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20,662 | Solve the system of equations:
$$
\left\{\begin{array}{l}
X^{2} Y^{2}+X Y^{2}+X^{2} Y+X Y+X+Y+3=0 \\
X^{2} Y+X Y+1=0
\end{array}\right.
$$ | (-2, -\dfrac{1}{2}) |
20,671 | 1. The rays forming a right angle with the vertex at the coordinate origin intersect the parabola $y^{2}=2 x$ at points $X$ and $Y$. Find the geometric locus of the midpoints of the segments $X Y$. | y^2 = x - 2 |
20,679 | Ten children were given 100 pieces of macaroni each on their plates. Some children didn't want to eat and started playing. With one move, one child transfers one piece of macaroni from their plate to each of the other children's plates. What is the minimum number of moves needed such that all the children end up with a different number of pieces of macaroni on their plates? | 45 |
20,683 | Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards). | 89 |
20,695 | For 8 $\star \star$ positive integers $r, n$ satisfying $1 \leqslant r \leqslant n$, find the arithmetic mean $f(r, n)$ of the smallest numbers in all $r$-element subsets of $\{1,2, \cdots, n\}$. | \dfrac{n + 1}{r + 1} |
20,735 | The case of Brown, Jones, and Smith is being investigated. One of them committed a crime. During the investigation, each of them made two statements. Brown: “I didn't do it. Jones didn't do it.” Smith: “I didn't do it. Brown did it.” Jones: “Brown didn't do it. Smith did it.” It was later found out that one of them told the truth twice, another lied twice, and the third told the truth once and lied once. Who committed the crime? | Brown |
20,749 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{\sqrt{1-2 x+3 x^{2}}-(1+x)}{\sqrt[3]{x}}$ | 0 |
20,766 | 1. Compute $\sin 18^{\circ}$ without tables. | \dfrac{\sqrt{5} - 1}{4} |
20,776 | 5. (6 points) 12 question setters make guesses about the answer to this question, with their guesses being “not less than 1”, “not greater than 2”, “not less than 3”, “not greater than 4”, “not less than 11”, “not greater than 12” (“not less than” followed by an odd number, “not greater than” followed by an even number). The number of question setters who guessed correctly is $\qquad$ people. | 7 |
20,779 | Example 5. Calculate the double integral $I=\int_{D} \int(x+y) d x d y$, where $D-$ is the region bounded by the lines $x=0, y=x^{2}+x-3$, $2 y=3 x(x \geqslant 0)$. | \dfrac{14}{5} |
20,807 | 3. Given real numbers $x_{1}, \ldots, x_{n}$. Find the maximum value of the expression
$$
A=\left(\sin x_{1}+\ldots+\sin x_{n}\right) \cdot\left(\cos x_{1}+\ldots+\cos x_{n}\right)
$$ | \dfrac{n^2}{2} |
20,825 | The angle bisector of angle \(ABC\) forms an angle with its sides that is three times smaller than the adjacent angle to \(ABC\). Find the measure of angle \(ABC\). | 72 |
20,837 | 3. Suppose that each of the two banks $A$ and $B$ will have a constant annual interest rate over the next two years. If we deposit 5/6 of our savings in bank $A$ and the remainder in bank $B$, our savings will grow to 67000 Kč after one year and to 74900 Kč after two years. However, if we deposit 5/6 of our savings in bank $B$ and the remainder in bank $A$, our savings will grow to 71000 Kč after one year. To what amount would our savings increase after two years in the latter case? | 84100 |
20,863 | In the triangle \(ABC\), \(AB = 8\), \(BC = 7\), and \(CA = 6\). Let \(E\) be the point on \(BC\) such that \(\angle BAE = 3 \angle EAC\). Find \(4AE^2\). | 135 |
20,882 | AC is a diameter of a circle. AB is a tangent. BC meets the circle again at D. AC = 1, AB = a, CD = b. Show that \( \frac{1}{a^2 + ½} < \frac{b}{a} < \frac{1}{a^2} \). | \frac{1}{a^2 + \frac{1}{2}} < \frac{b}{a} < \frac{1}{a^2} |
20,889 | 1. A group of schoolchildren heading to a school camp was to be seated in buses so that each bus had the same number of passengers. Initially, 22 people were to be seated in each bus, but it turned out that three schoolchildren could not be seated. When one bus left empty, all the schoolchildren were able to sit evenly in the remaining buses. How many schoolchildren were in the group, given that no more than 18 buses were allocated for transporting the schoolchildren, and no more than 36 people can fit in each bus? Provide the answer as a number without specifying the unit.
$(2$ points) | 135 |
20,945 | Example 2. Find the mass of the plate $D$ with surface density $\mu=x^{2} /\left(x^{2}+y^{2}\right)$, bounded by the curves
$$
y^{2}-4 y+x^{2}=0, \quad y^{2}-8 y+x^{2}=0, \quad y=\frac{x}{\sqrt{3}}, \quad x=0
$$ | \pi + \dfrac{3\sqrt{3}}{8} |
20,951 | How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? | 576 |
20,953 | 20. We can find sets of 13 distinct positive integers that add up to 2142. Find the largest possible greatest common divisor of these 13 distinct positive integers. | 21 |
20,964 | In triangle \(ABC\), altitude \(AD\) is drawn. A circle is tangent to \(BC\) at point \(D\), intersects side \(AB\) at points \(M\) and \(N\), and side \(AC\) at points \(P\) and \(Q\). Prove that:
$$
\frac{AM + AN}{AC} = \frac{AP + AQ}{AB}
$$ | \frac{AM + AN}{AC} = \frac{AP + AQ}{AB} |
20,969 | Given \( n \) new students such that among any 3 students, there are at least 2 students who know each other, and among any 4 students, there are at least 2 students who do not know each other. Determine the maximum value of \( n \). | 8 |
21,010 | Example 5. Find the integral $\int \frac{d x}{\sqrt{5-4 x-x^{2}}}$. | \arcsin\left(\frac{x + 2}{3}\right) + C |
21,012 | The base of a rectangular parallelepiped is a square with a side length of \(2 \sqrt{3}\). The diagonal of a lateral face forms an angle of \(30^\circ\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped. | 72 |
21,055 | 8. 4 people pass the ball to each other, with the requirement that each person passes the ball to someone else immediately after receiving it. Starting with person A, and counting this as the first pass, find the total number of different ways the ball can be passed such that after 10 passes, the ball is back in the hands of the starting player A. | 14763 |
21,073 | Mrs. Kucera was on a seven-day vacation, and Káta walked her dog and fed her rabbits for the entire duration. For this, she received a large cake and 700 CZK. After another vacation, this time a four-day one, Káta received the same cake and 340 CZK for walking and feeding according to the same rules.
What was the value of the cake? | 140 |
21,106 | Determine the geometric place of those points from which the parabola $y^{2}-2 p x=0$ is seen at an angle of $45^{\circ}$. | \left(x + \frac{3p}{2}\right)^2 - y^2 = 2p^2 |
21,114 | 2. A three-digit number is 33 times greater than the sum of its digits. Prove that this number is divisible by 9, and then determine this three-digit number. | 594 |
21,123 | 37. Simplify $\prod_{k=1}^{2004} \sin (2 \pi k / 4009)$. | \dfrac{\sqrt{4009}}{2^{2004}} |
21,127 | Solve the following equation in the set of real numbers:
$$
x^{2}+5 y^{2}+5 z^{2}-4 x z-2 y-4 y z+1=0 .
$$ | (4, 1, 2) |
21,137 | 10.251. Five circles are inscribed in an angle of $60^{\circ}$ such that each subsequent circle (starting from the second) touches the previous one. By what factor is the sum of the areas of all five corresponding circles greater than the area of the smallest circle? | 7381 |
21,154 | 4. Find all roots of the equation
$1-\frac{x}{1}+\frac{x(x-1)}{2!}-\frac{x(x-1)(x-2)}{3!}+\frac{x(x-1)(x-2)(x-3)}{4!}-\frac{x(x-1)(x-2)(x-3)(x-4)}{5!}+$
$+\frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!}=0 . \quad($ (here $n!=1 \cdot 2 \cdot 3 . . . n)$
In the Answer, indicate the sum of the found roots. | 21 |
21,174 | A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters. | 108 |
21,205 | $[$ Arithmetic of residues (miscellaneous).]
Solve the equation $x^{2}+y^{2}+z^{2}=2 x y z$ in integers. | (0, 0, 0) |
21,227 | 40. Find a five-digit number that has the following property: when multiplied by 9, the result is a number composed of the same digits but in reverse order, i.e., the number is reversed. | 10989 |
21,228 | On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\). | 18 |
21,231 | Given the equation $\lg (x-1) + \lg (3-x) = \lg (a-x)$ with respect to $x$ has two distinct real roots, find the range of values for $a$. | \left(3, \dfrac{13}{4}\right) |
21,244 | The numbers $96, 28, 6, 20$ were written on the board. One of them was multiplied, another was divided, another was increased, and another was decreased by the same number. As a result, all the numbers became equal to a single number. What is that number? | 24 |
21,306 | Team A and Team B have a table tennis team match. Each team has three players, and each player plays once. Team A's three players are \( A_{1}, A_{2}, A_{3} \) and Team B's three players are \( B_{1}, B_{2}, B_{3} \). The winning probability of \( A_{i} \) against \( B_{j} \) is \( \frac{i}{i+j} \) for \( 1 \leq i, j \leq 3 \). The winner gets 1 point. What is the maximum possible expected score for Team A? | \dfrac{91}{60} |
21,313 | Determine for how many natural numbers greater than 900 and less than 1001 the digital sum of the digital sum of their digital sum is equal to 1.
(E. Semerádová)
Hint. What is the largest digital sum of numbers from 900 to 1001? | 12 |
21,327 | Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$ . | 83 |
21,339 | Example 2. How many sets of positive integer solutions does the equation $x+2 y+3 z=2000$ have? | 332334 |
21,340 | A Baranya IC overtakes a freight train traveling on a parallel track, then they pass each other in the opposite direction. The ratio of the IC's speed to the freight train's speed is the same as the ratio of the time it takes to overtake to the time it takes to pass each other. How many times faster is the IC than the freight train, if both trains maintain a constant speed? | 1 + \sqrt{2} |
21,368 | Given positive real numbers \(a\) and \(b\) that satisfy \(ab(a+b) = 4\), find the minimum value of \(2a + b\). | 2\sqrt{3} |
21,375 | 6. (10 points) Two cubes of different sizes are glued together to form the solid figure shown below, where the four vertices of the smaller cube's bottom face are exactly the midpoints of the edges of the larger cube's top face. If the edge length of the larger cube is 2, then the surface area of this solid figure is . $\qquad$ | 32 |
21,397 | The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by the neural network are then rechecked manually by experts who do not make mistakes. The neural network can misclassify good work as not accepted and bad work as accepted. It is known that among all the submitted works, 20% are actually bad. What is the minimum percentage of bad works among those rechecked by the experts after the selection by the neural network? Indicate the integer part of the number in your answer. | 66 |
21,414 | On the base \(AC\) of an isosceles triangle \(ABC\), a point \(E\) is taken, and on the sides \(AB\) and \(BC\), points \(K\) and \(M\) are taken such that \(KE \parallel BC\) and \(EM \parallel AB\). What fraction of the area of triangle \(\mathrm{ABC}\) is occupied by the area of triangle \(KEM\) if \(BM:EM = 2:3\)? | \dfrac{6}{25} |
21,454 | 18. A positive integer is said to be 'good' if each digit is 1 or 2 and there is neither four consecutive 1 's nor three consecutive 2 's. Let $a_{n}$ denote the number of $n$-digit positive integers that are 'good'. Find the value of $\frac{a_{10}-a_{8}-a_{5}}{a_{7}+a_{6}}$.
(2 marks)
若某正整數的每個數字均為 1 或 2 , 且當中既沒有四個連續的 $\ulcorner 1$ 亦沒有三個連續的 22 , 便稱為 「好數」。設 $a_{n}$ 表示 $n$ 位「好數」的數目。求 $\frac{a_{10}-a_{8}-a_{5}}{a_{7}+a_{6}}$ 的值。 | 2 |
21,464 | ## Problem Statement
Calculate the definite integral:
$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{d x}{(6-\operatorname{tg} x) \sin 2 x}
$$ | \dfrac{\ln 5}{6} |
21,475 | 10. A kilogram of one type of candy is 80 k more expensive than another. Andrey and Yura bought 150 g of candies, which included both types of candies, with Andrey having twice as much of the first type as the second, and Yura having an equal amount of each. Who paid more for their purchase and by how much?
14 | 2 |
21,497 | Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 12 |
21,520 | 19 Let $p$ and $q$ represent two consecutive prime numbers. For some fixed integer $n$, the set $\{n-1,3 n-19,38-5 n, 7 n-45\}$ represents $\{p, 2 p, q, 2 q\}$, but not necessarily in that order. Find the value of $n$. | 7 |
21,539 | 83. Given two concentric circles with radii $r$ and $R (r < R)$. Through a point $P$ on the smaller circle, a line is drawn intersecting the larger circle at points $B$ and $C$. The perpendicular to $BC$ at point $P$ intersects the smaller circle at point $A$. Find $|PA|^2 + |PB|^2 + |PC|^2$. | 2(R^2 + r^2) |
21,576 | 6. Let the function be
$$
f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\left(k \in \mathbf{Z}_{+}\right) \text {. }
$$
If for any real number $a$, we have
$$
\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbf{R}\},
$$
then the minimum value of $k$ is $\qquad$ | 16 |
21,581 | # Task 1. (10 points)
Six consecutive natural numbers from 10 to 15 are written in circles on the sides of a triangle in such a way that the sums of the three numbers on each side are equal.
What is the maximum value that this sum can take?
 | 39 |
21,585 | Calculate the area of the figure bounded by the parabola \( y = -x^2 + 3x - 2 \) and the coordinate axes. | 1 |
21,627 | 2. Find the last three digits of $7^{2014}$.
. | 849 |
21,648 | Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$ .
*Proposed by David Tang* | 10 |
21,652 | 12. Let $M$ be a subset of the set $N=\{1,2,3, \cdots, 1998 \mid$ such that each natural number (element) in $M$ has exactly 1 zero. Then the set $M$ can contain at most $\qquad$ elements. | 414 |
21,653 | 4. Given that $[x]$ represents the greatest integer not exceeding $x$, the number of integer solutions to the equation $3^{2 x}-\left[10 \cdot 3^{x+1}\right]+ \sqrt{3^{2 x}-\left[10 \cdot 3^{x+1}\right]+82}=-80$ is $\qquad$ | 2 |
21,689 | 1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 64827. The answer should be presented as an integer. | 1120 |
21,692 | Given a positive integer $n$ that is not divisible by 2 or 3, and there do not exist non-negative integers $a$ and $b$ such that $\left|2^{a}-3^{b}\right|=n$, find the smallest value of $n$. | 35 |
21,738 | The numerical sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) is defined such that \(a_{1}=\log _{2}\left(\log _{2} f(2)\right)\), \(a_{2}=\log _{2}\left(\log _{2} f(f(2))\right)\), \(\ldots, a_{n}=\log _{2}(\log _{2} \underbrace{f(f(\ldots f}_{n}(2)))), \ldots\), where \(f(x)=x^{x}\). Determine the number \(n\) for which \(a_{n}=2059+2^{2059}\). | 5 |
21,741 | In the infinite decimal expansion of a real number \( a \), assume \( v_{a} \) is the number of different digit sequences of length \( n \) in this expansion. Prove: if for some \( n \), the condition \( V_{n} \leqslant n+8 \) holds, then the number \( a \) is rational. | a \text{ is rational} |
21,755 | Example 1. Let $z$ be a complex number, solve the equation
$$
\frac{1}{2}(z-1)=\frac{\sqrt{3}}{2}(1+z) \text { i. }
$$ | -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i |
21,770 | Lines parallel to the sides of a square form a small square whose center coincides with the center of the original square. It is known that the area of the cross, formed by the small square, is 17 times larger than the area of the small square. By how many times is the area of the original square larger than the area of the small square? | 81 |
21,785 |
An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vertices fill the leg. Determine the sum of the areas of these equilateral triangles. | 324 |
21,807 | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-6 n+7}{3 n^{2}+20 n-1}\right)^{-n+1}
$$ | e^{\frac{26}{3}} |
21,831 | 4.2. Find the sum of all roots of the equation $x^{2}-41 x+330=3^{x}\left(41-2 x-3^{x}\right)$. | 5 |
21,833 | Let $S$ be a square of side length $1$ . Two points are chosen independently at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$ , where $a$ $b$ , and $c$ are positive integers with $\gcd(a,b,c)=1$ . What is $a+b+c$ | 59 |
21,852 | Points \( O, A, B, \) and \( C \) do not lie on the same plane. Prove that the point \( X \) lies on the plane \( ABC \) if and only if \(\overrightarrow{OX} = p \overrightarrow{OA} + q \overrightarrow{OB} + r \overrightarrow{OC}\), where \( p + q + r = 1 \). Additionally, if the point \( X \) belongs to the triangle \( ABC \), then \( p : q : r = S_{BXC} : S_{CXA} : S_{AXB} \). | p : q : r = S_{BXC} : S_{CXA} : S_{AXB} |
21,860 | In triangle \(A B C\), side \(B C\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \(A\) and is tangent to \(B C\), with one tangent at point \(B\) and the other at point \(C\). | \dfrac{5}{3} |
21,890 | 6.51. $\lim _{x \rightarrow 1} \frac{\sin 3 \pi x}{\sin 2 \pi x}$. | -\dfrac{3}{2} |
21,913 | 3. Find the number of natural numbers $k$, not exceeding 454500, such that $k^{2}-k$ is divisible by 505. | 3600 |
21,914 | 4[ Numerical inequalities. Comparisons of numbers.]
What is greater: $\log _{3} 4$ or $\log _{4} 5$ ? | \log_{3}4 |
21,917 | 17. Let $[x]$ denote the greatest integer not exceeding $x$, for example $[3.15]=3,[3.7]=3,[3]=3$, then $[\sqrt[3]{1 \cdot 2 \cdot 3}]+[\sqrt[3]{2 \cdot 3 \cdot 4}]+[\sqrt[3]{3 \cdot 4 \cdot 5}]+\cdots+[\sqrt[3]{2000 \cdot 2001 \cdot 2002}]=$ $\qquad$ | 2001000 |
21,956 | $ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| 30 |
21,979 | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(2-3^{\operatorname{arctg}^{2} \sqrt{x}}\right)^{\frac{2}{\sin x}}$ | \dfrac{1}{9} |
22,107 | Which fourth-degree polynomial equation with integer coefficients has the roots: $\pm \sqrt{8+\sqrt{13}}$? | x^4 - 16x^2 + 51 = 0 |
22,111 | 1.14. Using elementary transformations, find the rank of the matrix
$$
A=\left(\begin{array}{ccccc}
5 & 7 & 12 & 48 & -14 \\
9 & 16 & 24 & 98 & -31 \\
14 & 24 & 25 & 146 & -45 \\
11 & 12 & 24 & 94 & -25
\end{array}\right)
$$ | 3 |
22,219 | $a$ , $b$ , $c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$ | \dfrac{17}{3} |
22,272 | Find the largest number such that when each of the fractions \(\frac{154}{195}\), \(\frac{385}{156}\), and \(\frac{231}{130}\) is divided by it, the results are natural numbers. | \dfrac{77}{780} |
22,293 | Prove that if \(a, b, c\) are the sides of a triangle and \(a^{4}+b^{4}=c^{4}\), then the triangle is acute-angled. | \text{The triangle is acute-angled.} |
22,367 | 19. Let
$$
F(x)=\frac{1}{\left(2-x-x^{5}\right)^{2011}},
$$
and note that $F$ may be expanded as a power series so that $F(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$. Find an ordered pair of positive real numbers $(c, d)$ such that $\lim _{n \rightarrow \infty} \frac{a_{n}}{n^{d}}=c$. | \left( \dfrac{1}{6^{2011} \cdot 2010!}, 2010 \right) |
22,371 | If the height on the base of an isosceles triangle is $18 \mathrm{~cm}$ and the median on the leg is $15 \mathrm{~cm}$, what is the area of this isosceles triangle? | 144 |
22,384 | 129. Form the equation of the curve passing through the point ( $3 ; 4$ ), if the slope of the tangent to this curve at any point ( $x ; y$ ) is equal to $x^{2}-2 x$. | y = \dfrac{1}{3}x^3 - x^2 + 4 |
22,434 | ## Task 2 - 030912
Wolfgang is on a train, whose own speed he has measured at $60 \mathrm{~km} / \mathrm{h}$. He wants to determine the speed of an oncoming double-decker articulated train. He knows that this double-decker articulated train, including the locomotive, is approximately $120 \mathrm{~m}$ long, and he times the duration it takes for the train to pass by, which is exactly 3.0 s.
At what speed is the oncoming train traveling? | 84 |
22,440 | A regular triangular prism \( ABC A_1 B_1 C_1 \) with a base \( ABC \) and side edges \( AA_1, BB_1, CC_1 \) is inscribed in a sphere of radius 3. The segment \( CD \) is the diameter of this sphere. Find the volume of the prism, given that \( AD = 2 \sqrt{6} \). | 6\sqrt{15} |
22,484 | The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
- $D(1) = 0$ ;
- $D(p)=1$ for all primes $p$ ;
- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$ . | 31 |
22,494 | 4. Compute $\sum_{k=1}^{\infty} \frac{k^{4}}{k!}$. | 15e |
22,499 | 2. Find the value of the expression $\sin ^{4} \frac{5 \pi}{24}+\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{19 \pi}{24}$. | \dfrac{6 - \sqrt{3}}{4} |
22,505 | Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \). | 1003 |
22,520 | 13.326. Three cars are dispatched from $A$ to $B$ at equal time intervals. They arrive in $B$ simultaneously, then proceed to point $C$, which is 120 km away from $B$. The first car arrives there one hour after the second. The third car, upon arriving at $C$, immediately turns back and meets the first car 40 km from $C$. Determine the speed of the first car, assuming that the speed of each car was constant throughout the route.
Fig. 13.16 | 30 |
22,527 | In a right-angled triangle, let \( s_{a} \) and \( s_{b} \) be the medians to the legs, and \( s_{c} \) be the median to the hypotenuse. Determine the maximum value of the expression \( \frac{s_{a} + s_{b}}{s_{c}} \). | \sqrt{10} |
22,537 | The quadratic \( x^2 + ax + b + 1 \) has roots which are positive integers. Show that \( a^2 + b^2 \) is composite. | a^2 + b^2 \text{ is composite} |
22,538 | The sum of three positive angles is $90^{\circ}$. Can the sum of the cosines of two of them be equal to the cosine of the third one? | \text{No} |
22,564 | 3. $\mathbf{C}$ is the set of complex numbers, let the set $A=\left\{z \mid z^{18}=1, z \in \mathbf{C}\right\}, B=\left\{w \mid w^{48}=1, w \in \mathbf{C}\right\}, D=\{z w \mid z \in A$ ， $w \in B\}$. Find the number of elements in $D$. | 144 |
22,565 | Example 2. Solve the equation $y^{\prime \prime \prime}+y^{\prime}=x^{4}$. | y = C_1 + C_2 \cos x + C_3 \sin x + \frac{1}{5}x^5 - 4x^3 + 24x |
22,603 | 4. Write the equation of a polynomial of degree 3 (in factored form if possible), whose graph passes through the points $A(4,-5), B(-1,0), C(0,5)$ and $D(5,0)$. Sketch the graph of the polynomial. | \frac{1}{2}(x + 1)(x - 2)(x - 5) |
22,619 | Let " $\sum$ " denote the cyclic sum. Given positive real numbers \(a\), \(b\), and \(c\) such that \(abc = 1\), prove for any integer \(n \geq 2\) that:
$$
\sum \frac{a}{\sqrt[n]{b+c}} \geq \frac{3}{\sqrt[n]{2}}
$$ | \frac{3}{\sqrt[n]{2}} |
22,620 | Given that the odd function \( f(x) \) is increasing on the interval \( (-\infty, 0) \), and that \( f(-2) = -1 \), \( f(1) = 0 \). When \( x_{1} > 0 \) and \( x_{2} > 0 \), it holds that \( f\left( x_{1} x_{2} \right) = f\left( x_{1} \right) + f\left( x_{2} \right) \). Find the solution set for the inequality \( \log _{2}|f(x)+1|<0 \). | (-4, -2) \cup (-2, -1) \cup \left( \frac{1}{4}, \frac{1}{2} \right) \cup \left( \frac{1}{2}, 1 \right) |
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