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 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\frac{\arcsin x}{x}\right)^{\frac{2}{x+5}}$ | 1 | 44 | 1 |
math | 8. Let $M=\{1,2, \cdots, 2017\}$ be the set of the first 2017 positive integers. If one element is removed from $M$, and the sum of the remaining elements in $M$ is exactly a perfect square, then the removed element is $\qquad$ . | 1677 | 72 | 4 |
math | Three. (25 points) Given that $a$, $b$, and $c$ are positive integers, and $a < b < c$. If
$$
a b c \mid (a b - 1)(b c - 1)(c a - 1),
$$
then, can three segments of lengths $\sqrt{a}$, $\sqrt{b}$, and $\sqrt{c}$ form a triangle? If so, find the area of the triangle; if not, explain the reason. | \frac{\sqrt{6}}{2} | 107 | 10 |
math | 11. Let $f(x)=\max \left|x^{3}-a x^{2}-b x-c\right|(1 \leqslant x \leqslant 3)$, find the minimum value of $f(x)$ when $a, b, c$ take all real numbers. | \frac{1}{4} | 65 | 7 |
math | 29. Given that the real numbers $x, y$ and $z$ satisfies the condition $x+y+z=3$, find the maximum value of $f(x y z)=\sqrt{2 x+13}+\sqrt[3]{3 y+5}+\sqrt[4]{8 z+12}$. | 8 | 69 | 1 |
math | 3. Determine all pairs $(m, n)$ of positive integers for which the number $4(m n+1)$ is divisible by the number $(m+n)^{2}$. | (1,1),(n+2,n),(,+2) | 37 | 13 |
math | Let's determine $k$ such that the expression
$$
a x^{2}+2 b x y+c y^{2}-k\left(x^{2}+y^{2}\right)
$$
is a perfect square. | \frac{+}{2}\\frac{1}{2}\sqrt{(-)^{2}+4b^{2}} | 49 | 26 |
math | Task B-2.4. Let $f(x)=x^{2}+b x+c, b, c \in \mathbb{R}$. If $f(0)+f(1)=\frac{1}{2}$, calculate $f\left(\frac{1}{2}\right)$. | 0 | 66 | 1 |
math | ## Task 1 - 200621
An airplane flying from A to B at a constant speed was still $2100 \mathrm{~km}$ away from B at 10:05 AM, and only $600 \mathrm{~km}$ away at 11:20 AM.
At what time will it arrive in B if it continues to fly at the same speed? | 11:50 | 89 | 5 |
math | 3. Find all functions $g: \mathbf{N}_{+} \rightarrow \mathbf{N}_{+}$, such that for all $m, n \in \mathbf{N}_{+}, (g(m) + n)(m + g(n))$ is a perfect square.
(US Proposal) | g(n)=n+c | 67 | 5 |
math | 8. Solve the system $\left\{\begin{array}{l}\log _{4} x-\log _{2} y=0 \\ x^{2}-5 y^{2}+4=0 .\end{array}\right.$ | {1;1},{4;2} | 52 | 9 |
math | Find all integer triples $(a,b,c)$ with $a>0>b>c$ whose sum equal $0$ such that the number
$$N=2017-a^3b-b^3c-c^3a$$ is a perfect square of an integer. | (36, -12, -24) | 56 | 12 |
math | Determine all positive integers $n \geqslant 3$ such that the inequality
$$
a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n-1} a_{n}+a_{n} a_{1} \leqslant 0
$$
holds for all real numbers $a_{1}, a_{2}, \ldots, a_{n}$ which satisfy $a_{1}+\cdots+a_{n}=0$. | n=3n=4 | 109 | 6 |
math | Example 8 Find the real root of the equation $\sqrt{x+2 \sqrt{x+2 \sqrt{x+2+\cdots+2 \sqrt{x+2 \sqrt{3 x}}}}}=x$.
| x_{1}=3,x_{2}=0 | 46 | 10 |
math | 10. Three points on the same straight line $A(x, 5), B(-2, y), C(1,1)$ satisfy $|B C|=2|A C|$, then $x+y=$ | -\frac{9}{2} | 46 | 7 |
math | (7) The function $f(x)=\frac{\sin \left(x+45^{\circ}\right)}{\sin \left(x+60^{\circ}\right)}, x \in\left[0^{\circ}, 90^{\circ}\right]$, then the product of the maximum and minimum values of $f(x)$ is . $\qquad$ | \frac{2\sqrt{3}}{3} | 80 | 12 |
math | 6.212. $\left\{\begin{array}{l}\frac{3}{uv}+\frac{15}{vw}=2, \\ \frac{15}{vw}+\frac{5}{wu}=2, \\ \frac{5}{wu}+\frac{3}{uv}=2 .\end{array}\right.$ | (-1,-3,-5),(1,3,5) | 74 | 13 |
math | Four, $E, F$ are
on the sides $B C$ and $C D$
of rectangle $A B C D$,
if the areas of $\triangle C E F$,
$\triangle A B E$, $\triangle A D F$
are 3,
4, 5 respectively. Find the area $S$ of $\triangle A E F$. | 8 | 76 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 2} \frac{\ln (x-\sqrt[3]{2 x-3})}{\sin \left(\frac{\pi x}{2}\right)-\sin ((x-1) \pi)}
$$ | \frac{2}{3\pi} | 65 | 9 |
math | 9. If the length of a rectangle is reduced by 3 cm and the width is increased by 2 cm, a square with the same area as the original rectangle is obtained, then the perimeter of the rectangle is $\qquad$ cm. | 26 | 50 | 2 |
math | $(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$ | f(4) = 2, f(5) = 14, f(6) = 90 | 98 | 26 |
math | Find all integers $b, n \in \mathbb{N}$ such that $b^{2}-8=3^{n}$. | b=3,n=0 | 29 | 6 |
math | 7.072. $10^{\frac{2}{x}}+25^{\frac{1}{x}}=4.25 \cdot 50^{\frac{1}{x}}$. | x_{1}=-\frac{1}{2};x_{2}=\frac{1}{2} | 47 | 23 |
math | Three, (15 points) Given real numbers $x, y$ satisfy the conditions:
$$
\left\{\begin{array}{l}
x+2 y>0, \\
x-2 y>0, \\
(x+2 y)(x-2 y)=4 .
\end{array}\right.
$$
Find the minimum value of $|x|-|y|$. | \sqrt{3} | 81 | 5 |
math | What is the measure, in degrees, of the smallest positive angle $x$ for which $4^{\sin ^{2} x} \cdot 2^{\cos ^{2} x}=2 \sqrt[4]{8}$ ? | 60 | 51 | 2 |
math | 1. (15 points) In a bookstore, there is a rule for "summing" discounts: "if different types of discounts apply to an item, they are applied sequentially one after another." For example, if two discounts A% and B% apply to an item, the first discount is applied to the original price, and the second discount is applied t... | 50 | 178 | 2 |
math | 2. The front tire of a bicycle wears out after $250 \mathrm{~km}$, while the rear tire wears out after $150 \mathrm{~km}$. After how many kilometers should the tires be swapped so that they wear out simultaneously? | 93.75\mathrm{~} | 56 | 10 |
math | 9.4. From Zlatoust to Miass, a "GAZ", a "MAZ", and a "KAMAZ" set off simultaneously. The "KAMAZ", having reached Miass, immediately turned back and met the "MAZ" 18 km from Miass, and the "GAZ" - 25 km from Miass. The "MAZ", having reached Miass, also immediately turned back and met the "GAZ" 8 km from Miass. What is t... | 60 | 117 | 2 |
math | Consider the numbers arranged in the following way:
\[\begin{array}{ccccccc} 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 2 & 5 & 9 & 14 & 20 & \cdots & \cdots \\ 4 & 8 & 13 & 19 & \cdots & \cdots & \cdots \\ 7 & 12 & 18 & \cdots & \cdots & \cdots & \cdots \\ 11 & 17 & \cdots & \cdots & \cdots & \cdots & \cdots \\ 16 & \cdots ... | (5, 196) | 237 | 8 |
math | 10.401 Find the radius of the circle if the area of the circle is $Q$ square units greater than the area of the inscribed regular dodecagon. | \sqrt{\frac{Q}{\pi-3}} | 38 | 12 |
math | For which values of the parameters $a$ and $b$ is the solution of the following equation real? Provide some pairs of $a, b$ values to numerically illustrate the different cases.
$$
\sqrt{2 a+b+2 x}+\sqrt{10 a+9 b-6 x}=2 \sqrt{2 a+b-2 x}
$$ | \begin{pmatrix}-<b\leqq-\frac{8}{9}<0\text{case}&\\sqrt{(+b)}\\-\frac{8}{9}<b\leqq0\text{case}&-\sqrt{(+b)}\end{pmatrix} | 77 | 60 |
math | 380. Two bodies simultaneously started linear motion from a certain point in the same direction with speeds $v_{1}=$ $=\left(6 t^{2}+4 t\right) \mathrm{m} /$ s and $v_{2}=4 t \mathrm{m} / \mathrm{s}$. After how many seconds will the distance between them be 250 m? | 5 | 84 | 1 |
math | Problem 1. A train consists of 20 cars, numbered from 1 to 20, starting from the beginning of the train. Some of the cars are postal cars. It is known that
- the total number of postal cars is even;
- the number of the nearest postal car to the beginning of the train is equal to the total number of postal cars;
- the ... | 4,5,15,16 | 119 | 9 |
math | 5. Find the number of 9-digit numbers in which each digit from 1 to 9 appears exactly once, the digits 1, 2, 3, 4, 5 are arranged in ascending order, and the digit 6 appears before the digit 1 (for example, 916238457). | 504 | 72 | 3 |
math | Princess Telassim cut a rectangular sheet of paper into 9 squares with sides of 1, 4, 7, 8, 9, 10, 14, 15, and 18 centimeters.
a) What was the area of the sheet before it was cut?
b) What were the dimensions of the sheet before it was cut?
c) Princess Telassim needs to reassemble the sheet. Help her by showing, with... | =32,b=33 | 109 | 7 |
math | Problem 4.5. On the meadow, there were 12 cows. The shepherds brought a flock of sheep. There were more sheep than the cows' ears, but fewer than the cows' legs. How many sheep were there if there were 12 times more sheep than shepherds? | 36 | 65 | 2 |
math | 7. For a regular quadrilateral pyramid $P-ABCD$ with base and lateral edge lengths all being $a$, $M$ and $N$ are moving points on the base edges $CD$ and $CB$ respectively, and $CM = CN$. When the volume of the tetrahedron $P-AMN$ is maximized, the angle between line $PA$ and plane $PMN$ is $\qquad$ | \frac{\pi}{4} | 91 | 7 |
math | 8. Given $x, y \in [0,+\infty)$. Then the minimum value of $x^{3}+y^{3}-5 x y$ is $\qquad$ . | -\frac{125}{27} | 42 | 10 |
math | 6. Given real numbers $x, y$ satisfy $x^{2}-x y-2 y^{2}=1$. Then the minimum value of $2 x^{2}+y^{2}$ is $\qquad$ .
| \frac{2+2 \sqrt{3}}{3} | 48 | 14 |
math | Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$? | 171 | 78 | 3 |
math | [ Extreme properties. Problems on maximum and minimum.] Concerning circles
Chord $A B$ is seen from the center of a circle of radius $R$ at an angle of $120^{\circ}$. Find the radii of the largest circles inscribed in the segments into which chord $A B$ divides the given circle. | \frac{1}{4}R\frac{3}{4}R | 71 | 16 |
math | If $\frac{8}{24}=\frac{4}{x+3}$, what is the value of $x$ ? | 9 | 28 | 1 |
math | Example 5 If real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ satisfy the system of equations
$$
\left\{\begin{array}{l}
x_{1} x_{2}+x_{1} x_{3}+x_{1} x_{4}+x_{1} x_{5}=-1, \\
x_{2} x_{1}+x_{2} x_{3}+x_{2} x_{4}+x_{2} x_{5}=-1, \\
x_{3} x_{1}+x_{3} x_{2}+x_{3} x_{4}+x_{3} x_{5}=-1, \\
x_{4} x_{1}+x_{4} x_{2}+x_{4} x_{3... | \pm \sqrt{2}, \pm \frac{\sqrt{2}}{2} | 274 | 19 |
math | Example 4. Find the radius of convergence of the power series
$$
\sum_{n=0}^{\infty}(1+i)^{n} z^{n}
$$ | \frac{1}{\sqrt{2}} | 39 | 10 |
math | 3. The maximum value of the function
$y=\frac{\sin x \cos x}{1+\sin x+\cos x}$ is $\qquad$ . | \frac{\sqrt{2}-1}{2} | 34 | 11 |
math | Given that there are only three positive integers between the fractions $\frac{112}{19}$ and $\frac{112+x}{19+x}$. Find the sum of all possible integer values of $x$. | 2310 | 47 | 4 |
math | 3. [6 points] On the plane $O x y$, the equation $2 a^{2}-2 a x-6 a y+x^{2}+2 x y+5 y^{2}=0$ defines the coordinates of point $A$, and the equation $a x^{2}+4 a^{2} x-a y+4 a^{3}+2=0$ defines a parabola with vertex at point $B$. Find all values of the parameter $a$ for which points $A$ and $B$ lie on the same side of t... | (-2;0)\cup(\frac{1}{2};3) | 143 | 15 |
math | Condition of the problem
Find the $n$-th order derivative.
$y=2^{k x}$ | 2^{kx}\cdotk^{n}\ln^{n}2 | 23 | 15 |
math | 3. The fraction $\frac{2 x+3}{2 x-7}$ is given, with $x \in \mathbb{N}, x \geq 4$. Let $x_{1}, x_{2}, \ldots, x_{1000}$ be the first 1000 natural numbers for which the fraction can be simplified.
a) Calculate $x_{1000}$.
b) Show that $x_{p}+x_{q}-x_{p+q}=1$ for any $p, q \in\{1,2, \ldots, 1000\}$.
## Note: All subje... | x_{1000}=5001 | 237 | 11 |
math | 9. The function $f(n)$ is defined for all positive integer $n$ and take on non-negative integer values such that $f(2)=0, f(3)>0$ and $f(9999)=3333$. Also, for all $m, n$,
$$
f(m+n)-f(m)-f(n)=0 \quad \text { or } \quad 1 .
$$
Determine $f(2005)$. | 668 | 101 | 3 |
math | \section*{Problem 1 - 071221}
Let \(x_{k}\) and \(y_{k}\) be integers that satisfy the conditions \(0 \leq x_{k} \leq 2\) and \(0 \leq y_{k} \leq 2\). a) Determine the number of all (non-degenerate) triangles with vertices \(P_{k}\left(x_{k} ; y_{k}\right)\), where \(x_{k}, y_{k}\) are the Cartesian coordinates of \(\... | 76 | 185 | 2 |
math |
Problem 10.1. Find the least natural number $a$ such that the equation $\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0$ has a root.
| 6 | 84 | 1 |
math | Example 2 If $a>0, b>0, c>0$, compare $a^{3}+b^{3}+c^{3}$ and $a^{2} b+b^{2} c+c^{2} a$. | a^{3}+b^{3}+c^{3} \geq a^{2} b+b^{2} c+c^{2} a | 51 | 32 |
math | How many ways are there to shuffle a deck of 32 cards? | 32! | 15 | 3 |
math | One, (20 points) Given real numbers $a, b, c$ satisfy the inequalities
$$
|a| \geqslant|b+c|,|b| \geqslant|c+a|,|r| \geqslant|a+b| .
$$
Find the value of $a+1+r$.
| a+b+c=0 | 75 | 5 |
math | 5. Find the largest three-digit number that is divisible by the sum of its digits and in which the first digit matches the third, but does not match the second. | 828 | 34 | 3 |
math | In convex quadrilateral $A B C D$,
\[
\begin{array}{l}
\angle A B D=\angle C B D=45^{\circ}, \\
\angle A D B=30^{\circ}, \angle C D B=15^{\circ} .
\end{array}
\]
Find the ratio of the sides of quadrilateral $A B C D$. | \sqrt{3}: 1:(1+\sqrt{3}): \sqrt{6} | 85 | 19 |
math | Example 3: A certain area currently has 10,000 hectares of arable land. It is planned that in 10 years, the grain yield per unit area will increase by $22\%$, and the per capita grain possession will increase by $10\%$. If the annual population growth rate is $1\%$, try to find the maximum number of hectares by which t... | 4 | 95 | 1 |
math | 73. Find three numbers such that the sums of all three and each pair are squares. | I=80,II=320,III=41 | 19 | 15 |
math | Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$. | (8, 3), (2, 9), (-7, 9), (-7, 3), (2, -6), (8, -6) | 50 | 36 |
math | 3. Ivan collects 2-kuna and 5-kuna coins. He has collected 39 coins and now has 144 kuna. How many 2-kuna coins and how many 5-kuna coins does he have? | 22 | 51 | 2 |
math | 10. Let $x$ and $y$ be real numbers, and satisfy $\left\{\begin{array}{l}(x-1)^{3}+2003(x-1)=-1 \\ (y-1)^{3}+2003(y-1)=1,\end{array} x+y=\right.$ | 2 | 74 | 1 |
math | [Example 3.5.6] Given that $p$ is a prime number, $r$ is the remainder when $p$ is divided by 210. If $r$ is a composite number and can be expressed as the sum of two perfect squares, find $r$.
保留源文本的换行和格式，直接输出翻译结果。 | 169 | 76 | 3 |
math | Example 5 Color each vertex of a 2003-gon with one of three colors: red, blue, or green, such that adjacent vertices have different colors. How many such colorings are there? ${ }^{[3]}$
(2002-2003, Hungarian Mathematical Olympiad) | 2^{2003}-2 | 67 | 8 |
math | A22 (17-6, UK) Determine the polynomial $P$ that satisfies the following conditions:
(a) $P$ is a homogeneous polynomial of degree $n$ in $x, y$, i.e., for all real numbers $t, x, y$, we have $P(t x, t y)=t^{n} P(x, y)$, where $n$ is a positive integer;
(b) For all real numbers $a, b, c$, we have $P(a+b, c)+P(b+c, a)+P... | P(x,y)=(x-2y)(x+y)^{n-1} | 136 | 17 |
math | $4 \cdot 229$ A decimal natural number $a$ consists of $n$ identical digits $x$, and number $b$ consists of $n$ identical digits $y$, number $c$ consists of $2n$ identical digits $z$. For any $n \geqslant 2$, find the digits $x, y, z$ that make $a^{2}+b=c$ true. | 3,2,1,n | 91 | 6 |
math | ## 17. Paris - Deauville
Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They leave together and arrive in Deauville simultaneously. However, Monsieur Dubois spent one-third of the time his wife continued driving on stops, while Madame Dubois ... | \frac{9}{8}r^{\} | 102 | 11 |
math | Find the smallest positive integer $n$ such that there exists a positive integer $k$ for which $\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13}$ holds. | 15 | 45 | 2 |
math | $\mathbf{F 1 8}$ (40-6, Japan) Determine all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, where $\mathbf{R}$ is the set of real numbers, such that for all $x, y \in \mathbf{R}$, the following equation always holds:
$$
f(x-f(y))=f(f(y))+x f(y)+f(x)-1
$$ | f(x)=1-\frac{x^{2}}{2} | 98 | 13 |
math | Find two simple solutions to the equation: $(x-1)^{2}+(x+1)^{2}=y^{2}+1$, where $x$ and $y$ are non-negative integers.
Find three integers $a, b$, and $c$ such that if $(x, y)$ is a solution to the equation, then $(a x+b y, c x+a y)$ is also a solution.
Deduce that the equation has infinitely many solutions. | (3x+2y,4x+3y) | 99 | 13 |
math | 3. (7 points) Calculate the sum
$$
1 \cdot 2+2 \cdot 3+3 \cdot 4+\ldots+2015 \cdot 2016
$$
Answer: 2731179360 | 2731179360 | 60 | 10 |
math | 1. If you open the cold water tap, the bathtub will fill up in 10 minutes, if you open the hot water tap, it will take 15 minutes. If you pull the plug, the bathtub will completely drain in 12 minutes. How long will it take to fill the bathtub if you open both taps and pull the plug?
# | 12 | 74 | 2 |
math | Maria Petrovna is walking down the road at a speed of 4 km/h. Seeing a stump, she sits on it and rests for the same whole number of minutes. Mikhail Potapovich is walking down the same road at a speed of 5 km/h, but sits on each stump twice as long as Maria Petrovna. They both set out and arrived at the same time. The ... | 1,3,11,33 | 102 | 9 |
math | In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$. | \frac{\sqrt{2}}{4} | 47 | 10 |
math | Example 11. Find the number of consecutive zeros at the end of 1987!.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated text:
Example 11. Find the number of consecutive zeros at the end of 1987!.
Note: The note at the end is not part of the original text and is provided for context. | 494 | 93 | 3 |
math | 2. Simplify $(\mathrm{i}+1)^{2016}+(\mathrm{i}-1)^{2016}=$ | 2^{1009} | 31 | 7 |
math | Determine all positive integers that cannot be written as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a, b$ positive and integral. | 1 \text{ and the numbers of the form } 2^m + 2 \text{ with } m \geq 0 | 38 | 29 |
math | 11.17 For what values of $p$ does the equation
$x^{2}-\left(2^{p}-1\right) x-3\left(4^{p-1}-2^{p-2}\right)=0$
have equal roots? | p=-2p=0 | 57 | 6 |
math | 5. For which integers $n \geq 2$ can we arrange the numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \leq k \leq n$ the sum of the first $k$ numbers in the row is divisible by $k$ ?
Answer: This is only possible for $n=3$. | 3 | 80 | 1 |
math | Bakayev E.V.
In each cell of a $1000 \times 1000$ square, a number is inscribed such that in any rectangle of area $s$ that does not extend beyond the square and whose sides lie along the cell boundaries, the sum of the numbers is the same. For which $s$ will the numbers in all cells necessarily be the same? | 1 | 83 | 1 |
math | 5. (10 points) At the end of the term, Teacher XiXi bought the same number of ballpoint pens, fountain pens, and erasers to distribute to the students in the class. After giving each student 2 fountain pens, 3 ballpoint pens, and 4 erasers, she found that there were 48 ballpoint pens left, and the remaining number of f... | 16 | 112 | 2 |
math | $7 \cdot 75$ A state issues license plates consisting of 6 digits (made up of the digits 0-9), and stipulates that any two license plates must differ in at least two digits (thus, the plate numbers 027592 and 020592 cannot both be used). Try to find the maximum number of license plates possible? | 10^5 | 81 | 4 |
math | 295. Find the coefficient of $x^{50}$ after expanding the brackets and combining like terms in the expressions:
a) $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+\ldots+x^{1000}$;
b) $(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+1000(1+x)^{1000}$. | C_{1001}^{50} | 115 | 11 |
math | For $ n \in \mathbb{N}$, let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$. Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$ | \frac{8}{3} | 73 | 7 |
math | In a given circle, let $AB$ be a side of an inscribed square. Extend $AB$ to $C$ so that $BC = AB$. If we connect $C$ with the center $O$ of the circle and $OC$ intersects the circle at $D$, then $CD$ is equal to twice the side of a regular decagon inscribed in the circle. | r(\sqrt{5}-1) | 80 | 8 |
math | 7,8
In the Russian Football Championship, 16 teams participate. Each team plays 2 matches against each of the others.
a) How many matches should "Uralan" play in a season?
b) How many matches are played in total in one season? | )30;b)240 | 57 | 8 |
math | SG. 3 Let $P$ and $P+2$ be both prime numbers satisfying $P(P+2) \leq 2007$. If $S$ represents the sum of such possible values of $P$, find the value of $S$. | 106 | 56 | 3 |
math | 1. How many digits does the number $\left[1.125 \cdot\left(10^{9}\right)^{5}\right]:\left[\frac{3}{32} \cdot 10^{-4}\right]$ have? | 51 | 55 | 2 |
math | 6.51 Find all positive integers $n>1$, such that $\frac{2^{n}+1}{n^{2}}$ is an integer.
(31st International Mathematical Olympiad, 1990) | 3 | 49 | 1 |
math | 2.1. Solve the equation $\operatorname{GCD}(a, b)+\operatorname{LCM}(a, b)=a+b+2$ in natural numbers. | (3,2),(2,3) | 38 | 9 |
math | 21. $y=\frac{3}{x^{2}-4}$. | (-\infty,-2)\cup(-2,2)\cup(2,\infty) | 17 | 20 |
math | 7. If (1) $a, b, c, d$ all belong to $\{1,2,3,4\}$; (2) $a \neq b, b \neq c, c \neq d, d \neq a$; (3) $a$ is the smallest value among $a, b, c, d$, then the number of different four-digit numbers $\overline{a b c d}$ that can be formed is $\qquad$ . | 28 | 106 | 2 |
math | Let's determine $a, b, c$, and $d$ such that the fraction
$$
\frac{a x^{2}-b x+c}{x^{2}-d x-a}
$$
is maximal at $x=2$ and minimal at $x=5$, and that the maximum value of the fraction is 1, and the minimum value is 2.5. | =3,b=10,=5,=2 | 81 | 12 |
math | Let $a$ ,$b$ and $c$ be distinct real numbers.
$a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b} $
$b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + ... | \frac{3}{2} | 257 | 7 |
math | 3. Let $a, b, c \in\left[\frac{1}{2}, 1\right]$, and let $s=\frac{a+b}{1+c}+\frac{b+c}{1+a}+\frac{c+a}{1+b}$, then the range of $s$ is $\qquad$ | \in[2,3] | 70 | 7 |
math | 9.1. In the morning, a dandelion blooms, it flowers yellow for three days, on the fourth morning it turns white, and by the evening of the fifth day, it withers. On Monday afternoon, there were 20 yellow and 14 white dandelions on the meadow, and on Wednesday - 15 yellow and 11 white. How many white dandelions will the... | 6 | 95 | 1 |
math | Find all $f: R\rightarrow R$ such that
(i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite
(ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$ | f(x) = x | 62 | 6 |
math | # Problem 2. (2 points)
The sum of the sines of five angles from the interval $\left[0 ; \frac{\pi}{2}\right]$ is 3. What are the greatest and least integer values that the sum of their cosines can take?
# | 2;4 | 58 | 3 |
math | 6. Given point $A(0,1)$, curve $C: y=\log _{a} x$ always passes through point $B$. If $P$ is a moving point on curve $C$, and $\overrightarrow{A B} \cdot \overrightarrow{A P}$ has a minimum value of 2, then the real number $a=$ $\qquad$. | e | 81 | 1 |
math | A rectangular wooden block has a square top and bottom, its volume is $576$, and the surface area of its vertical sides is $384$. Find the sum of the lengths of all twelve of the edges of the block. | 112 | 49 | 3 |
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