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 | 14. For 155 boxes containing balls of three colors: red, yellow, and blue, there are three classification methods: for each color, boxes with the same number of balls of that color are grouped into one class. If all natural numbers from 1 to 30 are the number of boxes in some class of a classification, then, 1) what is... | 30 | 116 | 2 |
math | Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that t... | 53 | 103 | 2 |
math | How many numbers less than four digits (from 0 to 9999) are not divisible by 3, 5, or 7? | 4571 | 32 | 4 |
math | 4. The function $f(x)$ that satisfies the equation
$$
f(x)+(x-2) f(1)+3 f(0)=x^{3}+2 \quad (x \in \mathbf{R})
$$
is $f(x)=$ . $\qquad$ | x^{3}-x+1 | 62 | 7 |
math | 28. From the six digits $0,1,2, 3, 4,5$, select 2 odd numbers and 2 even numbers to form a 4-digit number without repeated digits. What is the probability of forming a 4-digit even number? | \frac{8}{15} | 56 | 8 |
math | 5. Find all functions $f: \mathbf{N}_{+} \rightarrow \mathbf{N}_{+}$, such that for all positive integers $a, b$, there exists a non-degenerate triangle with side lengths $a, f(b), f(b+f(a)-1)$ (a triangle is called non-degenerate if its three vertices are not collinear).
(France) | f(n) = n | 81 | 5 |
math | Solve the following equation:
$$
\sqrt{x-2}+\sqrt{3-x}=x^{2}-5 x+7
$$ | 2or3 | 30 | 3 |
math | ## Task Condition
Calculate approximately using the differential.
$$
y=\sqrt[3]{x}, x=26.46
$$ | 2.98 | 29 | 4 |
math | Problem 6.3. Vitya and his mother left home at the same time and walked in opposite directions at the same speed: Vitya - to school, and his mother - to work. After 10 minutes, Vitya realized he didn't have the keys to the house, and he would return from school earlier than his mother, so he started to catch up with he... | 5 | 110 | 1 |
math | 3. Find the values of the following expressions:
(1) $\sin 10^{\circ} \cdot \sin 30^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ}$;
(2) $\sin ^{2} 20^{\circ}+\cos ^{2} 80^{\circ}+\sqrt{3} \sin 20^{\circ} \cdot \cos 80^{\circ}$;
(3) $\cos ^{2} A+\cos ^{2}\left(60^{\circ}-A\right)+\cos ^{2}\left(60^{\circ}+A\right)... | \frac{1}{16},\frac{1}{4},\frac{3}{2},\frac{1}{128} | 257 | 31 |
math | SUBIECTUL II
a) Find the number of natural divisors of the number $15^{51}$ that are multiples of the number $225^{20}$.
b) Show that the number $A=\frac{21^{n}+23^{n}-2^{2 n}+2^{n+1} \cdot 3^{2}}{38}$ is a natural number for any non-zero natural number $n$.
Supliment Gazeta Matematică $3 / 2013$ | 144 | 116 | 3 |
math | Problem 6. The probability of the first shooter hitting the target is $p_{1}$, and the second shooter is $p_{2}$. The shooters fired simultaneously. What is the probability that:
a) both shooters hit the target;
b) only one hits;
c) at least one hits? | )p_{1}p_{2},\,b)p_{1}+p_{2}-2p_{1}p_{2},\,)p_{1}+p_{2}-p_{1}p_{2} | 63 | 48 |
math | 2. Let $A B C D$ be a regular tetrahedron with side length 2. The plane parallel to edges $A B$ and $C D$ and lying halfway between them cuts $A B C D$ into two pieces. Find the surface area of one of these pieces. | 1+2\sqrt{3} | 62 | 8 |
math | 3.074. $\sin ^{2}(\alpha+2 \beta)+\sin ^{2}(\alpha-2 \beta)-1$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
3.074. $\sin ^{2}(\alpha+2 \beta)+\sin ^{2}(\alpha-2 \beta)-1$. | -\cos2\alpha\cos4\beta | 93 | 10 |
math | Compute the sum $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{1-3 x_{i}+3 x_{i}^{2}}$ for $x_{i}=\frac{i}{101}$.
Answer: $S=51$. | 51 | 67 | 2 |
math | 7. Let $\triangle A B C$ have three interior angles $\angle A, \angle B, \angle C$ with corresponding side lengths $a, b, c$. If $a<b<c$, and
\[
\left\{\begin{array}{l}
\frac{b}{a}=\frac{\left|b^{2}+c^{2}-a^{2}\right|}{b c} \\
\frac{c}{b}=\frac{\left|c^{2}+a^{2}-b^{2}\right|}{c a} \\
\frac{a}{c}=\frac{\left|a^{2}+b^{2... | 1:2:4 | 186 | 5 |
math | 1. When purchasing goods for an amount of no less than 1000 rubles, the store provides a discount on subsequent purchases of $50 \%$. Having 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of 300 rubles per kg, and sugar - at a price... | 1200 | 127 | 4 |
math | Let $x$, $y$, and $z$ be real numbers such that $x+y+z=20$ and $x+2y+3z=16$. What is the value of $x+3y+5z$?
[i]Proposed by James Lin[/i] | 12 | 62 | 2 |
math | 2. Determine the remainder of the division of the number $3^{100}$ by 13. | 3 | 23 | 1 |
math | M2. Two real numbers $x$ and $y$ satisfy the equation $x^{2}+y^{2}+3 x y=2015$.
What is the maximum possible value of $x y$ ? | 403 | 49 | 3 |
math | Squares $BAXX^{'}$ and $CAYY^{'}$ are drawn in the exterior of a triangle $ABC$ with $AB = AC$. Let $D$ be the midpoint of $BC$, and $E$ and $F$ be the feet of the perpendiculars from an arbitrary point $K$ on the segment $BC$ to $BY$ and $CX$, respectively.
$(a)$ Prove that $DE = DF$ .
$(b)$ Find the locus of the midp... | y = -x + \frac{a}{2} | 109 | 13 |
math | 13. Place 1996 indistinguishable balls into 10 different boxes, such that the $i$-th box contains at least $i$ balls $(i=1,2, \cdots, 10)$. How many different ways are there to do this? | C_{1950}^{9} | 62 | 10 |
math | XI OM - I - Task 3
Each side of a triangle with a given area $ S $ is divided into three equal parts, and the points of division are connected by segments, skipping one point to form two triangles. Calculate the area of the hexagon that is the common part of these triangles. | \frac{2}{9}S | 62 | 8 |
math | 一、(20 points) On the square, 5 loudspeakers are to be installed, divided into two groups. The first group has 2 loudspeakers installed together, and the second group has 3 loudspeakers installed together. The distance between the two groups is 50 meters. Where should one stand to hear the sound from both groups equally... | 22.5 | 77 | 4 |
math | 1. Let $a_{1}, a_{2}, \cdots, a_{2015}$ be a sequence of numbers taking values from $-1, 0, 1$, satisfying
$$
\sum_{i=1}^{2015} a_{i}=5 \text {, and } \sum_{i=1}^{2015}\left(a_{i}+1\right)^{2}=3040,
$$
where $\sum_{i=1}^{n} a_{i}$ denotes the sum of $a_{1}, a_{2}, \cdots, a_{n}$.
Then the number of 1's in this sequenc... | 510 | 154 | 3 |
math | 2. For real numbers $x$ and $y$, the following holds:
$$
x^{3}+x^{2}+x y+x+y+2=0 \quad \text { and } \quad y^{3}-y^{2}+3 y-x=0
$$
Determine the value of the expression $x-y$. | -1 | 73 | 2 |
math | 2nd Chinese 1987 Problem B3 A set of distinct positive integers has sum 1987. What is the maximum possible value for three times the total number of integers plus the number of odd integers? | 221 | 46 | 3 |
math | A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, comp... | 2716 | 120 | 4 |
math | Example 1. The following numbers are all approximate numbers obtained by rounding. Find their absolute error bounds and relative error bounds:
$$
\begin{array}{l}
\text { (1) } \mathrm{a}_{1} \approx 12.5, \quad(2) \mathrm{a}_{2} \approx 1.25 \times 10^{4}, \\
\text { (3) } \mathrm{a}_{3} \approx 0.0125
\end{array}
$$ | 0.4\% | 114 | 5 |
math | 8. (12) Find all natural numbers $n$ that have a divisor $d$ such that $n^{2}+d^{2}$ is divisible by $n d+1$.
## Senior League | k^{3},k\in\mathbb{N} | 45 | 13 |
math | Example 3 A right-angled triangle with integer side lengths, if the lengths of its two legs are the roots of the equation
$$
x^{2}-(k+2) x+4 k=0
$$
find the value of $k$ and determine the lengths of the three sides of the right-angled triangle. ${ }^{[2]}$
(2010, National Junior High School Mathematics League, Jiangxi... | (a, b)=(5,12) \text{ or } (6,8) | 95 | 19 |
math | Example 10. Find $\lim _{x \rightarrow \pi} \frac{\sin ^{2} x}{1+\cos ^{3} x}$. | \frac{2}{3} | 36 | 7 |
math | 22. Let $S$ be the set of all non-zero real-valued functions $f$ defined on the set of all real numbers such that
$$
\mathrm{f}\left(x^{2}+y f(z)\right)=x \mathrm{f}(x)+z \mathrm{f}(y)
$$
for all real numbers $x, y$ and $z$. Find the maximum value of $\mathrm{f}(12345)$, where $\mathrm{f} \in S$. | 12345 | 109 | 5 |
math | Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers. | \frac{41}{2} | 95 | 8 |
math | Find all integers $n \geq 1$ such that $3^{n-1}+5^{n-1}$ divides $3^{n}+5^{n}$. | 1 | 39 | 1 |
math | Example 9. The function $f(x)=a e^{-|x|}$ is given. For what value of $a$ can it be considered as the probability density function of some random variable $X$? | \frac{1}{2} | 44 | 7 |
math | Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts... | 300 | 120 | 3 |
math | Example 4 consists of forming $n$-digit numbers using the digits $1,2,3$, and in this $n$-digit number, each of $1,2,3$ must appear at least once. How many such $n$-digit numbers are there? | 3^{n}-3 \times 2^{n}+3 | 59 | 14 |
math | 3.1. At a sumo wrestling tournament, 20 sumo-tori (sumo wrestlers) arrived. After weighing, it was found that the average weight of the sumo-tori is 125 kg. What is the maximum possible number of wrestlers who weigh more than 131 kg, given that according to sumo rules, people weighing less than 90 kg cannot participate... | 17 | 88 | 2 |
math | Determine all pairs of [i]distinct[/i] real numbers $(x, y)$ such that both of the following are true:
[list]
[*]$x^{100} - y^{100} = 2^{99} (x-y)$
[*]$x^{200} - y^{200} = 2^{199} (x-y)$
[/list] | (2, 0) \text{ and } (0, 2) | 88 | 17 |
math | 2. Laura has 2010 lamps and 2010 switches in front of her, with different switches controlling different lamps. She wants to find the correspondence between the switches and the lamps. For this, Charlie operates the switches. Each time Charlie presses some switches, and the number of lamps that light up is the same as ... | 11 | 151 | 2 |
math | For example, the five-digit numbers formed by the digits $1, 2, 3, 4, 5$ with repetition, arranged in ascending order. Ask:
(1) What are the positions of 22435 and 43512?
(2) What is the 200th number? ${ }^{[1]}$ | 12355 | 78 | 5 |
math | 7.219. $5^{1+x^{3}}-5^{1-x^{3}}=24$.
7.219. $5^{1+x^{3}}-5^{1-x^{3}}=24$. | 1 | 53 | 1 |
math | 10. Let $f(x)=\frac{1}{x^{3}-x}$, find the smallest positive integer $n$ that satisfies the inequality $f(2)+f(3)+\cdots+f(n)>\frac{499}{2020}$. | 13 | 60 | 2 |
math | Find the limit of the following sequence:
$$
u_{n}=\sum_{i=1}^{n} \frac{1}{F_{i} F_{i+2}}
$$ | 1 | 40 | 1 |
math | 2. If positive real numbers $a, b$ satisfy $a+b=1$, then the minimum value of $\left(1+\frac{1}{a}\right)\left(1+\frac{8}{b}\right)$ is
保留源文本的换行和格式，翻译结果如下：
2. If positive real numbers $a, b$ satisfy $a+b=1$, then the minimum value of $\left(1+\frac{1}{a}\right)\left(1+\frac{8}{b}\right)$ is | 50 | 112 | 2 |
math | II. (16 points) A plot of land can be covered by $n$ identical square tiles. If smaller identical square tiles are used, then $n+76$ such tiles are needed to cover the plot. It is known that $n$ and the side lengths of the tiles are integers. Find $n$.
| 324 | 68 | 3 |
math | Let's draw the circles that touch the sides and pass through the vertices of a right-angled triangle. What are the lengths of the sides of the triangle if the radius of the first circle is $8 \mathrm{~cm}$ and the radius of the second circle is $41 \mathrm{~cm}$? | \overline{AB}=18 | 65 | 8 |
math | 9. Given that point $P$ is a moving point on the line $x+2 y=4$, and through point $P$ two tangents are drawn to the ellipse $x^{2}+4 y^{2}=4$, with the points of tangency being $A$ and $B$, when point $P$ moves, the line $AB$ passes through a fixed point with coordinates | (1,\frac{1}{2}) | 83 | 9 |
math | # 2. CONDITION
Which of the numbers is greater: $2^{1997}$ or $5^{850}$? | 2^{1997}>5^{850} | 30 | 13 |
math | Find all positive integers $n$ such that $1!+2!+\ldots+n!$ is a perfect square. | n=1n=3 | 26 | 6 |
math | 13. Find the range of the function $f(x)=|\sin x|+|\cos x|$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the request is to translate the given text, here is the translation:
13. Find the range of the functi... | [1,\sqrt{2}] | 89 | 7 |
math | 4. The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}$. Then the integer part of the sum $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{200 B}+1}$ is $\qquad$ | 1 | 101 | 1 |
math | 9.7 Try to select 100 numbers such that they satisfy
$$x_{1}=1,0 \leqslant x_{k} \leqslant 2 x_{k-1}, k=2,3, \cdots, 100$$
and make $s=x_{1}-x_{2}+x_{3}-x_{4}+\cdots+x_{99}-x_{100}$ as large as possible | 1+2+2^{3}+\cdots+2^{97} | 101 | 17 |
math | Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$. | n = 13 | 43 | 6 |
math | Task 2. (10 points) It is known that the function $f(x)$ for each value of $x \in(-\infty ;+\infty)$ satisfies the equation $f(x)-(x-0.5) f(-x-1)=1$. Find all such functions $f(x)$. | f(x)={\begin{pmatrix}\frac{1}{0.5+x},x\neq-0.5,\\0.5,-0.50\end{pmatrix}.} | 66 | 44 |
math | Exercise 4. There are 2 ways to place two identical $1 \times 2$ dominoes to cover a $2 \times 2$ chessboard: either by placing both horizontally, or by placing both vertically.
In how many ways can a $2 \times 11$ chessboard be covered with 11 identical $1 \times 2$ dominoes? | 144 | 82 | 3 |
math | ## Aufgabe 24/75
Aus einer Tabelle der Fakultäten will jemand den Wert für 20! entnehmen. Dabei stellt er fest, dass zwei Ziffern unleserlich sind:
$$
20!=2 \bullet \bullet 2902008176640000
$$
Wie kann man die unleserlichen Ziffern ermitteln, ohne das Produkt auszurechnen?
| 2432902008176640000 | 101 | 19 |
math | 5-2. Solve the inequality
$$
\sqrt{6 x-13}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-19 x+26
$$
In your answer, specify the sum of all integer values of $x$ that satisfy the inequality. | 7 | 74 | 1 |
math | 5. A person is walking parallel to a railway track at a constant speed. A train also passes by him at a constant speed. The person noticed that depending on the direction of the train, it passes by him either in $t_{1}=1$ minute or in $t_{2}=2$ minutes. Determine how long it would take the person to walk from one end o... | 4 | 91 | 1 |
math | 1. Use interval notation to represent the domain of the function
$$
f(x)=\ln \left(\frac{1-x}{x+3}-1\right)
$$
as . $\qquad$ | (-3,-1) | 44 | 5 |
math | Example 5 (2005 National High School Mathematics Competition Question) Define the function
$$f(k)=\left\{\begin{array}{l}
0, \text { if } k \text { is a perfect square } \\
{\left[\frac{1}{\{\sqrt{k}\}}\right], \text { if } k \text { is not a perfect square }}
\end{array} \text {, find } \sum_{k=1}^{240} f(k)\right. \t... | 768 | 114 | 3 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x(x+y+z)=a^{2}, \\
& y(x+y+z)=b^{2}, \\
& z(x+y+z)=c^{2} .
\end{aligned}
$$ | \begin{aligned}&x_{12}=\\frac{^{2}}{\sqrt{^{2}+b^{2}+^{2}}}\\&y_{12}=\\frac{b^{2}}{\sqrt{^{2}+b^{2}+^{2}}}\\&z_{12}=\\frac{^{2}}{\sqrt{^{2}+b^{2} | 54 | 84 |
math | 7.109. $5^{2 x-1}+2^{2 x}-5^{2 x}+2^{2 x+2}=0$. | 1 | 35 | 1 |
math | 1. Let real numbers $a, b, c$ satisfy $a+b+c=0, a^{3}+b^{3}+c^{3}=0$, find the value of $a^{19}+b^{19}+c^{19}$ | 0 | 58 | 1 |
math | What is the probability that in a KENO draw, the digit 8 does not appear in any of the 20 numbers drawn from the integers ranging from 1 to 80? | 0.063748 | 39 | 8 |
math | Example 3 Given $2 x>3 y>0$, find the minimum value of $\sqrt{2} x^{3}+\frac{3}{2 x y-3 y^{2}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 5 \sqrt[5]{\frac{27}{2}} | 68 | 14 |
math | 18*. Each of the variables $a, b, c, d, e, f$ is defined on the set of numbers $\{1; -1\}$. On which set is the variable $x$ defined, if $x=a-b+c-d+e-f$? | {-6;-4;-2;0;2;4;6} | 59 | 15 |
math | 43. Find all three-digit numbers that are equal to the arithmetic mean of all numbers obtained from the given number by all possible permutations of its digits (including, of course, the "identity permutation" that leaves all the digits of the number in place). | 111,222,333,444,555,666,777,888,999,407,518,629,370,481,592 | 52 | 59 |
math | Example 9 Given that $f(x)$ is a function defined on the set of natural numbers $\mathrm{N}$, satisfying $f(1)=\frac{3}{2}$, and for any $x, y \in \mathrm{N}$, there is $f(x+y)=\left(1+\frac{y}{x+1}\right) f(x)+\left(1+\frac{x}{y+1}\right) f(y)+x^{2} y+x y+x y^{2}$. Find $f(x)$.
Translate the above text into English, ... | f(x)=\frac{1}{4}x(x+1)(2x+1) | 140 | 20 |
math | 8.34 The real number sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfies the following equation: $a_{0}=a$, where $a$ is a real number,
$$a_{n}=\frac{a_{n-1} \sqrt{3}+1}{\sqrt{3}-a_{n-1}}, n \in N .$$
Find $a_{1994}$. | \frac{a+\sqrt{3}}{1-a \sqrt{3}} | 104 | 17 |
math | 3. Let $S=\left\{r_{1}, r_{2}, r_{3}, \cdots, r_{n}\right\} \subseteq\{1,2,3, \cdots, 50\}$, and any two numbers in $S$ do not sum to a multiple of 7, then the maximum value of $n$ is $\qquad$ . | 23 | 84 | 2 |
math | 14. (15 points) Solve the equation: $[x] \times\{x\}+x=2\{x\}+9$, where $[x]$ represents the integer part of $x$, and $\{x\}$ represents the fractional part of $x$. For example, $[3.14]=3,\{3.14\}=0.14$. (List all solutions) | 9;8\frac{1}{7};7\frac{1}{3};6\frac{3}{5} | 91 | 26 |
math | Given $\triangle A B C$ has a perimeter of 20, area of $01 \sqrt{3}$, and $\angle A=60^{\circ}$. Find $\sin A: \sin B: \sin C$. | 7: 8: 5 | 51 | 7 |
math | Example 3 Find all pairs of positive integers $(m, n), m, n \geqslant 2$, such that for any $a \in\{1$, $2, \cdots, n\}$, we have $a^{n} \equiv 1(\bmod m)$. | (m, n)=(p, p-1) | 65 | 10 |
math | The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the [i]Zabrubic word[/i]. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a [i]prophetic word[/i] of length $ k$. It is known th... | 504 | 138 | 3 |
math | 1. Calculate $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+$ $\frac{1}{\sqrt{2003}+\sqrt{2004}}=$ $\qquad$ | 2\sqrt{501}-1 | 74 | 9 |
math | ## Task Condition
Find the derivative.
$$
y=x-\ln \left(1+e^{x}\right)-2 e^{-\frac{x}{2}} \cdot \operatorname{arctg} e^{\frac{x}{2}}-\left(\operatorname{arctg} e^{\frac{x}{2}}\right)^{2}
$$ | \frac{\operatorname{arctg}e^{x/2}}{e^{x/2}\cdot(1+e^{x})} | 77 | 32 |
math | 309*. Solve the equation:
$$
x^{3}+x^{2}+x=-\frac{1}{3}
$$ | -\frac{1}{1+\sqrt[3]{2}} | 30 | 13 |
math | Solve the following equation if $x$ and $y$ are real numbers:
$$
\left(16 x^{2}+1\right)\left(y^{2}+1\right)=16 x y .
$$ | (\frac{1}{4},1)(-\frac{1}{4},-1) | 49 | 19 |
math | 4.51 Find the smallest (greater than 1) natural number such that it is at least 600 times larger than each of its prime divisors?
(Leningrad Mathematical Olympiad, 1989) | 1944 | 48 | 4 |
math | Example 1.1. Find the integrals:
a) $\int \sin (7 x+3) d x$,
b) $\int \frac{d x}{x+5}$.
c) $\int(3 x+5)^{99} d x$,
d) $\int e^{x^{3}} x^{2} d x$
e) $\int \sin ^{3} x \cos x d x$. | \begin{aligned})&\quad-\frac{1}{7}\cos(7x+3)+C\\b)&\quad\ln|x+5|+C\\)&\quad\frac{(3x+5)^{100}}{300}+C\\)&\quad\frac{1}{3}e^{x^{3}}+C\\e)&\quad\frac | 94 | 86 |
math | 8.3.1. (12 points) Five numbers form an increasing arithmetic progression. The sum of their cubes is zero, and the sum of their squares is 70. Find the smallest of these numbers. | -2\sqrt{7} | 45 | 7 |
math | 12. In $\triangle A B C$, $\angle B A C=30^{\circ}, \angle A B C=70^{\circ}, M$ is a point inside the triangle, $\angle M A B=\angle M C A=20^{\circ}$, find the degree measure of $\angle M B A$. | 30 | 72 | 2 |
math | 3. Given that $x$ and $y$ are integers, $y=\sqrt{x+2003}-$ $\sqrt{x-2009}$. Then the minimum value of $y$ is $\qquad$ . | 2 | 50 | 1 |
math | 74. Find the probability $P(\bar{A} \bar{B})$ given the following probabilities:
$$
P(A)=a, P(B)=b . \quad P(A+B)=c
$$ | 1- | 45 | 2 |
math | 10 If the vector $\vec{a}+3 \vec{b}$ is perpendicular to the vector $7 \vec{a}-5 \vec{b}$, and the vector $\vec{a}-4 \vec{b}$ is perpendicular to the vector $7 \vec{a}-2 \vec{b}$, then the angle between vector $\vec{a}$ and $\vec{b}$ is $\qquad$. | \frac{\pi}{3} | 90 | 7 |
math | 166. The sum of the reciprocals of three positive integers is equal to one. What are these numbers? | \frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1;\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1;\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1 | 25 | 66 |
math | Example 4. Integrate the equation
$$
x(x+2 y) d x+\left(x^{2}-y^{2}\right) d y=0
$$ | x^{3}+3x^{2}y-y^{3}=C | 37 | 16 |
math | 9. (16 points) Given $x, y, z > 0$. Find
$$
f(x, y, z)=\frac{\sqrt{x^{2}+y^{2}}+\sqrt{y^{2}+4 z^{2}}+\sqrt{z^{2}+16 x^{2}}}{9 x+3 y+5 z}
$$
the minimum value. | \frac{\sqrt{5}}{5} | 85 | 10 |
math | Find all primes of the form $a^{2}-1$, with $a \geqslant 2$ natural. | 3 | 26 | 1 |
math | LIII OM - III - Task 1
Determine all such triples of natural numbers $ a $, $ b $, $ c $, such that the numbers $ a^2 +1 $ and $ b^2 +1 $ are prime and | (1,2,3)(2,1,3) | 52 | 13 |
math | 8. Find the sum of all integer values of $\mathrm{h}$ for which the equation ||$r+h|-r|-4 r=9|r-3|$ in terms of $r$ has no more than one solution.
# | -93 | 48 | 3 |
math | SG. 4 If $\log _{p} x=2, \log _{q} x=3, \log _{r} x=6$ and $\log _{p q r} x=d$, find the value of $d$. | 1 | 54 | 1 |
math | The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$. | 1 | 84 | 1 |
math | The number $2^{29}$ has a $9$-digit decimal representation that contains all but one of the $10$ (decimal) digits. Determine which digit is missing | 4 | 38 | 1 |
math | Find the distance from the point $D(1 ; 3 ; 2)$ to the plane passing through the points $A(-3 ; 0 ; 1), B(2 ; 1 ;-1)$ and $C(-2 ; 2 ; 0)$.
# | \frac{10}{\sqrt{11}} | 58 | 12 |
math | (United States 1998)
Consider a $m \times n$ chessboard whose squares are colored in the usual black and white pattern. A move consists of choosing a rectangle of squares and inverting the colors of all the squares in it. What is the minimum number of moves required to make the chessboard monochrome? | \lfloor\frac{n}{2}\rfloor+\lfloor\frac{}{2}\rfloor | 68 | 22 |
math | 3. Find the value of $\sin ^{2} 10^{\circ}+\cos ^{2} 40^{\circ}+\sin 10^{\circ} \cos 40^{\circ}$. | \frac{3}{4} | 51 | 7 |
math | 65. Let $a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers, and $n>2$ be a given positive integer. Find the largest positive number $K$ and the smallest positive number $G$ such that the following inequality holds: $K<\frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{2}}{a_{2}+a_{3}}+\cdots+\frac{a_{n}}{a_{n}+a_{1}}<G$. (1991 Jap... | 1 < S_n < n-1 | 131 | 8 |
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