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 | $6 \cdot 179$ Find all functions $f: R \rightarrow R$, such that $x f(x)-y f(y)=(x-y) f(x+y)$. | f(x) = ax + b, \quad a, b \in \mathbb{R} | 39 | 21 |
math | Task 1. In the first week, a squirrel collected 84 nuts, in the second week 96 nuts, and in the third week 65 nuts. Another squirrel in the fourth week collected three times more nuts than the first squirrel collected in the second and third weeks combined. How many nuts did the two squirrels collect in total over the ... | 728 | 76 | 3 |
math | 2. Find the sum of all four-digit numbers in which only the digits $1,2,3,4,5$ appear, and each digit appears no more than once. (8 points) | 399960 | 41 | 6 |
math | 37. Given that the integer part of the real number $\frac{2+\sqrt{2}}{2-\sqrt{2}}$ is $a$, and the fractional part is $1-b$. Then the value of $\frac{(b-1) \cdot(5-b)}{\sqrt{a^{2}-3^{2}}}$ is | -1 | 72 | 2 |
math | 8. One or two? Let's take all natural numbers from 1 to 1000000 and for each of them, calculate the sum of its digits. For all the resulting numbers, we will again find the sum of their digits. We will continue this process until all the resulting numbers are single-digit. Among the million resulting numbers, 1 and 2 w... | 1 | 94 | 1 |
math | 11. (20 points) Given the line $y=x$ intersects the ellipse $C$ : $\frac{x^{2}}{16}+\frac{y^{2}}{11}=1$ at points $A$ and $B$, and a line $l$ passing through the right focus $F$ of the ellipse $C$ with an inclination angle of $\alpha$ intersects the chord $AB$ at point $P$, and intersects the ellipse $C$ at points $M$ ... | y=-\frac{1}{2} x+\frac{\sqrt{5}}{2} | 160 | 20 |
math | 7. Find all three-digit numbers in the decimal system that are equal to one third of the number with the same representation in another number system. | 116,120,153,195,236,240,356,360,476,480,596 | 29 | 43 |
math | 1. There are 28 students in the class. On March 8th, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given if it is known that there were 4 times as many roses as daffodils, but 10 times fewer than tulips? (A. A. Tesler) | 16 | 82 | 2 |
math | 7. (5 points) A box contains many identical but red, yellow, and blue glass balls. Each time two balls are drawn. To ensure that the result of drawing is the same 5 times, at least $\qquad$ draws are needed. | 25 | 52 | 2 |
math | 3. (8 points) Select 5 different numbers from 1 to 9, such that the sum of the selected 5 numbers is exactly half the sum of the 4 numbers that were not selected. The sum of the selected numbers is $\qquad$.
| 15 | 55 | 2 |
math | In a wire with a cross-sectional area of $A$, a current $I$ flows during a short circuit. Before the short circuit, the conductor was at a temperature of $T_{0}$. Estimate how long it will take for the wire to melt! Ignore radiation losses! (The specific resistance of the material depends on the temperature: $\varrho=\... | 23,1\mathrm{~} | 228 | 9 |
math | 121. Find the mathematical expectation and variance of a random variable $X$ distributed uniformly for $x \in [a ; b]$. | M(X)=\frac{+b}{2},\quadD(X)=\frac{(b-)^{2}}{12} | 30 | 29 |
math | 117. Find $\lim _{x \rightarrow \infty} \frac{2 x^{3}+x}{x^{3}-1}$. | 2 | 34 | 1 |
math | 2. For any natural values $\quad m$ and $n(n>1)$, the function $f\left(\frac{m}{n}\right)=\frac{\sqrt[n]{3^{m}}}{\sqrt[n]{3^{m}}+3}$ is defined. Calculate the sum
$f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+f\left(\frac{3}{2020}\right)+\cdots+f\left(\frac{4039}{2020}\right)+f\left(\frac{4040}{2020}\righ... | 2020.25 | 147 | 7 |
math | Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality
$$
f(f(x))=x^2f(x)+ax^2
$$
for all real $x$. | a = 0 | 64 | 4 |
math | 3. On a wooden ruler, three marks are made: 0, 7, and 11 centimeters. How can you measure a segment of (a) 8 cm; (b) 5 cm using it? | 8=2\times45=3\times11-4\times7 | 48 | 18 |
math | ## Task 19
$34+a=40, 40: 10=b, b \cdot a=c ; c:(a-b)+44=56 \quad$ What numbers do you need to insert for $a, b$ and $c$? | =6;b=4;=24 | 60 | 9 |
math | 15. In a geometric sequence with a common ratio greater than 1, what is the maximum number of terms that are integers between 100 and 1000? | 6 | 38 | 1 |
math | $2016$ bugs are sitting in different places of $1$-meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of contacts... | 1008^2 | 82 | 6 |
math | 9. (This question is worth 16 points) Let $O$ be the circumcenter of acute $\triangle A B C$, and the areas of $\triangle B O C, \triangle C O A, \triangle A O B$ form an arithmetic sequence. Find the minimum value of $\tan A+2 \tan C$.
---
The translation is provided as requested, maintaining the original format and... | 2\sqrt{6} | 86 | 6 |
math | ## Task B-3.4.
Determine all pairs of natural numbers $a$ and $b$ such that $a^{2}-4 b^{2}=a-2 b+2^{2022}$. | =2^{2021}+1,b=2^{2020} | 47 | 19 |
math | 10 Let the real-coefficient quadratic equation $x^{2}+a x+2 b-2=0$ have two distinct real roots, one of which lies in the interval $(0,1)$ and the other in the interval $(1,2)$. Then the range of $\frac{b-4}{a-1}$ is $\qquad$ | \frac{1}{2}<\frac{b-4}{-1}<\frac{3}{2} | 75 | 24 |
math | 11. (20 points) Let $x, y$ be complex numbers, find the maximum value of $\frac{|3 x+4 y|}{\sqrt{|x|^{2}+|y|^{2}+\left|x^{2}+y^{2}\right|}}$.
| \frac{5\sqrt{2}}{2} | 64 | 12 |
math | ## Task A-4.3.
Determine all natural numbers $m$ and $n$ for which $2^{n}+5 \cdot 3^{m}$ is a square of some natural number. | (2,2) | 44 | 5 |
math | Three, to be factored into a product of rational integral expressions in $x$ and $y$
$$
x^{7}+x^{6} y+x^{5} y^{2}+x^{4} y^{3}+x^{3} y^{4}+x^{2} y^{5}+x y^{6}+y^{7}
$$ | (x+y)(x^{2}+y^{2})(x^{4}+y^{4}) | 80 | 21 |
math | The twelve students in an olympiad class went out to play soccer every day after their math class, forming two teams of six players each and playing against each other. Each day they formed two different teams from those formed in previous days. By the end of the year, they found that every group of five students had p... | 132 | 83 | 3 |
math | 6.222.
$$
\left\{\begin{array}{l}
\sqrt{\frac{x+1}{x+y}}+\sqrt{\frac{x+y}{x+1}}=2 \\
\sqrt{\frac{x+1}{y+2}}-\sqrt{\frac{y+2}{x+1}}=1.5
\end{array}\right.
$$ | (11;1) | 81 | 6 |
math | 1. When $x^{2}$ was added to the quadratic trinomial $f(x)$, its maximum value increased by $\frac{27}{2}$, and when $4 x^{2}$ was subtracted from it, its maximum value decreased by 9. How will the maximum value of $f(x)$ change if $2 x^{2}$ is subtracted from it? | \frac{27}{4} | 81 | 8 |
math | 6.001. $\frac{x^{2}+1}{x-4}-\frac{x^{2}-1}{x+3}=23$. | x_{1}=-\frac{55}{16},x_{2}=5 | 34 | 19 |
math | Example 3 If the digits of a four-digit number are reversed to form a new four-digit number, the new number is exactly four times the original number. Find the original number.
(1988, Nanjing Mathematical Olympiad Selection Contest) | 2178 | 51 | 4 |
math | One, (20 points) Let $a, b$ be real numbers, and
$$
4 a^{2}+16 b^{2}-16 a b+13 a-26 b+3=0 \text {. }
$$
Find the range of $a^{2}+b^{2}$. | \left[\frac{1}{80},+\infty\right) | 70 | 16 |
math | 10.3. The equation $P(x)=0$, where $P(x)=x^{2}+b x+c$, has a unique root, and the equation $P(P(P(x)))=0$ has exactly three distinct roots. Solve the equation $P(P(P(x)))=0$. | 1;1+\sqrt{2};1-\sqrt{2} | 62 | 14 |
math | 1. Task: Compare the numbers $\left(\frac{2}{3}\right)^{2016}$ and $\left(\frac{4}{3}\right)^{-1580}$. | (\frac{2}{3})^{2016}<(\frac{4}{3})^{-1580} | 43 | 26 |
math | Using a fireworks analogy, we launch a body upward with an initial velocity of $c=90 \mathrm{msec}^{-1}$; after $t=5$ seconds, we hear its explosion. At what height did it explode, if the speed of sound is $a=340 \mathrm{msec}^{-1}$? (Neglect air resistance.) | 289\mathrm{~} | 78 | 8 |
math | Writing successively the natural numbers, we obtain the sequence:
$$
12345678910111213141516171819202122 \ldots
$$
Which digit is in the $2009^{th}$ position of this sequence? | 0 | 73 | 1 |
math | 10. (20 points) Find all values of the parameter $a$ for which the equation
$$
6|x-4 a|+\left|x-a^{2}\right|+5 x-3 a=0
$$
has no solution. | (-\infty,-13)\cup(0,+\infty) | 54 | 16 |
math | 60.Yura left the house for school 5 minutes later than Lena, but walked at twice her speed. How long after leaving will Yura catch up to Lena? | 5 | 35 | 1 |
math | 8. The number of positive integer pairs $(x, y)$ that satisfy $y=\sqrt{x+51}+\sqrt{x+2019}$ is $\qquad$ pairs. | 6 | 40 | 1 |
math | 4. Bivariate function
$$
\begin{array}{l}
f(x, y) \\
=\sqrt{\cos 4 x+7}+\sqrt{\cos 4 y+7}+ \\
\quad \sqrt{\cos 4 x+\cos 4 y-8 \sin ^{2} x \cdot \sin ^{2} y+6}
\end{array}
$$
The maximum value of the function is | 6 \sqrt{2} | 92 | 6 |
math | Let $a$ and $b$ be positive integers such that $a > b$. Professor Fernando told student Raul that if he calculated the number $A = a^2 + 4b + 1$, the result would be a perfect square. Raul, by mistake, swapped the numbers $a$ and $b$ and calculated the number $B = b^2 + 4a + 1$, which, by chance, is also a perfect squa... | =8,b=4,A=81,B=49 | 130 | 13 |
math | 9. (16 points) For positive integers $n(n \geqslant 2)$, let
$$
a_{n}=\sum_{k=1}^{n-1} \frac{n}{(n-k) 2^{k-1}} \text {. }
$$
Find the maximum value in the sequence $\left\{a_{n}\right\}$. | \frac{10}{3} | 82 | 8 |
math | 5. Given real numbers $m, n$ satisfy $m-n=\sqrt{10}$, $m^{2}-3 n^{2}$ is a prime number. If the maximum value of $m^{2}-3 n^{2}$ is $a$, and the minimum value is $b$, then $a-b=$ $\qquad$ | 11 | 72 | 2 |
math | 2. Let the line $y=a$ intersect the curve $y=\sin x(0 \leqslant x \leqslant \pi)$ at points $A$ and $B$. If $|A B|=\frac{\pi}{5}$, then $a=$ $\qquad$ (to 0.0001$)$. | 0.9511 | 76 | 6 |
math | Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$ | (13, 4900) | 32 | 11 |
math | 1. Find all integer values of $x, y$ that satisfy the condition $\frac{x+y}{x^{2}-x y+y^{2}}=\frac{3}{7}$.
The above text has been translated into English, preserving the original text's line breaks and format. | (4,5),(5,4) | 58 | 9 |
math | 2. Solve the equation:
$$
\sin x=\sqrt{\sin ^{2} 3 x-\sin ^{2} 2 x}
$$ | x\in{k\pi,\frac{\pi}{2}+2k\pi,\frac{\pi}{6}+2k\pi,\frac{5\pi}{6}+2k\pi:k\in\mathbb{Z}} | 33 | 53 |
math | 3.32 For the transportation of cargo from one place to another, a certain number of trucks of the same capacity were required. Due to road damage, each truck had to be loaded with 0.5 tons less than planned, which is why 4 additional trucks of the same capacity were required. The mass of the transported cargo was no le... | 2.5 | 96 | 3 |
math | G2.3 設 $a_{1} 、 a_{2} 、 a_{3} 、 a_{4} 、 a_{5} 、 a_{6}$ 為非負整數, 並霂足 $\left\{\begin{array}{c}a_{1}+2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}+6 a_{6}=26 \\ a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=5\end{array}\right.$ 。若 $c$ 為方程系統的解的數量, 求 $c$ 的值。
Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be non-negative integers and satisfy
$$
\left... | 5 | 315 | 1 |
math | Problem 5. Solve the system of equations
$$
\left\{\begin{array}{l}
12 x^{2}+4 x y+3 y^{2}+16 x=-6 \\
4 x^{2}-12 x y+y^{2}+12 x-10 y=-7
\end{array}\right.
$$ | -\frac{3}{4},\frac{1}{2} | 77 | 14 |
math | KöMaL Survey on a total of 14 questions, with 4 possible answers for each question. The best result was 10 correct answers. What is the probability that we achieve this result by random filling out? | 0.00030205 | 47 | 10 |
math | A random number generator will always output $7$. Sam uses this random number generator once. What is the expected value of the output? | 7 | 27 | 1 |
math | 10. If real numbers $a, b, c, d, e$ satisfy $a+b+c+d+e=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$, then the sum of the maximum and minimum values of $e$ is $\qquad$ | \frac{16}{5} | 73 | 8 |
math | 32. Find the number of ordered 7-tuples of positive integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\right)$ that have both of the following properties:
(i) $a_{n}+a_{n+1}=a_{n+2}$ for $1 \leq n \leq 5$, and
(ii) $a_{6}=2005$. | 133 | 102 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\cos 5 x-\cos 3 x}{\sin ^{2} x}$ | 8 | 42 | 1 |
math | Example 4 On the line segment $A_{1} A_{n}(n \geqslant 2)$, there are $n$ points $A_{1}, A_{2}, \cdots, A_{n}$. Now, some of these points (or the empty set) are painted red, so that the colored points are not adjacent. Find the number of ways to color. | a_{n} = a_{n-1} + a_{n-2} | 83 | 18 |
math | 1. The product of three natural numbers is 240. The product of the first and second number is 60, and the product of the first and third number is 24. What are these numbers? | =6,b=10,=4 | 46 | 9 |
math | 3.16. Reduce the equation of the circle $x^{2}-2 x+y^{2}+$ $+6 y=6$ to its canonical form. Determine the coordinates of the center and the radius. | (x-1)^{2}+(y+3)^{2}=4^{2},centerO(1,-3),radiusR=4 | 45 | 30 |
math | Calculate $(k+1)^{3}-k^{3}$ then find the formula for $\sum_{k=1}^{n} k^{2}$.
## Other Counting Tools
Example 3. There are $d^{n}$ different rolls of $\mathrm{n}$ colored dice with $\mathrm{d}$ faces. | \frac{n(n+1)(2n+1)}{6} | 67 | 15 |
math | 4. Calculate $\sqrt{6+\sqrt{20}}-\sqrt{6-\sqrt{20}}$. | 2 | 24 | 1 |
math | 61 (1161). When the polynomial $2 x^{3}-5 x^{2}+7 x-8$ is multiplied by the polynomial $a x^{2}+b x+11$, the resulting polynomial does not contain either $x^{4}$ or $x^{3}$. Find the coefficients $a$ and $b$ and determine what polynomial results from the multiplication. | =4,b=10;8x^{5}-17x^{2}-3x-88 | 84 | 23 |
math | 1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all real numbers $x, y$, we have
$$
\begin{array}{l}
f(2 x y)+f(f(x+y)) \\
=x f(y)+y f(x)+f(x+y) .
\end{array}
$$ | f(x)=0\text{or}f(x)=x\text{or}f(x)=2-x | 77 | 23 |
math | 1.28 Given $x, y \in N$, find the largest $y$ value such that there exists a unique $x$ value satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15} \text {. }
$$
(Wuhan, Hubei Province, China Mathematical Summer Camp, 1987) | 112 | 85 | 3 |
math | Determine all positive integer numbers $k$ for which the numbers $k+9$ are perfect squares and the only prime factors of $k$ are 2 and 3. | \{16,27,72,216,432,2592\} | 37 | 25 |
math | 8. In $\triangle A B C$ and $\triangle A E F$, $B$ is the midpoint of $E F$, $A B=E F=1, B C=6, C A=\sqrt{33}$. If $A B \cdot A E + A C \cdot A F=2$, then the cosine value of the angle between $E F$ and $B C$ is $\qquad$ | \frac{2}{3} | 89 | 7 |
math | 1. Vasya has 8 cards with the digits 1, 2, 3, and 4 - two of each digit. He wants to form an eight-digit number such that there is one digit between the two 1s, two digits between the two 2s, three digits between the two 3s, and four digits between the two 4s. Provide any number that Vasya can form. | 41312432or23421314 | 89 | 17 |
math | 7. Let $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where
$a, b, c, d$ are real coefficients. Assume
$$
P(1)=7, P(2)=52, P(3)=97 \text {, }
$$
then $\frac{P(9)+P(-5)}{4}=$ $\qquad$ . (Vietnam) | 1202 | 97 | 4 |
math | Let $P(z)=z^3+az^2+bz+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$. | 136 | 80 | 3 |
math | 1. The product of four numbers - the roots of the equations $x^{2}+2 b x+c=0$ and $x^{2}+2 c x+b=0$, is equal to 1. It is known that the numbers $b$ and $c$ are positive. Find them. | b==1 | 65 | 3 |
math | Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$. | 4060 | 132 | 4 |
math | Example 8. Find the integral $\int \frac{d x}{\sin x}$. | \ln|\operatorname{tg}\frac{x}{2}|+C | 20 | 15 |
math | G7.1 There are $a$ zeros at the end of the product $1 \times 2 \times 3 \times \ldots \times 100$. Find $a$. | 24 | 42 | 2 |
math | What is the value of the following fraction:
$1234321234321 \cdot 2468642468641-1234321234320$
$\overline{1234321234320 \cdot 2468642468641+1234321234321}$. | 1 | 103 | 1 |
math | # Problem 4. (3 points)
Given three numbers: a five-digit number, a four-digit number, and a three-digit number. Each of them consists of identical digits (each from its own set). Can it be such that their sum is a five-digit number consisting of five different digits? | 55555+6666+888=63109 | 61 | 20 |
math | 1. Determine the integer numbers $x$ and $y$ for which $x^{2}-5^{y}=8$
Ovidiu Bădescu, RMCS Nr.40/2012 | (x,y)\in{(-3,0);(3,0)} | 45 | 15 |
math | 10.270. Find the ratio of the sum of the squares of all medians of a triangle to the sum of the squares of all its sides. | \frac{3}{4} | 34 | 7 |
math | Let's determine the distinct digits $A, B, C, D, E, F$ such that the following equalities hold:
$$
\begin{aligned}
& A B C^{2}=D A E C F B \quad \text { and } \\
& C B A^{2}=E D C A B F
\end{aligned}
$$ | A=3,B=6,C=4,D=1,E=2,F=9 | 75 | 18 |
math | 6. Let $S$ be the set of all rational numbers in the interval $\left(0, \frac{5}{8}\right)$, for the fraction $\frac{q}{p} \in S,(p, q)=1$, define the function $f\left(\frac{q}{p}\right)=\frac{q+1}{p}$. Then the number of roots of $f(x)=\frac{2}{3}$ in the set $S$ is $\qquad$ | 5 | 105 | 1 |
math | 3. The sequence of real numbers $\left(a_{n}\right)$ is defined by $a_{1}=5 ; a_{2}=19$ and for $n \geq 3$, $a_{n}=5 a_{n-1}-6 a_{n-2}$. Find $a_{2007}$. | 3^{2008}-2^{2008} | 72 | 14 |
math | $8.27 \quad \log _{2}\left(1+\log _{\frac{1}{9}} x-\log _{9} x\right)<1$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$8.27 \quad \log _{2}\left(1+\log _{\frac{1}{9}} x-\log _{9} x\right)<1$. | (\frac{1}{3},3) | 104 | 9 |
math | Example 9. Find $\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+9}-3}$. | 6 | 30 | 1 |
math | Suppose that for the positive integer $n$, $2^{n}+1$ is prime. What remainder can this prime give when divided by $240$? | 3,5,17 | 36 | 6 |
math | 8. Let the quadratic function $f(x)=a x^{2}+b x+c$, when $x=3$ it has a maximum value of 10, and the length of the segment it intercepts on the $x$-axis is 4, then $f(5)=$ $\qquad$ . | 0 | 68 | 1 |
math | Example 4. On the sides $AB$, $BC$, and $CA$ of an equilateral triangle $ABC$, there are moving points $D$, $E$, and $F$ respectively, such that $|AD| + |BE| + |CF| = |AB|$. If $|AB| = 1$, when does the area of $\triangle DEF$ reach its maximum value? What is this maximum value?
---
Translating the text as requested,... | \frac{\sqrt{3}}{12} | 107 | 11 |
math | 19. Let
$$
F(x)=\frac{1}{\left(2-x-x^{5}\right)^{2011}},
$$
and note that $F$ may be expanded as a power series so that $F(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$. Find an ordered pair of positive real numbers $(c, d)$ such that $\lim _{n \rightarrow \infty} \frac{a_{n}}{n^{d}}=c$. | (\frac{1}{6^{2011}2010!},2010) | 115 | 23 |
math | An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves?
[i]2020 CCA Math Bonanza Individual Round #1[/i] | 0 | 83 | 1 |
math | 6. (15 points) From a homogeneous straight rod, a piece of length $s=80 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod move as a result? | 40\mathrm{~} | 47 | 7 |
math | 9. (14 points) Let positive real numbers $x, y, z$ satisfy $xyz=1$. Try to find the maximum value of
$$
f(x, y, z)=(1-yz+z)(1-xz+x)(1-xy+y)
$$
and the values of $x, y, z$ at that time. | 1 | 73 | 1 |
math | 15. How many ordered pairs $(x, y)$ of positive integers $x$ and $y$ satisfy the relation $x y + 5(x + y) = 2005$? | 10 | 43 | 2 |
math | ## Problem 3
For the non-zero natural number $n$, there exists a natural number $k, k \geq 2$, and positive rational numbers $a_{1}, a_{2}, \ldots, a_{k}$ such that $a_{1}+a_{2}+\ldots+a_{k}=a_{1} a_{2} \cdot \ldots \cdot a_{k}=n$. Determine all possible values of the number $n$. | n\in\mathbb{N}^{*}-{1,2,3,5} | 100 | 20 |
math | 12. (20 points) Find the smallest positive integer $n$ such that
$$
[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4 n+2016}]
$$ | 254016 | 44 | 6 |
math | 4.1. Solve the equation $\left(x^{2}-2 x+4\right)^{x^{2}-2 x+3}=625$. In the answer, specify the sum of the squares of all its roots. If there are no roots, put 0. | 6 | 59 | 1 |
math | 1. Solve the equation $\left|\begin{array}{ccc}x+1 & 2 x & 1 \\ x & 3 x-2 & 2 x \\ 1 & 1 & x\end{array}\right|=0$ ; | x_1=1, x_2, x_3=-2 \pm \sqrt{6} | 54 | 22 |
math | 6.087. $\left\{\begin{array}{l}x^{4}-y^{4}=15 \\ x^{3} y-x y^{3}=6 .\end{array}\right.$ | (-2,-1),(2,1) | 46 | 9 |
math | 8-4. In a giraffe beauty contest, two giraffes, Tall and Spotted, made it to the final. 105 voters are divided into 5 districts, each district is divided into 7 precincts, and each precinct has 3 voters. The majority of voters on each precinct choose the winner of their precinct; in a district, the giraffe that wins th... | 24 | 128 | 2 |
math | 11. Let $S=\{1,2,3, \cdots \cdots n\}$, and $A$ be a subset of $S$. Arrange the elements of $A$ in descending order, then alternately subtract or add the subsequent numbers starting from the largest number to get the alternating sum of $A$. For example, if $A=\{1,4,9,6,2\}$, rearranging it gives $\{9,6,4,2,1\}$, and it... | n\cdot2^{n-1} | 134 | 9 |
math | 6.3. The hunter told a friend that he saw a wolf with a one-meter tail in the forest. That friend told another friend that a wolf with a two-meter tail had been seen in the forest. Passing on the news further, ordinary people doubled the length of the tail, while cowards tripled it. As a result, the 10th channel report... | 5 | 119 | 1 |
math | 2. For an integer $x$, the following holds
$$
|\ldots||| x-1|-10|-10^{2}\left|-\ldots-10^{2006}\right|=10^{2007}
$$
Find the hundredth digit of the number $|x|$. | 1 | 70 | 1 |
math | 1. All graduates of the mathematics school took the Unified State Exam (USE) in mathematics and physical education. Each student's result in mathematics turned out to be equal to the sum of the results of all other students in physical education. How many graduates are there in the school if the total number of points ... | 51 | 79 | 2 |
math | ## Task 6
How many minutes are $70 \cdot 30$ min? Convert to hours.
How many pfennigs are $87 \cdot 20 \mathrm{Pf}$? Convert to marks. | 2100 | 49 | 4 |
math | 12. On the Cartesian plane, the number of lattice points (i.e., points with both integer coordinates) on the circumference of a circle centered at $(199,0)$ with a radius of 199 is $\qquad$ . | 4 | 52 | 1 |
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