task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Task B-2.2. For what values of the variables $x$ and $y$ does the expression $\frac{4 x^{2}+2 y^{2}-4 y+4}{2 x^{2}+y^{2}-2 y+5}$ have the smallest value, and what is that smallest value? | \frac{1}{2} | 69 | 7 |
math | 16. Find the value of
$$
\frac{1}{1+11^{-2011}}+\frac{1}{1+11^{-2009}}+\frac{1}{1+11^{-2007}}+\cdots+\frac{1}{1+11^{2009}}+\frac{1}{1+11^{2011}} .
$$ | 1006 | 90 | 4 |
math | 12. (IMO-20-1) The last three digits of the numbers $1978^{n}$ and $1978^{m}$ are equal. Find the positive integers $n$ and $m$ such that $n+m$ is minimized, where $n>m \geqslant 1$. | 106 | 70 | 3 |
math | $2 \cdot 116$ Let $S=\{1,2, \cdots, 10\}$. If a subset $T$ of $S$ contains at least 2 elements and the absolute difference between any two elements is greater than 1, then $T$ is said to have property $P$. Find the number of different subsets of $S$ that have property $P$. | 133 | 86 | 3 |
math | Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that
$$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$. | \lambda = \frac{2^k}{k+1} | 76 | 15 |
math | Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$.
Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$. | \sqrt[3]{3} | 54 | 8 |
math | A point is "bouncing" inside a unit equilateral triangle with vertices $(0,0)$, $(1,0)$, and $(1/2,\sqrt{3}/2)$. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the dir... | 30 - 17\sqrt{3} | 113 | 11 |
math | 1020. Investigate the function for extremum
$$
f(x, y)=x^{2}+y^{2}-4 y+4
$$ | 0 | 35 | 1 |
math | Example 7 Given that $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}=a^{2}, \\
x+y+z=b,
\end{array}\right.
$$
where $a, b$ are real constants, and $|b| \leqslant \sqrt{3}|a|$. Try to find the range of values for $x$. | \left[\frac{b-\sqrt{2\left(3 a^{2}-b^{2}\right)}}{3}, \frac{b+\sqrt{2\left(3 a^{2}-b^{2}\right)}}{3}\right] | 98 | 54 |
math | Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C... | 3 | 177 | 1 |
math | 8. Let $a_{1}, a_{2}, a_{3}, a_{4}$ be 4 distinct numbers from $1,2, \cdots, 100$, satisfying
$$
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\right)=\left(a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}\right)^{2},
$$
then the number of such ordered quadruples $\left(a_{1}, a_{2}, a_{3}, a_{4}\rig... | 40 | 166 | 2 |
math | 10 Let $x, y, z \geqslant 0$, and $x+y+z=1$, find the maximum and minimum values of $x^{2} y+y^{2} z+z^{2} x$.
| \frac{4}{27} | 50 | 8 |
math | 73. An electronic computer can quickly calculate the sum
$$
\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\ldots+\frac{2}{99 \cdot 101}
$$
However, you can compete with it if you uncover the secret and rewrite the terms differently. How can this be done? | \frac{100}{101} | 90 | 11 |
math | Example 6. Express $6 \sin ^{2} \frac{\alpha}{2}-8 \sin ^{4} \frac{\alpha}{2}$ as a product. | 2 \sin \frac{3 \alpha}{2} \sin \frac{\alpha}{2} | 38 | 21 |
math | 12.119. A sphere is inscribed in a cone. The radius of the circle where the cone and the sphere touch is $r$. Find the volume of the cone if the angle between the height and the slant height of the cone is $\alpha$. | \frac{\pir^{3}\operatorname{ctg}^{3}(\frac{\pi}{4}-\frac{\alpha}{2})}{3\cos^{2}\alpha\sin\alpha} | 56 | 43 |
math | 16. A store received fewer than 600 but more than 500 plates. When they started arranging them in tens, they were short of three plates to make a complete number of tens, and when they started arranging them in dozens (12 plates), there were 7 plates left. How many plates were there? | 547 | 69 | 3 |
math | 13th APMO 2001 Problem 5 What is the largest n for which we can find n + 4 points in the plane, A, B, C, D, X 1 , ... , X n , so that AB is not equal to CD, but for each i the two triangles ABX i and CDX i are congruent? | 4 | 77 | 1 |
math | One, (20 points) If $x, y \in [0,1]$, try to find the maximum value of
$$
x \sqrt{1-y} + y \sqrt{1-x}
$$ | 1 | 46 | 1 |
math | [ Classical combinatorics (miscellaneous). ] [ Formulas of abbreviated multiplication (miscellaneous).]
Twenty-five coins are distributed into piles as follows. First, they are arbitrarily divided into two groups. Then any of the existing groups is again divided into two groups, and so on until each group consists of... | 300 | 100 | 3 |
math | [ Coordinate method on the plane $]$
Find the equation of the line passing through the intersection point of the lines $3 x+2 y-5=0$ and $x-3 y+2=0$ and parallel to the y-axis. | 1 | 52 | 1 |
math | ## 256. Math Puzzle $9 / 86$
In a quartz clock, the quartz crystal oscillates 32768 times in one second.
How much would a clock gain in a day if its quartz oscillated 32769 times, i.e., one more time per second? | 2.64 | 67 | 4 |
math | 1. (10 points) Calculate: $2 \frac{1}{6}+\left[\left(4.32-1.68-1 \frac{8}{25}\right) \times \frac{5}{11}-\frac{2}{7}\right] \div 1 \frac{9}{35}=$ | 2\frac{5}{12} | 76 | 9 |
math | Example 4 The function $f$ is defined on the set of ordered pairs of positive integers, and satisfies
$$
\begin{array}{c}
f(x, x)=x, f(x, y)=f(y, x), \\
(x+y) f(x, y)=y f(x, x+y) .
\end{array}
$$
Calculate $f(14,52)$. | 364 | 84 | 3 |
math | 9. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=a_{2}=a_{3}=1$. Let $b_{n}=a_{n}+a_{n+1}+a_{n+2}\left(n \in \mathbf{N}^{*}\right)$. If the sequence $\left\{b_{n}\right\}$ is a geometric sequence with a common ratio of 3, find the value of $a_{100}$. | \frac{3^{100}+10}{13} | 110 | 16 |
math | 3. Solve the equation
$$
\frac{x^{7}-1}{x^{5}-1}=\frac{x^{5}-1}{x^{3}-1}
$$ | 0,-1 | 37 | 3 |
math | 8.277. $\operatorname{ctg} x-\operatorname{tg} x-2 \operatorname{tg} 2 x-4 \operatorname{tg} 4 x+8=0$.
8.277. $\cot x - \tan x - 2 \tan 2x - 4 \tan 4x + 8 = 0$. | \frac{\pi}{32}(4k+3),k\in\mathbb{Z} | 85 | 22 |
math | Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$.
(D. Krekov) | 45^\circ | 73 | 4 |
math | We write down all the numbers from 1 to one million successively. Then we replace each number with the sum of its digits. Then we repeat this process until only single-digit numbers remain. Which digit appears most frequently? | 1 | 45 | 1 |
math | Consider a regular pentagon $ABCDE$, and let the intersection of diagonals $\overline{CA}$ and $\overline{EB}$ be $F$. Find $\angle AFB$.
[i]Proposed by Justin Chen[/i] | 108^\circ | 49 | 5 |
math | 19. (12 points) A bag contains 2 white balls and $n$ red balls $\left(n \geqslant 2\right.$, and $\left.n \in \mathbf{Z}_{+}\right)$, each time two balls are drawn from the bag (the two balls are returned to the bag after each draw). If the two balls drawn are of the same color, it is considered a win; otherwise, it is... | n=2 | 182 | 3 |
math | Example 2 Find all real-valued functions $f$ defined on the set of rational numbers, satisfying $f(x+y)=f(x)+f(y)+2xy$.
untranslated text remains unchanged. | f(x)=x^{2}+c x | 41 | 10 |
math | 492. The volumes obtained by rotating a rectangle about each of its sides are respectively $a$ cubic meters and $b$ cubic meters. Find the length of the diagonal of the rectangle. | \sqrt{\frac{^{2}+b^{2}}{}}\cdot\sqrt[6]{\frac{}{\pi^{2}}} | 40 | 30 |
math | Example 12 (2002 Hunan Provincial Competition Question) Let the two roots of the quadratic equation $2 x^{2}-t x-2=0$ with respect to $x$ be $\alpha, \beta(\alpha<\beta$.
(1) If $x_{1} 、 x_{2}$ are two different points in the interval $[\alpha, \beta]$, prove: $4 x^{2} x_{2}-t\left(x_{1}+x_{2}\right)-4<0$;
(2) Let $f(x... | 4 | 205 | 1 |
math | 32nd Putnam 1971 Problem A5 A player scores either A or B at each turn, where A and B are unequal positive integers. He notices that his cumulative score can take any positive integer value except for those in a finite set S, where |S| =35, and 58 ∈ S. Find A and B. Solution | A=11,B=8 | 75 | 7 |
math | Example 20 (1992 Canadian Mathematical Olympiad Training Problem) Find the positive integer solutions $(x, y, z)$ to the equation $8^{x}+15^{y}=17^{x}$. | (x,y,z)=(2,2,2) | 48 | 10 |
math | (4) Find all natural numbers $n$ greater than 3 such that $1+\mathrm{C}_{n}^{1}+\mathrm{C}_{n}^{2}+\mathrm{C}_{n}^{3}$ divides $2^{2000}$. | n=7n=23 | 58 | 7 |
math | Let's find the minimum of the expression $6 t^{2}+3 s^{2}-4 s t-8 t+6 s+5$.
---
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
---
Let's find the minimum of the expression $6 t^{2}+3 s^{2}-4 s t-8 t+6 s+5$. | \frac{8}{7} | 93 | 7 |
math | 8.5. 10 girls and 10 boys stood in a row such that girls and boys alternate, specifically from left to right: girl-boy-girl-boy and so on. Every minute, in one (any) pair of neighbors "girl-boy," the children can swap places, provided that the girl is to the left of the boy. Can such an "exchange process" continue for ... | 55 | 85 | 2 |
math | ## Task A-2.2.
Determine all triples $(a, b, c)$ of real numbers for which
$$
a^{2}+b^{2}+c^{2}=1 \quad \text { and } \quad(2 b-2 a-c) a \geqslant \frac{1}{2}
$$ | (\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}},-\frac{1}{\sqrt{6}})(-\frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}}) | 73 | 62 |
math | 9. (16 points) If real numbers $a$, $b$, $c$ satisfy
$$
2^{a}+4^{b}=2^{c}, 4^{a}+2^{b}=4^{c} \text {, }
$$
find the minimum value of $c$. | \log_{2}3-\frac{5}{3} | 65 | 13 |
math | 2. Every 20 minutes for a week, an exact amount of liters of water (always the same amount) is transferred from a tank with 25,000 liters to another initially empty reservoir. From this second reservoir, at regular intervals of time, 1 liter is extracted first, then 2 liters, then 3 liters, and so on. At the end of the... | 2016 | 160 | 4 |
math | The distance between the centers of circles with radii 2 and 3 is 8. Find the smallest and largest of the distances between points, one of which lies on the first circle, and the other on the second.
# | 313 | 47 | 3 |
math | Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which
\[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\]
is a rational number. | n = 1, 3 | 79 | 7 |
math | ## Task B-1.2.
If $\frac{a}{a+b^{2}}+\frac{b}{b+a^{2}}=\frac{4}{5}$ and $a b=-1$, what is $a^{9}+b^{9}$? | 1030 | 56 | 4 |
math | 3. [4] Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. A circle is tangent to the sides $A B$ and $A C$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $B C$. Given that $A B=6$, find the area of the portion of the circle that lies outsid... | \pi-2 | 104 | 4 |
math | 1. If the function
$$
f(x)=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)
$$
has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is . $\qquad$ | 2 \sqrt{3} | 92 | 6 |
math | Problem 5. The numbers $a, b$, and $c$ satisfy the equation $\sqrt{a}=\sqrt{b}+\sqrt{c}$. Find $a$, if $b=52-30 \sqrt{3}$ and $c=a-2$. | 27 | 59 | 2 |
math | 16. In Rt $\triangle A B C$, $\angle C=90^{\circ}, A C=11$, point $D$ is on side $A C$ and $A D=10, \angle B D C=$ $3 \angle B A C$. If the perimeter of $\triangle A B C$ is written in the form $\frac{1}{2}(m+\sqrt{n})$ (where $m, n$ are integers), then $m+n=$ $\qquad$ . | 638 | 108 | 3 |
math | 3B. If $a+b+c=0$ and $a^{2}+b^{2}+c^{2}=6$, calculate the value of the expression
$$
a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}
$$ | 9 | 65 | 1 |
math | Given a parameterized curve $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$.
Find the area bounded by the curve $ C$, the $ x$ axis and two lines $ x\equal{}\pm 1$. | \frac{5\sqrt{5}}{2} | 74 | 12 |
math | 1. A cooperative receives apple and grape juice in identical barrels and produces an apple-grape drink in identical cans. One barrel of apple juice is enough for exactly 6 cans of the drink, and one barrel of grape juice is enough for exactly 10 cans. When the recipe for the drink was changed, one barrel of apple juice... | 15 | 105 | 2 |
math | Example: Let $f(x)$ be a function defined on $\mathbf{R}$, for any $x, y \in \mathbf{R}$, we have
$$
f(x+3) \leqslant f(x)+3, f(x+2) \geqslant f(x)+2 .
$$
Let $g(x)=f(x)-x$.
(1) Prove: $g(x)$ is a periodic function;
(2) If $f(998)=1002$, find the value of $f(2000)$. | 2004 | 126 | 4 |
math | 10. A cube with an edge length of a certain integer is cut into 99 smaller cubes, 98 of which are unit cubes with an edge length of 1, and the other cube also has an integer edge length. Then its edge length is $\qquad$. | 3 | 58 | 1 |
math | 14.5.3 * Express the natural number $n$ as the sum of as many distinct natural numbers as possible. Find the maximum possible number of summands.
Express the natural number $n$ as the sum of as many distinct natural numbers as possible. Find the maximum possible number of summands. | [\frac{1}{2}(\sqrt{8n+1}-1)] | 65 | 17 |
math | 7. Simplify: (1) $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 47^{\circ} \sin 48^{\circ}}+\cdots+\frac{1}{\sin 133^{\circ} \sin 134^{\circ}}$;
(2) $\frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\cdots+\frac{1}{\sin 89^{\circ} \sin 90^{\circ}}$;
(3) $2 \sin 2^{\circ}+4 \sin ... | \frac{1}{\sin1},\frac{\cos1}{\sin1},90\cot1 | 208 | 24 |
math | Task B-3.4. If $\operatorname{tg} x=4+\operatorname{ctg} x$, calculate
$$
\frac{\operatorname{tg}^{2} x-\operatorname{ctg}^{2} x}{\operatorname{tg}^{3} x+\operatorname{ctg}^{3} x}
$$ | \frac{4}{17} | 77 | 8 |
math | $P$ is a polynomial of degree 4. $P(0)=P(1)=1, P(2)=4, P(3)=9, P(4)=16$. Calculate $P(-1)$ | 6 | 47 | 1 |
math | 21. Determine the maximum number of different sets consisting of three terms which form an arithmetic progressions that can be chosen from a sequence of real numbers $a_{1}, a_{2}, \ldots, a_{101}$, where
$$
a_{1}<a_{2}<a_{3}<\cdots<a_{101} .
$$ | 2500 | 77 | 4 |
math | 78. Find the distribution series of the random variable $X$ - the number of times 6 points appear when rolling a die once. | X:\begin{pmatrix}x_{i}&0&1\\p_{i}&5/6&1/6\end{pmatrix} | 29 | 32 |
math | 46. Calculate: $79 \cdot 81+1$. | 6400 | 16 | 4 |
math | 1. Calculate the remainder obtained when dividing $2^{4 n+2}+2^{4}$ by $2^{2 n+1}+2^{n+1}+1$ for every natural number $n \geq 2$. | 15 | 52 | 2 |
math | ## Task 2 - 210912
Mr. Schulze meets Mr. Lehmann after a long time and invites him to his home. On the way, he tells Mr. Lehmann that he is the father of three children. Mr. Lehmann wants to know how old they are; he is satisfied with age statements in full years.
Mr. Schulze replies: "The product of the three age st... | 6,6,2 | 244 | 5 |
math | 17. (CZE) ${ }^{\mathrm{DMO} 05}$ Find all pairs of integers $x, y \geq 1$ satisfying the equation $x^{y^{2}}=y^{x}$. | (1,1),(16,2),(27,3) | 51 | 15 |
math | 10. Tetrahedron $P-ABC, PA=BC=\sqrt{6}, PB=AC=\sqrt{8}, PC=AB=\sqrt{10}$, then the radius of the circumscribed sphere of the tetrahedron is $\qquad$ | \sqrt{3} | 58 | 5 |
math | 1768. From a batch of 5000 electric lamps, 300 were selected using a simple random sampling without replacement. The average burning duration of the lamps in the sample was 1450 hours, and the variance was 40000. What is the probability that the average burning duration of the lamps in the entire batch is within the ra... | 0.99964 | 94 | 7 |
math | Someone standing on the bank of a river sees a building on the opposite bank at an angle of $60^{\circ}$; if they move $30 \mathrm{~m}$ further, they see the building at an angle of $30^{\circ}$. Calculate the width of the river and the height of the building. | 15\mathrm{~},15\sqrt{3}=25.98\mathrm{~} | 70 | 24 |
math | Example 4 Find all positive integer tuples $(a, b, c, d)$ such that
$$\left\{\begin{array}{l}
b d > a d + b c, \\
(9 a c + b d)(a d + b c) = a^{2} d^{2} + 10 a b c d + b^{2} c^{2}
\end{array}\right.$$ | (a, 3a, c, 3c) | 88 | 12 |
math | KOVSHIN=BOTTLE+GLASS; $\quad$ TWO KOVSHINS=SEVEN GLASSES; BOTTLE=CUP+TWO GLASSES; $\quad$ BOTTLE=how many CUPS? | 5 | 48 | 1 |
math | Task B-4.1. Determine the natural numbers $k$ and $n$ such that
$$
\frac{((3!)!)!}{3!}=k \cdot n!
$$
where $n$ is the largest possible number with the given property. | k=120,n=719 | 56 | 10 |
math | 435. At time $t$, the body is at a distance of $s=\frac{1}{4} t^{4}+4 t^{3}+16 t^{2}$ km from the starting point. Find its acceleration after 2 hours. | 92 | 56 | 2 |
math | 2.22. Find $\operatorname{grad} U$, if:
a) $U=\sqrt{x^{2}+y^{2}+z^{2}}$;
b) $U=\frac{1}{x^{2}+y^{2}+z^{2}}$. | \begin{aligned}&)\operatorname{grad}U=\frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\ddot{i}+\frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\ddot{j}+\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\ddot | 62 | 87 |
math | 5. Given that $A$ took an integer number of hours to travel from location A to location B, and the number of kilometers $A$ walks per hour is the same as the number of hours it took to travel from location A to location B. $B$ walks 2 kilometers per hour from location A to location B, and rests for 1 hour after every 4... | 16 | 107 | 2 |
math | 3. (17 points) Find the smallest natural number that is simultaneously twice a perfect square and three times a perfect cube.
# | 648 | 27 | 3 |
math | 10. If $a^{2}+b^{2}=1, c^{2}+d^{2}=4$, then the range of values for $a d+b c$ is | [-2,2] | 40 | 5 |
math | 7.113. $x^{\lg x}=1000 x^{2}$. | 0.1;1000 | 22 | 8 |
math | $2 \cdot 73$ Let positive integers $a, b$ make $15a + 16b$ and $16a - 15b$ both squares of positive integers. Find the smallest value that the smaller of these two squares can take. | 481^2 | 58 | 5 |
math | (British MO) Determine the greatest value that the expression
$$
x^{2} y+y^{2} z+z^{2} x-x y^{2}-y z^{2}-z x^{2}
$$
can take for $0 \leqslant x, y, z \leqslant 1$. | \frac{1}{4} | 70 | 7 |
math | Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$.
| \lim_{n \to \infty} S_n = \frac{1}{2} | 69 | 20 |
math | ## Task B-4.4.
Given the complex number $w=-\frac{1}{2}+\frac{i \sqrt{3}}{2}$.
Calculate $(1+w)\left(1+w^{2}\right)\left(1+w^{3}\right) \ldots\left(1+w^{2019}\right)$. | 2^{673} | 74 | 6 |
math | 68*. All coefficients of the polynomial $p(x)$ are natural numbers less than 10. It is known that $p(10)=1248$. Find this polynomial. | p(x)=x^{3}+2x^{2}+4x+8 | 40 | 18 |
math | 2. The product of four different numbers, which are roots of the equations
$$
x^{2}+6 b x+c=0 \text{ and } x^{2}+6 c x+b=0
$$
is equal to 1. At least one of the numbers $\mathrm{b}$ and c is an integer. Find $\mathrm{b}+\mathrm{c}$ given that b and c are positive. | 2.5 | 90 | 3 |
math | 295. Find the number that is three times less than the square of the sum of its digits. | 3,27,48 | 22 | 7 |
math | Let $S$ be the set of all real numbers greater than or equal to $1$.
Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled. | f(x) = c \text{ constant for all } x \ge 1 | 81 | 17 |
math | 546. Two controllers $A$ and $B$ arranged samples of products made by nine masters in the order of decreasing quality (in parentheses are the ordinal numbers of products of the same quality):
(A)
$\begin{array}{llll}\text { (A) } & 1 & 2 \\ \text { (B) } & 2 & 1\end{array}$
$\left.\begin{array}{ccc}(3, & 4 & 5 \\ 4 & 3... | 0.93 | 164 | 4 |
math | Find the number of ways to choose 2005 red, green, and yellow balls such that the number of red balls is even or the number of green balls is odd. | \binom{2007}{2}-\binom{1004}{2} | 37 | 22 |
math | 5. Let $\left\{x_{n}\right\}$ be a sequence of positive real numbers. If $x_{1}=\frac{3}{2}$ and $x_{n+1}^{2}-x_{n}^{2}=\frac{1}{(n+2)^{2}}-\frac{1}{n^{2}}$ for all positive integers $n$, find $x_{1}+x_{2}+\cdots+x_{2009}$.
(1 mark)設 $\left\{x_{n}\right\}$ 爲一個正實數數列。若 $x_{1}=\frac{3}{2}$, 且對所有正整數 $n$ 皆有 $x_{n+1}^{2}-x_{... | \frac{4040099}{2010} | 222 | 16 |
math | Find all $n \in \mathbb{N}^{*}$ such that $n^{3}+n-2$ is a power of 2. | 2,5 | 34 | 3 |
math | 8. Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$, let $S=x^{2}+y^{2}$, then the value of $\frac{1}{S_{\max }}+\frac{1}{S_{\text {min }}}$ is | \frac{8}{5} | 68 | 7 |
math | 11. A sequence $U_{1}, U_{2}, U_{3}, \ldots$ is defined as follows:
- $U_{1}=2$;
- if $U_{n}$ is prime then $U_{n+1}$ is the smallest positive integer not yet in the sequence;
- if $U_{n}$ is not prime then $U_{n+1}$ is the smallest prime not yet in the sequence.
The integer $k$ is the smallest such that $U_{k+1}-U_{k}... | 270 | 131 | 3 |
math | 6. An aquarium in the shape of a rectangular parallelepiped has dimensions: length $1.5 m$, width - $400 mm$, height - 80 cm. It is filled with water at a rate of 2 liters/minute. How many seconds after the start of filling will the aquarium be completely filled? (10 points) | 14400 | 73 | 5 |
math | 2. Let $a, b, c \in \mathbf{R}$, and satisfy the system of equations
$$
\left\{\begin{array}{l}
a^{2}+b^{2}+c^{2}-10 a-11=0, \\
a^{2}-b c-4 a-5=0 .
\end{array}\right.
$$
Then the range of values for $a b+b c+c a$ is | [-40,72] | 99 | 7 |
math | A cylindrical container with a diameter of $36 \mathrm{~cm}$ has a cylindrical insert with a radius of $6 \mathrm{~cm}$ and another with a radius of $12 \mathrm{~cm}$ placed inside it. What is the maximum radius of another cylindrical insert that can be placed inside the container alongside the others? | \frac{36}{7}\approx5.14\mathrm{~} | 71 | 18 |
math | Problem 3. A pedestrian path with a rectangular shape has a length of $3030 \mathrm{~m}$ and a width of $180 \mathrm{~cm}$. The path needs to be covered with square tiles whose area is $9 \mathrm{~dm}^2$. How many such tiles are needed? | 60600 | 71 | 5 |
math | 4. Given the sequence $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$, and $a_{0}=3$. Then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$ $\qquad$ . | \frac{1}{3}\left(2^{n+2}-n-3\right) | 87 | 21 |
math | 8. Given that the volume of tetrahedron $P-ABC$ is $1$, $G$ and $K$ are the centroids of $\triangle ABC$ and $\triangle PBC$ respectively. A line through $G$ intersects $AB$ and $AC$ at points $M$ and $N$. Then the maximum volume of the quadrilateral pyramid $K-MNCB$ is | \frac{5}{27} | 82 | 8 |
math | 5. Variant 1.
It is known that
$$
\left(x^{2}-x+3\right)\left(y^{2}-6 y+41\right)\left(2 z^{2}-z+1\right)=77
$$
Find $\frac{x y}{z}$. | 6 | 65 | 1 |
math | 2. Calculate $A=\sqrt{\underbrace{111 \ldots 1}_{2 n}+\underbrace{111 \ldots 1}_{n+1}+\underbrace{666 \ldots 6}_{n}+8}$. | \underbrace{33\ldots33}_{n}6 | 59 | 15 |
math | 4. Quadrilateral $ABCD$ is inscribed in a circle of radius 7. The rays $AB$ and $DC$ intersect at point $P$, and the rays $BC$ and $AD$ intersect at point $Q$. It is known that triangles $ADP$ and $QAB$ are similar (vertices are not necessarily in the corresponding order).
a) Find $AC$.
b) Suppose additionally that t... | AC=14,\angleDAC=45,S_{ABCD}=94 | 167 | 17 |
math |
10.4. A positive integer is called simple if its ordinary decimal representation consists entirely of zeroes and ones. Find the least positive integer $k$ such that each positive integer $n$ can be written as $n=a_{1} \pm a_{2} \pm a_{3} \pm \cdots \pm a_{k}$, where $a_{1}, \ldots, a_{k}$ are simple.
| 9 | 91 | 1 |
math | ## Task 1 - 260621
In the following five-digit number, two digits have become illegible and have been replaced by asterisks.
$$
27 * * 7
$$
Instead of the asterisks, two digits are to be inserted so that the number is divisible by 9. Give all five-digit numbers that can arise from such insertion! Verify that all the... | 27027,27117,27207,27297,27387,27477,27567,27657,27747,27837,27927 | 91 | 65 |
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