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 | Several siblings are dividing the inheritance. According to the will, $A$ receives 2000 crowns and one-tenth of the remainder; $B$ receives 4000 crowns and one-tenth of the remainder; $C$ receives 6000 crowns and one-tenth of the remainder, and so on. After the division, it turned out that each sibling received an equa... | 9 | 107 | 1 |
math | A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes conta... | 2 | 130 | 3 |
math | 4. A circle is filled with 12 positive integers, each taken from $\{1,2, \cdots, 9\}$ (each number can appear multiple times on the circle). Let $S$ represent the sum of all 12 numbers on the circle. If the sum of any three consecutive numbers on the circle is a multiple of 7, then the number of possible values for $S$... | 9 | 93 | 1 |
math | Problem 6.6. On an island, there live knights who always tell the truth, and liars who always lie. One day, 65 inhabitants of the island gathered for a meeting. Each of them, in turn, made the statement: "Among the previously made statements, there are exactly 20 fewer true statements than false ones." How many knights... | 23 | 80 | 2 |
math | 3. There are two cylinders with a volume ratio of $5: 8$. The lateral surfaces of these cylinders unfold into the same rectangle. If the length and width of this rectangle are both increased by 6, its area increases by 114. What is the area of this rectangle? $\qquad$ . | 40 | 66 | 2 |
math | ## Task 3 - 250613
Dirk and Jörg met at the secondary raw materials collection center. Jörg has bundled his waste paper into several packages of $5 \mathrm{~kg}$ each and also has an additional $3 \mathrm{~kg}$ of loose paper.
Dirk delivers $32 \mathrm{~kg}$ of paper. When they compare their collection results, they ... | 4\cdot5+3=235\cdot5+3=28 | 148 | 18 |
math | Example 4 (Adapted from the 26th International Mathematical Olympiad 1994) Let $f:(-1,+\infty) \rightarrow (-1,+\infty)$ be a continuous and monotonic function, with $f(0)=0$, and satisfying
$$
f[x+f(y)+x f(y)] \geqslant y+f(x)+y f(x) \text {, for } \forall x, y \in(-1,+\infty),
$$
Find $f(x)$. | f(x)=-\frac{x}{1+x};f(x)=x | 114 | 15 |
math | 2. Given that $\alpha$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$, then the value of $\frac{\alpha^{3}-1}{\alpha^{5}+\alpha^{4}-\alpha^{3}-\alpha^{2}}$ is. $\qquad$ . | 20 | 70 | 2 |
math | 9. There are 1000 lamps and 1000 switches, each switch controls all lamps whose numbers are multiples of its own, initially all lamps are on. Now pull the 2, 3, 5 switches, then the number of lamps that are still on is $\qquad$. | 499 | 64 | 3 |
math | $A$ and $B$ are chess players who compete under the following conditions: The winner is the one who first reaches (at least) 2 points; if they both reach 2 points at the same time, the match is a draw. (A win is 1 point, a draw is $1 / 2$ point, a loss is 0 points) -
a) What is the expected number of games, if the pro... | 0.315 | 132 | 5 |
math | 13.430 A passenger can travel from Moscow to city $N$ by train. In this case, he will be on the way for 20 hours. If, however, he waits for the departure of the plane (and he will have to wait more than 5 hours after the train departs), the passenger will reach city $N$ in 10 hours, including the waiting time. How many... | 10 | 142 | 2 |
math | What is the value of $\frac{1}{10}-\frac{1}{100}+\frac{1}{1000}-\frac{1}{10000}$? | \frac{909}{10000} | 44 | 13 |
math | 9. Given that $x, y, z$ are integers from 1 to 9, and $x>y>z$, find the integer solutions $(x, y, z)$ for the equation $3(x-y)=4(y-z)$. | (8,4,1),(9,5,2) | 51 | 13 |
math | ## Task $2 / 78$
Determine the number $A(n)$ of at least two-digit natural numbers (in decimal notation) that have the following property:
Each digit with a higher place value is smaller than each digit with a lower place value. | 502 | 53 | 3 |
math | 8.2. It is known that $70 \%$ of mathematicians who have moved to IT regret their change of activity. At the same time, only $7 \%$ of all people who have moved to IT regret the change. What percentage of those who have moved to IT are mathematicians, if only they regret the change of activity? | 10 | 71 | 2 |
math | 4. [6] Find the real solution(s) to the equation $(x+y)^{2}=(x+1)(y-1)$. | (-1,1) | 31 | 5 |
math | 3. Given three vertices of a cube are
$$
P(4,7,4), Q(7,11,4), R(11,8,9) \text {. }
$$
then the coordinates of the center of the cube are $\qquad$ | \left(\frac{15}{2}, \frac{15}{2}, \frac{13}{2}\right) | 58 | 28 |
math | 1. Determine all integer solutions $x, y, z$ such that
$$
x^{2}\left(x^{2}+y\right)=y^{z+1}
$$ | (0,0,z),z\geq0 | 39 | 11 |
math | Let's determine $m$ such that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
---
Determine $m$ so that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$. | <-\frac{1}{3} | 79 | 8 |
math | $2 \cdot 61$ (1) For $n$ mutually independent numbers $x_{1}, x_{2}, \cdots, x_{n}$ each taking values 1, 0, or -1, find the minimum value of the sum of the pairwise products of these $n$ numbers.
(2) Find the minimum value of the sum of the pairwise products of $n$ numbers whose absolute values do not exceed 1. | -\frac{n}{2}forevenn,-\frac{n-1}{2}foroddn | 94 | 21 |
math | Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ .
(I. Voronovich) | f(x) = a - x \quad \forall x \in \mathbb{Q}, \text{ where } a \in \mathbb{Q} | 53 | 34 |
math | Example 12 (1976 Yugoslav Mathematical Olympiad) Let $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}, n=$ $1,2,3, \cdots$ Find $\sum_{k=1}^{99} a_{k}$. | \frac{9}{10} | 73 | 8 |
math | 1. Let's call a natural number $n$ squareable if the numbers from 1 to $n$ can be arranged in such an order that each member of the sequence, when added to its position, results in a perfect square. For example, the number 5 is squareable, as the numbers can be arranged as: 32154, in which $3+1=2+2=1+3=4$ and $5+4=4+5=... | 915 | 124 | 3 |
math | 2. How many three-digit natural numbers have an even number of distinct natural divisors? | 878 | 18 | 3 |
math | Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418{,}195{,}493$.
[i]Proposed by Eugene Chen[/i] | 2 | 72 | 1 |
math | $$
\begin{aligned}
M= & |2012 x-1|+|2012 x-2|+\cdots+ \\
& |2012 x-2012|
\end{aligned}
$$
The minimum value of the algebraic expression is . $\qquad$ | 1012036 | 68 | 7 |
math | Task 6 - 200736 During one day, students from each of the grades 6, 7, and 8 came to a lending library; in total, there were 85 students. Exactly one third of the students from grade 6, exactly one third of the students from grade 7, and exactly one quarter of the students from grade 8, which amounted to a total of 26 ... | 27,30,28 | 171 | 8 |
math | 77. Heat Flow. The temperature of three sides of a square metal sheet is maintained at $0^{\circ} \mathrm{C}$, while the temperature of the fourth side is maintained at $100^{\circ} \mathrm{C}$. Neglecting heat loss due to radiation, find the temperature at the center of the sheet. | 25\mathrm{C} | 74 | 7 |
math | 14. The sequence $\left\{a_{n}\right\}$ is an arithmetic sequence, and it satisfies $3 a_{5}=8 a_{12}>0$. The sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=a_{n} \cdot a_{n+1} \cdot a_{n+2}\left(n \in \mathbf{N}^{*}\right)$, and the sum of the first $n$ terms of $\left\{b_{n}\right\}$ is denoted as $S_{n}$. For what... | 16 | 147 | 2 |
math | (Example 3 Given $f(x)=\log _{a}\left(2-a^{x}\right)$ is a decreasing function of $x$ on $[0,1]$, find the range of real number $a$. | 0<<1or1<<2 | 49 | 7 |
math | An ice cream vendor offers three different flavors. A person buys an ice cream with 5 scoops. How many possibilities are there (the order of the scoops does not matter). | 21 | 37 | 2 |
math | 24. Let $\left\{x_{n}\right\}_{n=1}^{\infty}$ be a sequence of real numbers such that $x_{1}=3, x_{2}=24$ and
$$
x_{n+2}=\frac{1}{4} x_{n+1}+\frac{3}{4} x_{n}
$$
for every positive integers $n$. Determine the value of $\lim _{n \rightarrow \infty} x_{n}$. | 15 | 109 | 2 |
math | 1. Given complex numbers $z$ and $\omega$ satisfy the following two conditions:
(1) $z+\omega+3=0$;
(2) $|z|, 2, |\omega|$ form an arithmetic sequence.
Is there a maximum value for $\cos (\arg z - \arg \omega)$? If so, find it. | \frac{1}{8} | 74 | 7 |
math | Example 4 Determine all complex numbers $\alpha$ such that for any complex numbers $z_{1}, z_{2}\left(\left|z_{1}\right|<1,\left|z_{2}\right|<1, z_{1} \neq z_{2}\right)$, we have $\left(z_{1}+\alpha\right)^{2}+\alpha \overline{z_{1}} \neq\left(z_{2}+\alpha\right)^{2}+\alpha \overline{z_{2}}$. | {\alpha|\alpha\in{C},|\alpha\mid\geqslant2} | 117 | 20 |
math | Solve the following system of equations:
$$
\frac{2 x^{2}}{1+x^{2}}=y, \quad \frac{2 y^{2}}{1+y^{2}}=z, \quad \frac{2 z^{2}}{1+z^{2}}=x
$$ | 0or1 | 66 | 3 |
math | 3. In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $1,2,3,4,5,6,7,8,9$ (digits can repeat) so that the resulting 12-digit number is divisible by 18. In how many ways can this be done? | 26244 | 89 | 5 |
math | 5. By definition, $n!=1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$. Which factor should be removed from the product $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 20!$ so that the remaining product becomes a square of some natural number? | 10! | 72 | 3 |
math | Find all integers $n \in \mathbb{N}$ such that $(n+1) \mid\left(n^{2}+1\right)$. | 01 | 34 | 2 |
math | Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$, with $ x, y$ being integers. | 5 | 40 | 1 |
math | Problem 11.4. Find the least positive integer $a$ such that the system
$$
\left\lvert\, \begin{aligned}
& x+y+z=a \\
& x^{3}+y^{3}+z^{2}=a
\end{aligned}\right.
$$
has no an integer solution.
Oleg Mushkarov | 4 | 77 | 1 |
math | 3. For any $x \in[0,1]$, we have $|a x+b| \leqslant 1$.
Then the maximum value of $|b x+a|$ is $\qquad$ | 2 | 47 | 1 |
math | We remove the four corners of an $n \times n$ rectangle. Is it possible to cover it with L-tetrominoes? | 4k+2 | 29 | 4 |
math | ## Task $8 / 68$
A plane is divided into 56 parts by $n$ lines. None of the $n$ lines is parallel to another, and no more than two lines intersect at any point.
What is $n$? | 10 | 53 | 2 |
math | Example 3. Integrate the equation
$$
\left(y^{3}-2 x y\right) d x+\left(3 x y^{2}-x^{2}\right) d y=0
$$ | y^{3}x-x^{2}C | 46 | 10 |
math | Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$? | 1:2 | 69 | 5 |
math | $15 \cdot 18$ in simplest form has a denominator of 30, find the sum of all such positive rational numbers less than 10.
(10th American Invitational Mathematics Examination, 1992) | 400 | 51 | 3 |
math | In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament? | 12 | 72 | 2 |
math | 607. The sum of a two-digit number and its reverse is a perfect square. Find all such numbers.
(Definition of the number reversed to the given one, see § 12, p. 12.4.) | 29,38,47,56,65,74,83,92 | 49 | 23 |
math | 4. In the number $2016^{* * * *} 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,6,7,8$ (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done | 2160 | 78 | 4 |
math | 4.4.1. (12 points) A goat eats 1 hay wagon in 6 weeks, a sheep in 8 weeks, and a cow in 3 weeks. How many weeks will it take for 5 goats, 3 sheep, and 2 cows to eat 30 such hay wagons together? | 16 | 68 | 2 |
math | 2. A notebook costs 10 rubles. Eight children bought notebooks, and each had a different non-zero amount of rubles left, but none had enough for another notebook. The children pooled their remaining rubles, and it was exactly enough to buy several more notebooks. How much money did each child have left before pooling? | 1,2,3,4,6,7,8,9 | 67 | 15 |
math | ## Task Condition
Find the derivative of the specified order.
$y=e^{-x} \cdot(\cos 2 x-3 \sin 2 x), y^{IV}=?$ | -e^{-x}\cdot(79\cos2x+3\sin2x) | 39 | 19 |
math | Example 2. In Rt $\triangle A B C$, $\angle C=90^{\circ}, \angle A B C$ $=66^{\circ}, \triangle A B C$ is rotated around $C$ to the position of $\triangle A^{\prime} B^{\prime} C^{\prime}$, with vertex $B$ on the hypotenuse $A^{\prime} B^{\prime}$, and $A^{\prime} C$ intersects $A B$ at $D$. Find $\angle B D C$. (1993,... | 72^{\circ} | 137 | 6 |
math | Example X (China Mathematical Olympiad 2000) Given that $a, b, c$ are the sides of $\triangle ABC$, $a \leqslant b \leqslant c$, and $R$ and $r$ are the radii of the circumcircle and incircle of $\triangle ABC$, respectively. Let $f=a+b-2 R-2 r$, try to determine the sign of $f$ using the size of angle $C$.
| When\C\in[\frac{\pi}{3},\frac{\pi}{2}),\f>0;\when\C=\frac{\pi}{2},\f=0;\when\C\in(\frac{\pi}{2},\pi),\f<0 | 101 | 55 |
math | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=1000$, if $f(0)=1$ and for any $x$ the equality $f(x+2)=f(x)+4 x+2$ holds. | 999001 | 61 | 6 |
math | A sequence $u_{1}, u_{2}, \ldots, u_{n}$ composed of the integers $1,2, \ldots, n$ is quasi-increasing if, for every index $k, u_{k} \leq u_{k+1}+2$. For example, the sequence $1,6,4,2,5,3$ is quasi-increasing but the sequence $1,4,6,2,5,3$ is not. Determine the number of quasi-increasing sequences of $n$ terms. | s_{n}=2\times3^{n-2} | 117 | 13 |
math | 4. Vanya wrote down a four-digit number, subtracted a two-digit number from it, multiplied the result by a two-digit number, divided by the sum of two single-digit numbers, added a single-digit number, and then divided the result by the sum of three single-digit numbers. To write all the numbers, he used only one digit... | 2017 | 98 | 4 |
math | Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$, such that: no matter which $m$ of these numbers are erased, among the remaining $1000-m$ numbers, there will be two such that one divides the other. | 499 | 65 | 3 |
math | 10. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
a b+c+d=3 \\
b c+d+a=5 \\
c d+a+b=2 \\
d a+b+c=6
\end{array}\right.
$$ | (a, b, c, d)=(2,0,0,3) | 62 | 16 |
math | 404. A cylindrical glass is filled with mercury. Calculate the force of pressure of the mercury on the side surface of the glass, if its height is 0.1 m, and the radius of the base is 0.04 m. The density of mercury is $13600 \mathrm{kg} / \mathrm{m}^{3}$. | 167.6(\mathrm{N}) | 78 | 10 |
math | 3.5 On the first day of the sports competition, $\frac{1}{6}$ of the boys' team and $\frac{1}{7}$ of the girls' team did not meet the qualifying standards and were eliminated from further competition. Over the remaining period of the competition, an equal number of athletes from both teams were eliminated due to non-co... | 72 | 153 | 2 |
math | 13.089. A train traveled a certain distance at a speed of 120 km/h. After that, it traveled a distance 75 km longer at a speed of 150 km/h, and the remaining distance, 135 km shorter than the distance traveled, at a speed of 96 km/h. How long is the entire journey if the average speed of the train turned out to be 120 ... | 415 | 97 | 3 |
math | $12 \cdot 1$ There is a pile of 100 small weights, with a total weight of 500 grams. It is known that there are only 1-gram, 10-gram, and 50-gram weights. In this pile of weights, how many of each type of weight are there?
（9th All-Russian Mathematical Olympiad, 1983） | 60 | 88 | 2 |
math | # Problem 5. (3 points)
In trapezoid $A B C D$, the lateral side $C D$ is equal to the diagonal $A C$. On the smaller arc $B C$ of the circumscribed circle of triangle $B C D$, a point $E$ is chosen such that $C D=C E$. Find the angle $\angle A E B$. | 90 | 81 | 2 |
math | 11. (10 points) There are 20 piles of stones, each containing 2006 stones. The rule is: taking one stone from each of any 19 piles and placing them into another pile is considered one operation. After fewer than 20 such operations, one pile has 1990 stones, and another pile has between 2080 and 2100 stones. How many st... | 2090 | 97 | 4 |
math | Example 1. Solve the system of equations
$$
\left\{\begin{array}{l}
3 x+2 y-6=0, \\
2(x+2 y)+5(x-3)=0 .
\end{array}\right.
$$ | \left\{\begin{array}{l}x=3, \\ y=-\frac{3}{2} .\end{array}\right.} | 54 | 33 |
math | 12th Chinese 1997 Problem A1 The real numbers x 1 , x 2 , ... , x 1997 have sum -318 √3 and satisfy -1/√3 ≤ x i ≤ √3. What is the maximum possible value for the sum of their 12th powers? Solution | 1736/3^6+260\cdot3^6+2^{12}/3^6 | 73 | 26 |
math | You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime positiv... | 67 | 84 | 2 |
math | 7. Three distinct lines are drawn in the plane. Suppose there exist exactly $n$ circles in the plane tangent to all three lines. Find all possible values of $n$. | 0,2,4 | 36 | 5 |
math | 14. (3 points) There is a clock that now shows 10 o'clock. After $\qquad$ minutes, the minute hand and the hour hand overlap for the first time; after another $\qquad$ minutes, the minute hand and the hour hand overlap for the second time. | 54\frac{6}{11};65\frac{5}{11} | 61 | 20 |
math | 43. Find the general solution of the equation $1+y^{\prime}+y+x y^{\prime}=0$. | \frac{C}{1+x}-1 | 27 | 9 |
math | Determine the smallest positive real $K$ such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$.
[i]Proposed by Fajar Yuliawan, Indonesia[/i] | \frac{\sqrt{6}}{3} | 90 | 10 |
math | 4-181 Find all integer pairs $(a, b)$, where $a \geqslant 1, b \geqslant 1$, and satisfy the equation $a^{b^{2}}=b^{a}$. | (,b)=(1,1),(16,2),(27,3) | 52 | 18 |
math | Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies
\[
\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},
\]
compute the ordered triple $(a, b, c)$. | (16, -11, 7) | 122 | 11 |
math | 2. (2 points) Find the minimum value of the expression $4 x^{2}+4 x \sin y-\cos ^{2} y$. | -1 | 33 | 2 |
math | 3. Find all natural numbers $n$ for which the number $2^{10}+2^{13}+2^{14}+3 \cdot 2^{n}$ is a square of a natural number.
$(16$ points) | 13,15 | 54 | 5 |
math | 2. Let $a_{1}=1, a_{n+1}=2 a_{n}+n^{2}$, then the general term formula $a_{n}$ $=$ $\qquad$ | 7 \times 2^{n-1}-n^{2}-2 n-3 | 43 | 18 |
math | [ Extremal properties (miscellaneous).]
What is the largest number of points that can be placed on a segment of length 1 so that on any segment of length $d$, contained in this segment, there are no more than $1+1000 d^{2}$ points? | 32 | 61 | 2 |
math | 9. (1973 Kyiv Mathematical Olympiad) Find three prime numbers such that their product is five times their sum. | 2,5,7 | 27 | 5 |
math | For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$. Write all their divisors in an ascending order: $1=m_1<\ldots<m_k=m$, $1=n_1<\ldots<n_k=n$ and set
$$
m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k.
$$
Find all pairs of numbers $(m, n)$, $m\geqslant n$,... | (18, 20) | 128 | 9 |
math | ## Task B-3.6.
Determine all real numbers $x$ in the interval $[0,1]$ for which $\operatorname{tg}\left(2 \pi \sin ^{2}(2 \pi x)\right)=0$. | {0,\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{5}{8},\frac{3}{4},\frac{7}{8},1} | 53 | 53 |
math | If $x$, $y$, $k$ are positive reals such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\] find the maximum possible value of $k$. | \frac{\sqrt{7} - 1}{2} | 80 | 13 |
math | Find natural numbers $a$ and $b$ such that $7^{3}$ is a divisor of $a^{2}+a b+b^{2}$, but 7 is not a divisor of either $a$ or $b$. | =1,b=18 | 50 | 6 |
math | 9-8-1. In Midcity, houses stand along one side of a street, each house can have $1,2,3, \ldots, 9$ floors. According to an ancient law of Midcity, if two houses on one side of the street have the same number of floors, then no matter how far apart they are from each other, there must be a house with more floors between... | 511 | 105 | 3 |
math | A natural number $N$ greater than 20 can be represented as a palindrome in both base 14 and base 20 (a palindrome is a number that reads the same forward and backward, for example, $12321$ and $3443$ are palindromes, while 12331 is not a palindrome). The minimum value of $N$ is $\qquad$ (answer in decimal). | 105 | 94 | 3 |
math | 2. Solve the equation
$$
\log _{(3 x+7)}\left(9+12 x+4 x^{2}\right)+\log _{(2 x+3)}\left(6 x^{2}+23 x+21\right)=4
$$ | x_{3}=-\frac{1}{4} | 63 | 12 |
math | 6. [5] Let $A B C D$ be an isosceles trapezoid such that $A B=10, B C=15, C D=28$, and $D A=15$. There is a point $E$ such that $\triangle A E D$ and $\triangle A E B$ have the same area and such that $E C$ is minimal. Find $E C$. | \frac{216}{\sqrt{145}} | 92 | 14 |
math | Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\begin{align*}x_1(x_1+1)&=A\\x_2(x_2+1)&=A\\x_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*} | -7 | 106 | 2 |
math | 51 Given $\alpha, \beta>0, x, y, z \in \mathbf{R}^{+}, x y z=2004$. Find the maximum value of $u$, where, $u=$
$$\sum_{\mathrm{oc}} \frac{1}{2004^{\alpha+\beta}+x^{\alpha}\left(y^{2 \alpha+3 \beta}+z^{2 \alpha+3 \beta}\right)} .$$ | 2004^{-(\alpha+\beta)} | 106 | 11 |
math | 2.4. (SRP, 81). Solve the equation
$$
x^{6}+3 x^{3}+1=y^{4}
$$
in integers. | 0,\1 | 39 | 3 |
math |
3.360. $\frac{\cos 67^{\circ} \cos 7^{\circ}-\cos 83^{\circ} \cos 23^{\circ}}{\cos 128^{\circ} \cos 68^{\circ}-\cos 38^{\circ} \cos 22^{\circ}}-\operatorname{tg} 164^{\circ}$.
| 0 | 96 | 1 |
math | 9. Given a point $P(3,1)$ and two lines $l_{1}: x+2 y+3=0, l_{2}: x+2 y-7=0$, find the equation of the circle passing through $P$ and tangent to $l_{1}, l_{2}$. | (x-\frac{4}{5})^{2}+(y-\frac{3}{5})^{2}=5or(x-4)^{2}+(y+1)^{2}=5 | 66 | 41 |
math | A cube with an edge of $12 \mathrm{~cm}$ was divided into smaller, identical cubes such that the sum of the surface areas of all the smaller cubes was eight times greater than the surface area of the original cube.
Determine how many small cubes there were and what the length of their edges was.
(M. Volfová) | 512 | 71 | 3 |
math | 3. In recent years, the use of mortgage loans by young families has become quite popular. Consider the possibility of obtaining a mortgage with a constant, fixed interest rate. Assume that the repayment of such a loan is made through equal (annuity) payments at the end of each payment period stipulated by the contract.... | S-2T+rS-0.5rT+(0.5rS)^2<S-2T+rS | 271 | 26 |
math | 3. [15] Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{0}$, where each $a_{i}$ is either 1 or -1 . Let $r$ be a root of $p$. If $|r|>\frac{15}{8}$, what is the minimum possible value of $n$ ? | 4 | 91 | 1 |
math | 8. Let the three-digit number $n=\overline{a b c}$, if the lengths $a, b, c$ can form an isosceles (including equilateral) triangle, then the number of such three-digit numbers is $\qquad$.
| 165 | 56 | 3 |
math | Example 10 Given that $a, b, x, y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=16, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$
Find the value of $a x^{5}+b y^{5}$. | 20 | 110 | 2 |
math | An empty swimming pool was filled with water by two faucets A and B, both with constant flow rates. For four hours, both faucets were open and filled $50 \%$ of the pool. Then, faucet B was turned off and, for two hours, faucet A filled $15 \%$ of the pool's volume. After this period, faucet A was turned off and faucet... | 7 | 104 | 1 |
math | [Example 4.5.6] Given that $p, q$ are both integers greater than 1, and $2 p-1$ is a multiple of $q$, $2 q-1$ is a multiple of $p$, find the value of $p+q$. | 8 | 60 | 1 |
math | Let's calculate the sides and angles of the triangle, given that the radius of the circumscribed circle around the triangle is \( R = 8.5 \, \text{m} \), one of the angles of the triangle is \( \alpha = 47^\circ \), and the difference between the heights \( m_b \) and \( m_c \), which are perpendicular to the sides of ... | 12.433\, | 111 | 8 |
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