrt{3}} | 38 | 13 |
math | 4. Determine all three-digit numbers that are divisible by 11, and the sum of their digits is 10. | 154,253,352,451,550 | 26 | 19 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{4}}\left(\frac{\ln (\operatorname{tg} x)}{1-\operatorname{ctg} x}\right)^{1 /\left(x+\frac{\pi}{4}\right)}$ | 1 | 67 | 1 |
math | What is the sum of those three-digit numbers, all of whose digits are odd? | 69375 | 17 | 5 |
math | 6. A. Let $a=\sqrt[3]{3}, b$ be the fractional part of $a^{2}$. Then $(b+2)^{3}=$ $\qquad$ | 9 | 41 | 1 |
math | 1. According to the inverse theorem of Vieta's theorem, we form a quadratic equation. We get $x^{2}-\sqrt{2019} x+248.75=0$.
Next, solving it, we find the roots $a$ and $b$: $a=\frac{\sqrt{2019}}{2}+\frac{32}{2}$ and $b=\frac{\sqrt{2019}}{2}-\frac{32}{2}$, and consequently, the distance between the points $a$ and $b$:... | 32 | 132 | 2 |
math | 4. (ROM) Solve the equation
$$
\cos ^{2} x+\cos ^{2} 2 x+\cos ^{2} 3 x=1 .
$$ | x\in{\pi/2+\pi,\pi/4+\pi/2,\pi/6+\pi/3\mid\inZ} | 40 | 31 |
math | (V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this
angle (There was an error in published condition of this problem). | 120^\circ | 50 | 5 |
math | 16. The book has 568 pages. How many digits in total have been used for numbering from the first to the last page of the book? | 1596 | 33 | 4 |
math | Example. Calculate the definite integral
$$
\int_{1}^{2} x \ln ^{2} x d x
$$ | 2\ln^22-2\ln2+\frac{3}{4} | 29 | 18 |
math | 8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=64
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 32 | 102 | 2 |
math | Example 11 Given $-1 \leqslant 2 x+y-z \leqslant 8, 2 \leqslant x$ $-y+z \leqslant 9, -3 \leqslant x+2 y-z \leqslant 7$. Find the maximum and minimum values of the function $u=7 x+5 y-2 z$. | -6 \leqslant u \leqslant 47 | 88 | 16 |
math | Example 1. Solve the inequality
$$
25^{x}>125^{3 x-2}
$$ | x<\frac{6}{7} | 25 | 9 |
math | A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle? | 400 | 57 | 3 |
math | 14. (15 points) The distance between location A and location B is 360 kilometers. A truck loaded with 6 boxes of medicine is driving from location A to location B, while at the same time, a motorcycle starts from location B and heads towards the truck. The truck's speed is 40 kilometers/hour, and the motorcycle's speed... | 8\frac{2}{3} | 175 | 8 |
math | 17. $[\mathbf{1 0}]$ Determine the last two digits of $17^{17}$, written in base 10 . | 77 | 34 | 2 |
math | Subject 1 Determine the real numbers a and b such that: $\lim _{x \rightarrow 2} \frac{\sqrt{x^{2}-x+a}-b}{x-2}=\frac{3}{4}$. | =2,b=2 | 48 | 5 |
math | 5. Given $\odot O: x^{2}+y^{2}=1$ with a moving chord $A B=\sqrt{3}$, and a moving point $P$ on $\odot C:(x-2)^{2}+(y-3)^{2}=2$. Then the minimum value of $\sqrt{\overrightarrow{P A} \cdot \overrightarrow{P B}+\frac{3}{4}}$ is $\qquad$. | \sqrt{13}-\frac{1}{2}-\sqrt{2} | 98 | 18 |
math | The set of numbers $(p, a, b, c)$ of positive integers is called [i]Sozopolian[/i] when:
[b]* [/b]p is an odd prime number
[b]*[/b] $a$, $b$ and $c$ are different and
[b]*[/b] $ab + 1$, $bc + 1$ and $ca + 1$ are a multiple of $p$.
a) Prove that each [i]Sozopolian[/i] set satisfies the inequality $p... | p = 5 | 172 | 5 |
math | Problem 2. One third of the total quantity of a certain commodity is sold at a 10% profit, and half of the same commodity is sold at a 15% loss. By how much does the price of the remaining part of the commodity need to be increased to compensate for the loss? | 25 | 63 | 2 |
math | 3. let $p$ be an odd prime number. Find all natural numbers $k$ such that
$$
\sqrt{k^{2}-p k}
$$
is a positive integer.
## Solution | (\frac{p+1}{2})^2 | 42 | 11 |
math | Example 1 Given positive integers $r$ and $n$, find the smallest positive integer $m$, satisfying: if the set $S=\{1,2, \cdots, m\}$ is arbitrarily divided into $r$ pairwise disjoint subsets $A_{1}, A_{2}, \cdots, A_{r}$, then there must exist two positive integers $a, b$ belonging to the same subset $A_{i}(1 \leqslant... | m_{0}=(n+1)r | 124 | 9 |
math | 4A. Calculate the value of the expression
$$
\cos x \cos 2 x \cos 3 x \ldots \cos 1010 x \text {, for } x=\frac{2 \pi}{2021}
$$ | -\frac{1}{2^{1010}} | 56 | 12 |
math | N1. The sequence $a_{0}, a_{1}, a_{2}, \ldots$ of positive integers satisfies
$$
a_{n+1}=\left\{\begin{array}{ll}
\sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { is an integer } \\
a_{n}+3, & \text { otherwise }
\end{array} \quad \text { for every } n \geqslant 0 .\right.
$$
Determine all values of $a_{0}>1$ for whi... | Allpositivemultiplesof3 | 153 | 8 |
math | Let $n_1,n_2,\ldots,n_s$ be distinct integers such that
$$(n_1+k)(n_2+k)\cdots(n_s+k)$$is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. For each of the following assertions give a proof or a counterexample:
$(\text a)$ $|n_i|=1$ for some $i$
$(\text b)$ If further all $n_i$ are positive, then
$$\{n... | \{n_1, n_2, \ldots, n_s\} = \{1, 2, \ldots, s\} | 140 | 34 |
math | Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x+y)\leq f(x^2+y)$$ for all $x,y$. | f(x) = c | 40 | 6 |
math | 7. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
\cdots \cdots \cdots \cdots \cdots \\
a_{n 1} a_{n 2} a_{n 3} a_{n 4} \cdots a... | 2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}} | 274 | 20 |
math | 9. find all pairs $(a, b)$ of natural numbers such that
$$
\frac{a^{3}+1}{2 a b^{2}+1}
$$
is an integer. | (,b)=(2n^{2}+1,n) | 43 | 13 |
math | A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that
(i) no three points in $P$ lie on a line and
(ii) no two points in $P$ lie on a line through the origin.
A triangle with vertices in $P$ is fat, if $O$ is strictly inside the triangle. Find the maximum number of ... | 2021 \cdot 505 \cdot 337 | 125 | 16 |
math | 10. Let $S$ be the area of a triangle inscribed in a circle of radius 1. Then the minimum value of $4 S+\frac{9}{S}$ is $\qquad$ . | 7 \sqrt{3} | 44 | 6 |
math | 18. N4 (BUL) Find all positive integers $x$ and $y$ such that $x+y^{2}+z^{3}=x y z$. where $z$ is the greatest common divisor of $x$ and $y$ - | (4,2),(4,6),(5,2),(5,3) | 55 | 17 |
math | 6. In the Cartesian coordinate system, the "rectilinear distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as
$$
d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right| \text {. }
$$
If point $C(x, y)$ has equal rectilinear distances to $A(1,3)$ and $B(6,9)$, where the ... | 5(\sqrt{2}+1) | 173 | 9 |
math | 607. The sum of a two-digit number and its reverse is a perfect square. Find all such numbers.
(Definition of the number reversed to the given one, see § 12, p. 12.4.) | 29,38,47,56,65,74,83,92 | 49 | 23 |
math | ## Task A-3.5.
Let $n$ be a natural number. A sequence of $2 n$ real numbers is good if for every natural number $1 \leqslant m \leqslant 2 n$, the sum of the first $m$ or the sum of the last $m$ terms of the sequence is an integer. Determine the smallest possible number of integers in a good sequence. | 2 | 87 | 1 |
math | 3. Given $a>1$. Then the minimum value of $\log _{a} 16+2 \log _{4} a$ is . $\qquad$ | 4 | 38 | 1 |
math | Folklore
In Chicago, there are 36 gangsters, some of whom are at odds with each other. Each gangster is a member of several gangs, and there are no two gangs with the same membership. It turned out that gangsters who are in the same gang do not feud, but if a gangster is not a member of a certain gang, then he feuds wi... | 3^{12} | 101 | 5 |
math | ## Problem 2.
Solve, in the set of real numbers, the system of equations
$$
\left.\begin{array}{l}
y^{3}-6 x^{2}+12 x-8=0 \\
z^{3}-6 y^{2}+12 y-8=0 \\
x^{3}-6 z^{2}+12 z-8=0
\end{array}\right\}
$$ | (2,2,2) | 94 | 7 |
math | 3. (10 points) The expressway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. The distance from Lishan Town to the county town is 54 kilometers. At 8:30 AM, a bus departs from Lishan Town to the county town, arriving at 9:15. After a 15-minute stop, it continues to th... | 10:08 | 152 | 5 |
math | 12. Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{2014}}{3}\right]$.
(2 marks)
Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}... | 15 | 159 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{h \rightarrow 0} \frac{\sin (x+h)-\sin (x-h)}{h}$ | 2\cosx | 38 | 4 |
math | ## Task Condition
Find the derivative.
$$
y=x-\ln \left(2+e^{x}+2 \sqrt{e^{2 x}+e^{x}+1}\right)
$$ | \frac{1}{\sqrt{e^{2x}+e^{x}+1}} | 44 | 21 |
math | Cátia leaves school every day at the same time and returns home by bicycle. When she cycles at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $16 \mathrm{~h} 30 \mathrm{~m}$. If she cycles at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at $17 \mathrm{~h} 15 \mathrm{~m}$. At what speed should she cycle to arri... | 12\mathrm{~}/\mathrm{} | 123 | 10 |
math | 7. (10 points) There is a sufficiently deep water tank with a rectangular base that is 16 cm long and 12 cm wide. Originally, the tank contains 6 cm of water and 6 cm of oil (the oil is above the water). If a block of iron with dimensions 8 cm in length, 8 cm in width, and 12 cm in height is placed in the tank, the hei... | 7 | 102 | 1 |
math | 7. For what values of $a$ does the equation
$$
[x]^{2}+2012 x+a=0
$$
(where $[x]$ is the integer part of $x$, i.e., the greatest integer not exceeding $x$) have the maximum number of solutions? What is this number? | 89 | 69 | 2 |
math | Example 3 Find the minimum value of the function $\sqrt{2 x^{2}-4 x+4}+\sqrt{2 x^{2}-16 x+\log _{2}^{2} x-2 x \log _{2} x+2 \log _{2} x+50}$. | 7 | 68 | 1 |
math | 2. Let $\mathrm{i}=\sqrt{-1}$ be the imaginary unit, then $\mathrm{i}+2 \mathrm{i}^{2}+3 \mathrm{i}^{3}+\cdots+2013 \mathrm{i}^{2013}=$ | 1006+1007\mathrm{i} | 59 | 13 |
math | 2. Given $a, b, c \in \mathbf{R}$, and satisfy $a>b>c$, $a+b+c=0$. Then, the range of $\frac{c}{a}$ is $\qquad$ | \left(-2,-\frac{1}{2}\right) | 49 | 14 |
math | 45. A peasant bought a cow, a goat, a sheep, and a pig, paying 1325 rubles. The goat, pig, and sheep together cost 425 rubles. The cow, pig, and sheep together cost 1225 rubles, and the goat and pig together cost 275 rubles. Find the price of each animal. | 900,150,100,175 | 82 | 15 |
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