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 | Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(... | 126 | 112 | 3 |
math | 14. Teacher Li went to the toy store to buy balls. The money he brought was just enough to buy 60 plastic balls. If he didn't buy plastic balls, he could buy 36 glass balls or 45 wooden balls. Teacher Li finally decided to buy 10 plastic balls and 10 glass balls, and use the remaining money to buy wooden balls. How man... | 45 | 90 | 2 |
math | 16. On Sunday, after Little Fat got up, he found that his alarm clock was broken. He estimated the time and set the alarm clock to 7:00, then he walked to the science museum and saw that the electronic clock on the roof pointed to 8:50. He visited the museum for one and a half hours and returned home at the same speed.... | 12:00 | 121 | 5 |
math | 1.037. $\frac{\left(\left(4,625-\frac{13}{18} \cdot \frac{9}{26}\right): \frac{9}{4}+2.5: 1.25: 6.75\right): 1 \frac{53}{68}}{\left(\frac{1}{2}-0.375\right): 0.125+\left(\frac{5}{6}-\frac{7}{12}\right):(0.358-1.4796: 13.7)}$. | \frac{17}{27} | 138 | 9 |
math | ## Task 6 - 130736
A train traveling at a constant speed took exactly 27 s to cross a 225 m long bridge (calculated from the moment the locomotive entered the bridge until the last car left the bridge).
The train passed a pedestrian walking in the opposite direction of the train in exactly 9 s. During this time, the ... | 126\, | 108 | 5 |
math | Example 4. Find the integral $\int \cos ^{5} x d x$. | \sinx-\frac{2}{3}\sin^{3}x+\frac{1}{5}\sin^{5}x+C | 19 | 28 |
math | 5. Papa, Masha, and Yasha are going to school. While Papa takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they made 400 steps. How many steps did Papa take? | 90 | 67 | 2 |
math | 10. $1^{1}+2^{2}+3^{3}+4^{4}+5^{5}+6^{6}+7^{7}+8^{8}+9^{9}$ divided by 3 has a remainder of what? Why? | 1 | 60 | 1 |
math | 7. A mowing team had to mow two meadows, one twice as large as the other. For half a day, the team mowed the larger meadow. After that, they split in half: one half stayed on the large meadow and finished it by evening, while the other half mowed the smaller meadow but did not finish it. How many mowers were in the tea... | 8 | 101 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2} \frac{x^{3}-3 x-2}{x-2}$ | 9 | 36 | 1 |
math | IMO 1977 Problem B2 Let a and b be positive integers. When a 2 + b 2 is divided by a + b, the quotient is q and the remainder is r. Find all pairs a, b such that q 2 + r = 1977. | (50,37)(50,7) | 62 | 12 |
math | 13.131. The beacon keeper, inspecting his river section in an ordinary rowboat, traveled 12.5 km upstream and then returned to the starting point along the same route. In this voyage, he overcame every 3 km against the current and every 5 km with the current in the same average time intervals, and was on the way for ex... | \frac{5}{6} | 101 | 7 |
math |
Zadatak B-4.7. Točkom $M(6,2)$ unutar kružnice $(x-3)^{2}+(y-4)^{2}=36$ povučena je tetiva duljine $6 \sqrt{3}$. Napišite jednadžbu pravca na kojem leži ta tetiva.
| \frac{5}{12}x-\frac{1}{2}6 | 84 | 17 |
math | IX OM - II - Task 2
Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to... | 4 | 74 | 1 |
math | 5. (5 points) Find all solutions of the system of equations $\left\{\begin{array}{l}\frac{y^{3}}{x^{2}}+\frac{x}{y}=1 \\ \frac{2 x}{y^{2}}+\frac{4 y}{x}=1\end{array}\right.$. | x_{1}=-8,\quady_{1}=-4,\quadx_{2}=-54,\quady_{2}=-18 | 69 | 32 |
math | Example 4. The three vertices of a triangle are $(0,0)$, $(2,3)$, and $(3,-2)$. When the triangle is translated, the translation path of $(0,0)$ is $\mathrm{y}^{2}=2 \mathrm{x}$. Find the translation paths of the other two points. | (y-3)^{2}=2(x-2) \text{ and } (y+2)^{2}=2(x-3) | 70 | 30 |
math | 79. A shooter makes three attempts. The success (hitting the target) and failure (miss) of each of them are independent of the outcomes of the other attempts, and the probability of successful completion of each attempt is constant and equal to p. Find the probability of successful completion of two attempts out of thr... | 3p^{2}(1-p) | 65 | 8 |
math | 13.373 A computing machine was given the task to solve several problems sequentially. Registering the time spent on the assignment, it was noticed that the machine spent the same multiple of time less on solving each subsequent problem compared to the previous one. How many problems were proposed and how much time did ... | 8 | 133 | 1 |
math | 7.2. There are nuts in three boxes. In the first box, there are six nuts fewer than in the other two boxes combined, and in the second box, there are ten fewer nuts than in the other two boxes combined. How many nuts are in the third box? Justify your answer. | 8 | 62 | 1 |
math | $7.1 \quad \sqrt{5} \cdot 0.2^{\frac{1}{2 x}}-0.04^{1-x}=0$. | x_{1}=1,x_{2}=0.25 | 37 | 13 |
math | 3. Given that the perimeter of $\triangle A B C$ is 20, the radius of the inscribed circle is $\sqrt{3}$, and $B C=7$. Then the value of $\tan A$ is | \sqrt{3} | 48 | 5 |
math | Example 20([37.4]) Let positive integers $a, b$ be such that $15a + 16b$ and $16a - 15b$ are both squares of positive integers. Find the smallest possible values of these two squares. | (13 \cdot 37)^2 | 59 | 10 |
math | Find all prime numbers $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.
543 | 3,7 | 30 | 3 |
math | B1. The sum of four fractions is less than 1. Three of these fractions are $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{10}$. The fourth fraction is $\frac{1}{n}$, where $n$ is a positive integer. What values could $n$ take? | n>15 | 72 | 4 |
math | Find all real numbers $x, y, z$ satisfying:
$$
\left\{\begin{array}{l}
(x+1) y z=12 \\
(y+1) z x=4 \\
(z+1) x y=4
\end{array}\right.
$$
Elementary Symmetric Polynomials In this section, we are interested in the links between the coefficients of a polynomial and its roots.
## Viète's Formulas
Proposition (Viète's Fo... | (2,-2,-2)(\frac{1}{3},3,3) | 272 | 18 |
math | 2. On a sheet of lined paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, while the second has the opposite. Find the maximum possible area of their common part, if the first rectangle contains 2015 cells, and the second - 2016. | 1302 | 66 | 4 |
math | 6. Given a rectangle $A B C D$. Point $M$ is the midpoint of side $A B$, point $K$ is the midpoint of side $B C$. Segments $A K$ and $C M$ intersect at point $E$. How many times smaller is the area of quadrilateral $M B K E$ compared to the area of quadrilateral $A E C D$? | 4 | 84 | 1 |
math | 38. Find the largest number by which each of the fractions $\frac{154}{195}$, $\frac{385}{156}$, and $\frac{231}{130}$ can be divided to yield natural numbers. | \frac{77}{780} | 57 | 10 |
math | A side of triangle $H_{1}$ is equal to the arithmetic mean of the other two sides. The sides of triangle $H_{2}$ are each 10 units smaller, and the sides of triangle $H_{3}$ are each 14 units larger than the sides of $H_{1}$. The radius of the inscribed circle of $H_{2}$ is 5 units smaller than that of $H_{1}$, and the... | =25,b=38,=51 | 122 | 11 |
math | Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$.
(1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$.
(2) Find the value of $ nx_nx_{n \minus{} 1}$.
(3) Show that a sequence $ \{x_n\}$ is monotone decreasing.
(4) Find $ \lim_{n\to\inf... | \frac{\pi}{2} | 144 | 7 |
math | 3. The sum of positive numbers $a, b, c$ and $d$ does not exceed 4. Find the maximum value of the expression
$$
\sqrt[4]{2 a^{2}+a^{2} b}+\sqrt[4]{2 b^{2}+b^{2} c}+\sqrt[4]{2 c^{2}+c^{2} d}+\sqrt[4]{2 d^{2}+d^{2} a}
$$ | 4\sqrt[4]{3} | 103 | 8 |
math | The sum of Zipporah's age and Dina's age is 51. The sum of Julio's age and Dina's age is 54. Zipporah is 7 years old. How old is Julio? | 10 | 50 | 2 |
math | 187. System with three unknowns. Solve the following system of equations:
$$
\left\{\begin{aligned}
x+y+z & =6 \\
x y+y z+z x & =11 \\
x y z & =6
\end{aligned}\right.
$$ | (1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1) | 60 | 37 |
math | Shapovalov A.V.
There are three piles of 40 stones each. Petya and Vasya take turns, Petya starts. On a turn, one must combine two piles, then divide these stones into four piles. The player who cannot make a move loses. Which of the players (Petya or Vasya) can win, regardless of how the opponent plays? | Vasya | 83 | 3 |
math | $\square$ Example 5 Let $a_{1}, a_{2}, \cdots, a_{n}$ be given non-zero real numbers. If the inequality
$$r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant \sqrt[m]{x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}}$$
$-\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}$ (where $... | r_{i}=\left(\frac{a_{i}}{\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}}\right)^{m-1}(i=1,2, \cdots, n) | 238 | 62 |
math | 13. Let $A$ and $B$ be two distinct points on the parabola
$$
y^{2}=2 p x(p>0)
$$
Then the minimum value of $|\overrightarrow{O A}+\overrightarrow{O B}|^{2}-|\overrightarrow{A B}|^{2}$ is $\qquad$. | -4 p^{2} | 74 | 6 |
math | Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$. Determine all $n<2001$ with the property that
$d_9-d_8=22$. | n = 1995 | 86 | 8 |
math | ## Task 1.
Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ such that
$$
f\left(x^{2}(f(y))^{2}\right)=(f(x))^{2} f(y), \quad \text { for all } x, y \in \mathbb{Q}^{+}
$$
$\left(\mathbb{Q}^{+}\right.$ is the notation for the set of all positive rational numbers.) | f(x)=1,x\in\mathbb{Q}^{+} | 110 | 16 |
math | A circle intersects the coordinate axes at points $A(a, 0), B(b, 0) C(0, c)$ and $D(0, d)$. Find the coordinates of its center. | (\frac{+b}{2};\frac{+}{2}) | 43 | 15 |
math | 2. Let $n \geqslant 2$, find the maximum and minimum value of the product $x_{1} x_{2} \cdots x_{n}$ under the conditions: (1) $x_{i} \geqslant \frac{1}{n}, i=1,2, \cdots, n$; (2) $x_{1}^{2}+x_{2}^{2}+\cdots+$ $x_{n}^{2}=1$. | \frac{\sqrt{n^{2}-n+1}}{n^{n}} | 108 | 17 |
math | For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i] | 2 | 66 | 1 |
math | Example 2 Given that $x, y, z$ are positive real numbers, and satisfy $x^{4}+y^{4}+z^{4}=1$, find the minimum value of $\frac{x^{3}}{1-x^{8}}+\frac{y^{3}}{1-y^{8}}$ + $\frac{z^{3}}{1-z^{8}}$. (1999 Jiangsu Province Mathematical Winter Camp Problem) | \frac{9}{8} \sqrt[4]{3} | 95 | 14 |
math | [b]a)[/b] Solve the equation $ x^2-x+2\equiv 0\pmod 7. $
[b]b)[/b] Determine the natural numbers $ n\ge 2 $ for which the equation $ x^2-x+2\equiv 0\pmod n $ has an unique solution modulo $ n. $ | n = 7 | 75 | 5 |
math | Problem 1 Let real numbers $a, b, c \geqslant 1$, and satisfy
$$a b c+2 a^{2}+2 b^{2}+2 c^{2}+c a-b c-4 a+4 b-c=28$$
Find the maximum value of $a+b+c$. | 6 | 72 | 1 |
math | 1. Given $0<x, y<1$. Find the maximum value of
$$
\frac{x y(1-x-y)}{(x+y)(1-x)(1-y)}
$$
(Liu Shixiong) | \frac{1}{8} | 47 | 7 |
math | Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying
$a + b - c - d = 3$
$ab - 3bc + cd - 3da = 4$
$3ab - bc + 3cd - da = 5$
Find $11(a - c)^2 + 17(b -d)^2$. | 63 | 81 | 2 |
math | Prove that for all real numbers $ a,b$ with $ ab>0$ we have:
$ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$
and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$
$ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}... | \sqrt[3]{\frac{a^2 b^2 (a+b)^2}{4}} \le \frac{a^2 + ab + b^2}{3} | 149 | 40 |
math |
N4. Find all integers $x, y$ such that
$$
x^{3}(y+1)+y^{3}(x+1)=19
$$
| (2,1),(1,2),(-1,-20),(-20,-1) | 38 | 21 |
math | Find all prime numbers $p$ for which there exist positive integers $x, y$ and $z$ such that the number
$$
x^{p}+y^{p}+z^{p}-x-y-z
$$
is a product of exactly three distinct prime numbers. | p = 2, 3, 5 | 59 | 10 |
math | 2. Find all pairs of positive integers $n, m$ that satisfy the equation $3 \cdot 2^{m}+1=n^{2}$. | (n,)=(7,4),(5,3) | 33 | 11 |
math | Problem 9.1. Find all values of $a$ such that the equation
$$
\left(a^{2}-a-9\right) x^{2}-6 x-a=0
$$
has two distinct positive roots.
Ivan Landjev | \in(-3,(1-\sqrt{37})/2) | 55 | 15 |
math | 2. Given that $a$ is a real number, and makes the quadratic equation in $x$, $x^{2}+a^{2} x+a=0$, have real roots. Then the maximum value that $x$ can take is $\qquad$ | \frac{\sqrt[3]{2}}{2} | 55 | 12 |
math | SI. 4 In the coordinate plane, the distance from the point $(-C, 0)$ to the straight line $y=x$ is $\sqrt{D}$, find the value of $D$. | 8 | 43 | 1 |
math | Let's determine the area of the right-angled triangle, given that $\alpha=38^{\circ} 40^{\prime}$ and the angle bisector of $\alpha$ is $f_{a}=7.8 \mathrm{~cm}$. | 21.67\mathrm{~}^{2} | 55 | 13 |
math | 1. Given quadratic trinomials $f_{1}(x)=x^{2}-x+2 a, f_{2}(x)=x^{2}+2 b x+3, f_{3}(x)=4 x^{2}+(2 b-3) x+6 a+3$ and $f_{4}(x)=4 x^{2}+(6 b-1) x+9+2 a$. Let the differences of their roots be $A, B, C$ and $D$ respectively. It is known that $|A| \neq|B|$. Find the ratio $\frac{C^{2}-D^{2}}{A^{2}-B^{2}}$. The values of $A,... | \frac{1}{2} | 170 | 7 |
math | 2. From the 99 natural numbers $1,2,3, \cdots, 99$, the number of ways to choose two different numbers such that their sum is less than 99 is $\qquad$ ways. | 2352 | 50 | 4 |
math | 4. If acute angles $\alpha, \beta$ satisfy
$$
\sin \alpha=\cos (\alpha+\beta) \cdot \sin \beta,
$$
then the maximum value of $\tan \alpha$ is $\qquad$ | \frac{\sqrt{2}}{4} | 50 | 10 |
math | ерешин D.A.
Once, Rabbit was in a hurry to meet with Donkey Eeyore, but Winnie-the-Pooh and Piglet unexpectedly came to him. Being well-mannered, Rabbit offered his guests some refreshments. Pooh tied a napkin around Piglet's mouth and single-handedly ate 10 pots of honey and 22 cans of condensed milk, with each pot o... | 30 | 213 | 2 |
math | ## 71. Math Puzzle $4 / 71$
The side lengths of a triangle are $a=4 \mathrm{~cm}, b=13 \mathrm{~cm}$ and $c=15 \mathrm{~cm}$.
What are a) the area, b) the three heights, c) the radius of the incircle, and d) the radii of the three excircles? | A=24\mathrm{~}^{2},h_{}=12\mathrm{~},h_{b}=3.7\mathrm{~},h_{}=3.2\mathrm{~},\rho=1.5\mathrm{~},\rho_{}=2\mathrm{~},\rho_{b}=8\mathrm{~},\rho_{}=24\mathrm{} | 88 | 90 |
math | 1. What relation that does not depend on $m$ exists between the solutions of the equation
$$
\left(x^{2}-6 x+5\right)+m\left(x^{2}-5 x+6\right)=0 ?
$$ | x_{1}+x_{2}+x_{1}x_{2}=11 | 52 | 20 |
math | ## Task B-1.2.
With which digit does the number $2^{2022}+3^{2022}+7^{2022}$ end? | 2 | 40 | 1 |
math | 2. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then
$$
[\sqrt{2010+\sqrt{2009+\sqrt{\cdots+\sqrt{3+\sqrt{2}}}}}]
$$
is equal to
(there are a total of 2009 square roots). | 45 | 73 | 2 |
math | 5. Given a circle with its center at the origin and radius $R$ intersects with the sides of $\triangle A B C$, where $A(4,0), B(6,8)$, $C(2,4)$. Then the range of values for $R$ is $\qquad$. | [\frac{8\sqrt{5}}{5},10] | 64 | 15 |
math | Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct. | 46 | 112 | 2 |
math | A convex polyhedron has more faces than vertices. What is the minimum number of triangles among the faces? | 6 | 22 | 1 |
math | 21 Find the integer part of
$$
\frac{1}{\frac{1}{2003}+\frac{1}{2004}+\frac{1}{2005}+\frac{1}{2006}+\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}} .
$$ | 286 | 87 | 3 |
math | 4. Find all integers $x$ and $y$ that satisfy the equation $x^{4}-2 y^{2}=1$. | x=\1,y=0 | 28 | 6 |
math | 7. (6 points) A sequence of numbers arranged according to a rule: $2,5,9,14,20,27, \cdots$, the 1995th number in this sequence is | 1993005 | 48 | 7 |
math | 2. The following equation is to be solved in the set of natural numbers
$$
2^{x}+2^{y}+2^{z}=2336
$$ | 5,8,11 | 38 | 6 |
math | 433. Establish a one-to-one correspondence by formula between all positive real numbers \( x (0 < x < +\infty) \) and real numbers \( y \) confined within the interval between 0 and 1 \( (0 < y < 1) \).
## V. Functions of an integer variable. | \frac{y}{1-y} | 68 | 8 |
math | Problem 2. Two sidi clocks were set to show the exact time on March 21, 2022, at 9 PM. One works accurately, while the other is fast by 3 minutes every hour. On which date and at what time will the hands of the two clocks be in the same position as on March 21, 2022, at 9 PM? | March\31,\2022,\9\PM | 85 | 13 |
math | In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest
side has length $12$. What is the length of the altitude (height) of the triangle that
is perpendicular to the longest side? | 3 | 53 | 1 |
math | Example 1 Try to find the interval of monotonic increase for the function $y=\log _{0.5}\left(x^{2}+4 x+4\right)$. | (-\infty,-2) | 39 | 7 |
math | Example 2 Find all positive integers $x>1, y>1, z>1$, such that $1!+2!+\cdots+x!=y^{z}$. | x=y=3, z=2 | 38 | 8 |
math | Let's determine the maximum of the function $f(x)=\sqrt{x-2}+\sqrt{2 x-7}+\sqrt{18-3 x}$. | 3\sqrt{3} | 36 | 6 |
math | 6. If $\frac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)}=\frac{2004}{2005}$, find $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}$.
(1 mark)
若 $\frac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)}=\frac{2004}{2005}$, 求 $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}$ 。 | \frac{4011}{4010} | 134 | 13 |
math | 847. Find the limits:
1) $\lim _{x \rightarrow 0} x^{x}$; 2) $\lim _{x \rightarrow 0}\left(\frac{1}{x}\right)^{\sin x}$; 3) $\lim _{x \rightarrow 1} x^{\frac{1}{x-1}}$ | 1,1,e | 78 | 4 |
math | A quadrilateral is drawn on a sheet of transparent paper. What is the minimum number of times the sheet needs to be folded to ensure that it is a square?
# | 2 | 34 | 1 |
math | 12. Let $n$ be a positive integer, when $n>100$, the first two digits of the decimal part of $\sqrt{n^{2}+3 n+1}$ are | 49 | 42 | 2 |
math | 4. Find all solutions of the equation $x^{2}-12 \cdot[x]+20=0$, where $[x]-$ is the greatest integer not exceeding $x$. | 2,2\sqrt{19},2\sqrt{22},10 | 38 | 18 |
math | 1. In a regular polygon with 67 sides, we draw all segments connecting two vertices, including the sides of the polygon. We choose $n$ of these segments and assign each one of them a color from 10 possible colors. Find the minimum value of $n$ that guarantees, regardless of which $n$ segments are chosen and how the col... | 2011 | 126 | 4 |
math | ## Task 3 - 170623
In the turning shop of a company, individual parts are turned from lead rods. Each lead rod produces one individual part.
The swarf obtained from the production of every 6 individual parts can be melted down to make one lead rod. (Any smaller amount of swarf is insufficient for this purpose.)
What... | 43 | 97 | 2 |
math | B2. Solve the equation $\log _{2}\left(2 \cos ^{2} x-2 \cos x+1\right)=-1$. | \\frac{\pi}{3}+2k\pi | 35 | 12 |
math | The sequence $\{c_{n}\}$ is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let $\lambda$ be the limit value $\lim_{n\to\infty}c_{n}.$ Find $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ | 1 | 112 | 1 |
math | Example 5. The random variable $X$ is given by the probability density function
$$
p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-x^{2} / 2 \sigma^{2}}
$$
Find the probability density function of $Y=X^{3}$. | (y)=\frac{1}{3\sigmay^{2/3}\sqrt{2\pi}}e^{-y^{2/3}/2\sigma^{2}} | 66 | 36 |
math | Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$ | (2, 2, 2) | 35 | 9 |
math | 5. If for any $x \in[0,1]$, we have
$$
f(x)=k\left(x^{2}-x+1\right)-x^{4}(1-x)^{4} \geqslant 0 \text {, }
$$
then the minimum value of $k$ is $\qquad$ . | \frac{1}{192} | 74 | 9 |
math | 119. For what values of $n$ is the sum $n^{2}+(n+1)^{2}+(n+2)^{2}+$ $(n+3)^{2}$ divisible by 10 ( $n$ is a natural number)? | 5t+1 | 58 | 4 |
math | 19. Football-2. (from 10th grade. 6 points) A regular football match is in progress. A draw is possible. The waiting time for the next goal does not depend on previous events in the match. It is known that the expected number of goals in football matches between these teams is 2.8. Find the probability that an even num... | 0.502 | 86 | 5 |
math | 12. Let $f(x)=\frac{1+10 x}{10-100 x}$. Set $f^{n}=f \circ f \circ \cdots \circ f$. Find the value of
$$
f\left(\frac{1}{2}\right)+f^{2}\left(\frac{1}{2}\right)+f^{3}\left(\frac{1}{2}\right)+\cdots+f^{6000}\left(\frac{1}{2}\right)
$$ | 595 | 113 | 3 |
math | 1. Given a sequence of positive numbers $\left\{a_{n}\right\}$ satisfying $a_{n+1} \geqslant 2 a_{n}+1$, and $a_{n}<2^{n+1}$ for $n \in \mathbf{Z}_{+}$. Then the range of $a_{1}$ is $\qquad$ . | 0<a_{1}\leqslant3 | 82 | 10 |
math | 7.1.1. (12 points) Find the greatest negative root of the equation
$$
\frac{\sin \pi x - \cos 2 \pi x}{(\sin \pi x - 1)^2 + \cos^2 \pi x - 1} = 0
$$ | -0.5 | 65 | 4 |
math | For how many one-digit positive integers $k$ is the product $k \cdot 234$ divisible by 12 ? | 4 | 28 | 1 |
math | The complex numbers $z=x+i y$, where $x$ and $y$ are integers, are called Gaussian integers. Determine the Gaussian integers that lie inside the circle described by the equation
$$
x^{2}+y^{2}-4 x-10 y+20=0
$$
in their geometric representation. | \begin{pmatrix}3i&4i&5i&6i&7i;\\1+3i&1+4i&1+5i&1+6i&1+7i\\2+3i&2+4i&2+5i&2+6i&2+7i\\3+3i&3+4i | 69 | 81 |
math | Example 8 Given $x+y=2, x^{2}+y^{2}=\frac{5}{2}$. Then $x^{4}+y^{4}=$ $\qquad$
(1995, Chongqing and Four Other Cities Junior High School Mathematics League) | \frac{41}{8} | 62 | 8 |
math | 10th Australian 1989 Problem B3 Let N be the positive integers. The function f : N → N satisfies f(1) = 5, f( f(n) ) = 4n + 9 and f(2 n ) = 2 n+1 + 3 for all n. Find f(1789). Solution | 3581 | 76 | 4 |
math | Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$ | 30^\circ | 53 | 4 |
math | \section*{Problem 18}
Find the minimum value of \(x y / z+y z / x+z x / y\) for positive reals \(x, y, z\) with \(x^{2}+y^{2}+z^{2}=1\).
| \sqrt{3} | 58 | 5 |
math | 455. Based on a sample of size $n=41$, a biased estimate $D_{\text {v }}=3$ of the population variance was found. Find the unbiased estimate of the population variance. | 3.075 | 46 | 5 |
math | In a circle, intersecting chords $A B$ and $C D$ are perpendicular, $A D=m, B C=n$. Find the diameter of the circle. | \sqrt{^{2}+n^{2}} | 35 | 11 |
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