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4,760 | In the number $2016 * * * * 02 *$, each of the 5 stars must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 11-digit number is divisible by 15. How many ways can this be done? | 864 |
4,795 | ## Problem Statement
Calculate the volumes of the bodies bounded by the surfaces.
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{64}=1, z=4, z=0
$$ | 44\pi |
4,801 | Given a sequence \(\left\{a_{n}\right\}\) with the sum of the first \(n\) terms \(S_{n}\) satisfying the equation \(S_{n} + a_{n} = \frac{n-1}{n(n+1)}\) for \(n = 1, 2, \cdots\), find the general term \(a_{n}\). | \dfrac{1}{2^n} - \dfrac{1}{n(n+1)} |
4,839 | 14. $[8]$ Evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^{4}+4}$. | \dfrac{3}{8} |
4,849 | 8. Petya saw a bank advertisement: "Deposit 'Super-Income' - up to $10 \%$ annually!". He became interested in this offer and found it on the bank's website. The conditions turned out to be slightly different, as reflected in the table.
| Interest period | from 1 to 3 months | from 4 to 6 months | from 7 to 9 months | from 10 to 12 months |
| :---: | :---: | :---: | :---: | :---: |
| Rate | $10 \%$ | $8 \%$ | $8 \%$ | $8 \%$ |
* - interest is credited at the end of the period.
What is the actual effective annual interest rate of this deposit? Round your answer to the tenths. | 8.8 |
4,857 | 3. Find the value of the expression $4 \sin 40^{\circ}-\operatorname{tg} 40^{\circ}$. | \sqrt{3} |
4,863 | In triangle \( \triangle ABC \), \( AB = AC \), \( \angle A = 80^\circ \), and point \( D \) is inside the triangle such that \( \angle DAB = \angle DBA = 10^\circ \). Find the measure of \( \angle ACD \). | 30 |
4,881 | In a certain class, there are 28 boys and 22 girls. If 5 students are to be elected to different class committee positions, and it's desired that both boys and girls are represented among the 5 students, how many different election outcomes are possible? | 239297520 |
4,895 | Find the number of digit of $\sum_{n=0}^{99} 3^n$ .
You may use $\log_{10} 3=0.4771$ .
2012 Tokyo Institute of Technology entrance exam, problem 2-A | 48 |
4,953 | Inscribed Spheres. If the faces of a certain hexahedron are equilateral triangles, congruent to the faces of a regular octahedron, then what is the ratio of the radii of the spheres inscribed in these polyhedra? | \dfrac{2}{3} |
4,994 | 3. Solve the inequality:
$$
x^{2}+4 x \sqrt{x+6} \leqslant 5(x+6)
$$ | [-5, 3] |
5,057 | Let \( x_{i} \in \mathbf{R} \), \( x_{i} \geqslant 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \). Find the minimum value of \( \max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4} + x_{5}\right\} \). | \dfrac{1}{3} |
5,093 | 8. Car A and Car B start from locations $A$ and $B$ respectively at the same time, heading towards each other. They meet after 3 hours, then Car A turns around and heads back to $A$, while Car B continues on. After Car A reaches $A$ and turns around to head towards B, it meets Car B half an hour later. How many minutes does it take for Car B to travel from $A$ to B? | 432 |
5,118 | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\sqrt{\left(n^{2}+1\right)\left(n^{2}+2\right)}-\sqrt{\left(n^{2}-1\right)\left(n^{2}-2\right)}\right)
$$ | 3 |
5,127 | Inside triangle $A B C$, there are points $P$ and $Q$ such that point $P$ is at distances 6, 7, and 12 from the lines $A B, B C, C A$ respectively, and point $Q$ is at distances 10, 9, and 4 from the lines $A B, B C, C A$ respectively. Find the radius of the inscribed circle of triangle $A B C$. | 8 |
5,132 | 24. [10] Compute, in terms of $n$,
$$
\sum_{k=0}^{n}\binom{n-k}{k} 2^{k}
$$
Note that whenever $s<t,\binom{s}{t}=0$. | \dfrac{2^{n+1} + (-1)^n}{3} |
5,155 | \[\sin ^{3} x(1+\operatorname{ctg} x) + \cos ^{3} x(1+\operatorname{tg} x) = 2 \sqrt{\sin x \cos x}.\] | \frac{\pi}{4} + 2\pi n |
5,181 | ## 29. Umbrella
One Saturday evening, two brothers, Jean and Pierre, exchanged their observations about the behavior of their neighbor, Madame Martin. They agreed that every Sunday she goes out of the house once (and only once), that in two out of three cases she takes an umbrella with her, and therefore even on good weather days, she goes for a walk with an umbrella half the time. However, Jean believed that on rainy days she never forgets to take an umbrella, while Pierre, on the contrary, claimed that sometimes she goes for a walk without an umbrella even on rainy days.
Which of the brothers, in your opinion, is right and which is wrong, if in their region, on average, half of the days are rainy? | Pierre |
5,195 | Given a regular rectangular pyramid \(\Gamma\) with a height of 3 and an angle of \(\frac{\pi}{3}\) between its lateral faces and its base, first inscribe a sphere \(O_{1}\) inside \(\Gamma\). Then, sequentially inscribe spheres \(O_{2}, O_{3}, O_{4}, \ldots\) such that each subsequent sphere is tangent to the preceding sphere and to the four lateral faces of \(\Gamma\). Find the sum of the volumes of all the inscribed spheres. | \dfrac{18}{13}\pi |
5,211 | The median of a triangle bisects its perimeter. Prove that the triangle is isosceles. | \text{The triangle is isosceles.} |
5,253 | 5. In how many ways can we pay an amount of 20 kuna using coins of $5 \mathrm{kn}$, $2 \mathrm{kn}$, and $1 \mathrm{kn}$, so that each type of coin is used at least once?
## Tasks worth 10 points: | 13 |
5,302 | 5. On each face of an opaque cube, a natural number is written. If several (one, two, or three) faces of the cube can be seen at the same time, then find the sum of the numbers on these faces. Using this method, the maximum number of different sums that can be obtained is _.
翻译结果如下：
5. On each face of an opaque cube, a natural number is written. If several (one, two, or three) faces of the cube can be seen at the same time, then find the sum of the numbers on these faces. Using this method, the maximum number of different sums that can be obtained is _. | 26 |
5,342 | Points $\boldsymbol{A}$ and $\boldsymbol{B}$ are located on a straight highway running from west to east. Point B is 9 km east of A. A car departs from point A heading east at a speed of 40 km/h. Simultaneously, from point B, a motorcycle starts traveling in the same direction with a constant acceleration of 32 km/h². Determine the greatest distance that can be between the car and the motorcycle during the first two hours of their movement. | 16 |
5,352 | 9. (10 points) At 8:10 AM, Feifei walked from home to school. 3 minutes later, the dog started running to catch up with her, and caught up with her 200 meters away from home; after catching up, it immediately ran back home, and then immediately turned around to catch Feifei again, catching up with her 400 meters away from home. After catching up, it ran back home again, and then went to catch Feifei, finally catching up with her at school. Feifei arrived at school at 8: $\qquad$ minutes. | 28 |
5,365 | Seven fishermen are standing in a circle. Each fisherman has a professional habit of exaggerating numbers, with a distinct measure of exaggeration (an integer) indicating by how many times the number mentioned by the fisherman exceeds the true value. For example, if a fisherman with an exaggeration measure of 3 catches two fish, he will claim to have caught six fish. When asked: "How many fish did your left neighbor catch?" the answers were (not necessarily in the seated order) $12, 12, 20, 24, 32, 42,$ and $56$. When asked: "How many fish did your right neighbor catch?" six of the fishermen answered $12, 14, 18, 32,$ $48,$ and $70$. What did the seventh fisherman answer? | 16 |
5,391 | 6. (10 points) A water pool contains one-eighteenth of its capacity. Two water pipes start filling the pool simultaneously. When the water level reaches two-ninths of the pool's capacity, the first pipe continues to fill the pool alone for 81 minutes, during which it injects an amount of water equal to what the second pipe has already injected. Then, the second pipe fills the pool alone for 49 minutes, at which point the total amount of water injected by both pipes is the same. After that, both pipes continue to fill the pool. How many more minutes do the two pipes need to fill the pool together? | 231 |
5,420 | How many ways can a pile of 100 stones be divided into smaller piles such that the number of stones in any two piles differs by no more than one? | 99 |
5,453 | Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$? | 2 - \sqrt{2} |
5,463 | A circle is drawn through the vertex \( A \) of an isosceles triangle \( ABC \) with base \( AC \). The circle touches the side \( BC \) at point \( M \) and intersects side \( AB \) at point \( N \). Prove that \( AN > CM \). | AN > CM |
5,464 | 18. (6 points) December 31, 2013, was a Tuesday, so, June 1, 2014, is $\qquad$. (Answer with a number: Monday is represented by 1, Tuesday by 2, Wednesday by 3, Thursday by 4, Friday by 5, Saturday by 6, and Sunday by 7.) | 7 |
5,486 | 3. The parabolas $y=-x^{2}+b x+c, b, c \in \mathbb{R}$ are tangent to the parabola $y=x^{2}$. Determine the equation of the curve on which the vertices of the parabolas $y=-x^{2}+b x+c$ lie. | y = \dfrac{1}{2}x^2 |
5,499 | [ Average values $\quad]$ [ Area of a circle, sector, and segment ]
At a familiar factory, metal disks with a diameter of 1 m are cut out. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius is 10 mm. Engineer Sidorov believes that a stack of 100 disks will on average weigh 10000 kg. By how much is Engineer Sidorov mistaken? | 4 |
5,561 | 11. A and B repeatedly play a game. In each game, the players take turns tossing a fair coin, and the first player to toss heads wins. In the first game, A tosses first, and in subsequent games, the loser of the previous game starts first. The probability that A wins the 6th game is $\qquad$. | \dfrac{364}{729} |
5,590 | Example 1. Plot the graph of the function $y=\frac{1}{2} x^{2}+2 x-1$. | y = \frac{1}{2}x^2 + 2x - 1 |
5,609 | Compare the following values: $\frac{3^{2000}+2}{3^{2001}+2}, \frac{3^{2001}+2}{3^{2002}+2}, \frac{3^{2002}+2}{3^{2003}+2}$. | \frac{3^{2000}+2}{3^{2001}+2} > \frac{3^{2001}+2}{3^{2002}+2} > \frac{3^{2002}+2}{3^{2003}+2} |
5,614 | A king has eight sons, and they are all fools. Each night, the king sends three of them to guard the golden apples from the Firebird. The princes cannot catch the Firebird and blame each other, so no two of them agree to go on guard together a second time. What is the maximum number of nights this can continue? | 8 |
5,618 | A function $f$ , defined on the set of real numbers $\mathbb{R}$ is said to have a *horizontal chord* of length $a>0$ if there is a real number $x$ such that $f(a+x)=f(x)$ . Show that the cubic $$ f(x)=x^3-x\quad \quad \quad \quad (x\in \mathbb{R}) $$ has a horizontal chord of length $a$ if, and only if, $0<a\le 2$ . | (0, 2] |
5,623 | All natural numbers whose digits sum up to 5 are ordered in increasing order. What number is in the 125th place? | 41000 |
5,660 | ## Aufgabe 25/78
Man ermittle alle Paare ganzer Zahlen $(m ; n)$, die der Gleichung $\sqrt{2^{n}+1}=m$ genügen.
| (3, 3) |
5,707 | 10. (10 points) Two buildings in a residential area are 220 meters apart. After planting 10 trees at equal distances between them, the distance between the 1st tree and the 6th tree is $\qquad$ meters. | 100 |
5,745 |
An archipelago consisting of an infinite number of islands stretches along the southern shore of a boundless sea. The islands are connected by an infinite chain of bridges, with each island connected to the shore by a bridge. In the event of a severe earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that after the severe earthquake, it will be possible to travel from the first island to the shore using the remaining intact bridges? | \dfrac{2}{3} |
5,754 | 假设有5个不同的任务（编号为1到5）需要分配给3个不同的团队（编号为A到C），要求满足以下条件：1. 每个团队至少分配一个任务。2. 任务1和任务2不能分配给同一个团队。3. 任务3和任务4只能分配给同一个团队。问有多少种不同的分配方法满足这些条件？ | 30 |
5,766 | ## Task Condition
Find the derivative of the specified order.
$y=\frac{\ln x}{x^{3}}, y^{IV}=?$ | \dfrac{360 \ln x - 342}{x^7} |
5,769 | 2. Usually, Nikita leaves home at 8:00 AM, gets into Uncle Vanya's car, who drives him to school by a certain time. But on Friday, Nikita left home at 7:10 and ran in the opposite direction. Uncle Vanya waited for him and at $8: 10$ drove after him, caught up with Nikita, turned around, and delivered him to school 20 minutes late. How many times faster was Uncle Vanya's car speed compared to Nikita's running speed? | 13 |
5,785 | The integers $a_{1}, a_{2}, \ldots, a_{10}$ are greater than 1 and their sum is 2006. What is the smallest possible value of
$$
\binom{a_{1}}{2}+\ldots+\binom{a_{10}}{2}
$$
sum? | 200200 |
5,818 | Positive numbers \( x, y, \) and \( z \) satisfy the condition \( x y z \geq x y + y z + z x \). Prove the inequality:
\[
\sqrt{x y z} \geq \sqrt{x} + \sqrt{y} + \sqrt{z}
\]
(A. Khrabrov) | \sqrt{x y z} \geq \sqrt{x} + \sqrt{y} + \sqrt{z} |
5,853 | ## Task B-4.4.
Determine the set of all points from which the hyperbola $16 x^{2}-25 y^{2}=400$ is seen at a right angle. | x^2 + y^2 = 9 |
5,906 | 8. Given an even function $y=f(x)$ defined on $\mathbf{R}$ that satisfies $f(x)=f(2-x)$, when $x \in[0,1]$, $f(x)=x^{2}$, then for $2 k-1 \leqslant x \leqslant 2 k+1(k \in \mathbf{Z})$, the expression for $f(x)$ is $\qquad$. | (x - 2k)^2 |
5,951 | 10. (5 points) There are several apples and pears. If each bag contains 5 apples and 3 pears, then when the pears are exactly finished, there are still 4 apples left; if each bag contains 7 apples and 3 pears, then when the apples are exactly finished, there are still 12 pears left. How many apples and pears are there in total? $\qquad$ | 132 |
5,956 | 1. The sum of 100 positive integers is 101101. What is the maximum possible value of their greatest common divisor? Prove your conclusion. | 1001 |
5,972 | 1. A mathematician and a physicist started running on a running track towards the finish line at the same time. After finishing, the mathematician said: "If I had run twice as fast, I would have beaten the physicist by 12 seconds." And after finishing, the physicist said: "If I had run twice as fast, I would have beaten the mathematician by 36 seconds." By how many seconds did the winner beat the second participant? | 16 |
5,997 | Task 3. (15 points) At the quality control department of an oil refinery, Engineer Valentina Ivanovna received a research object consisting of about 200 oil samples (a container designed for 200 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or high-sulfur, and density - either light or heavy. The relative frequency (statistical probability) that a randomly selected sample will be a heavy oil sample is $\frac{1}{9}$. The relative frequency that a randomly selected sample will be a light low-sulfur oil sample is $\frac{11}{18}$. How many high-sulfur oil samples does the object contain, if there were no low-sulfur samples among the heavy oil samples? | 77 |
6,022 | Folklore
Compare: $\sin 3$ and $\sin 3^{\circ}$. | \sin 3 > \sin 3^{\circ} |
6,033 | 7. A cone and a cylinder, with their bottom faces on the same plane, have a common inscribed sphere. Let the volume of the cone be $V_{1}$, and the volume of the cylinder be $V_{2}$, and $V_{1}=k V_{2}$, then $k_{\text {min }}=$ $\qquad$ | \dfrac{4}{3} |
6,053 | Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikita, turned around, and took him to school, arriving 20 minutes late. By how many times does the speed of Uncle Vanya's car exceed Nikita's running speed? | 13 |
6,061 | Given \( f(x) = 2x - \frac{2}{x^2} + \frac{a}{x} \), where the constant \( a \in (0, 4] \). Find all real numbers \( k \) such that for \( x_1, x_2 \in \mathbf{R}^+ \), it always holds that \( \left| f(x_1) - f(x_2) \right| > k \left| x_1 - x_2 \right| \). | (-\infty, 2 - \frac{a^3}{108}) |
6,064 | Problem 8. Candies are in the shape of $1 \times 1 \times 1$ cubes. The teacher arranged them into a parallelepiped $3 \times 4 \times 5$ and offered the children to enjoy. In the first minute, Petya took one of the corner candies (see figure). Each subsequent minute, the children took all the candies that had a neighboring face with already missing candies (for example, in the second minute, they took 3 candies). How many minutes did it take for the children to take all the candies?
Answer: It took 10 minutes. | 10 |
6,092 | A circle with its center on side $AB$ of triangle $ABC$ touches the other two sides. Find the area of the circle if $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm, where $a$, $b$, and $c$ are the lengths of the sides of the triangle. | \dfrac{3136}{81}\pi |
6,098 | Example 9 (1994, 12th American Invitational Mathematics Examination) For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} n\right]=1994$. | 312 |
6,100 | Given \( x_{i} \in \mathbf{R} \) (for \( i = 1, 2, \cdots, n \)), satisfying \( \sum_{i=1}^{n} \left| x_{i} \right| = 1 \) and \( \sum_{i=1}^{n} x_{i} = 0 \). Prove that:
\[ \left| \sum_{i=1}^{n} \frac{x_{i}}{i} \right| \leq \frac{1}{2} - \frac{1}{2n} \] | \frac{1}{2} - \frac{1}{2n} |
6,104 | Example 6. Find $\int e^{\alpha x} \sin \beta x d x$. | \dfrac{e^{\alpha x}(\alpha \sin \beta x - \beta \cos \beta x)}{\alpha^2 + \beta^2} + C |
6,108 | After the celebration, Mom put aside the last piece of cake for Aunt. When Aunt finally arrived, she found only a dirty plate instead of the treat. Mom tried to find out what happened and got the following answers from her four children:
Adam: "Blanka or Cyril ate it."
Blanka: "Adam or Cyril ate it."
Cyril: "None of us is lying."
Dana: "Everyone except me is lying."
In the end, it turned out that one of the children ate the cake and that this child was telling the truth.
Find out which child it was.
(E. Novotná) | Dana |
6,113 | 4. If the sum of 12 distinct positive integers is 2016, then the maximum value of the greatest common divisor of these positive integers is $\qquad$ . | 24 |
6,114 | Given \( x \in [0, \pi] \), find the range of values for the function
$$
f(x)=2 \sin 3x + 3 \sin x + 3 \sqrt{3} \cos x
$$
| [-3\sqrt{3}, 8] |
6,123 | Four. (20 points) Given the function $f_{1}(x)=\sqrt{x^{2}+48}$, when $n \in \mathbf{Z}_{+}, n \geqslant 2$,
$$
f_{n}(x)=\sqrt{x^{2}+6 f_{n-1}(x)} \text {. }
$$
Find the real solutions to the equation $f_{n}(x)=2 x$. | 4 |
6,124 | 2. If the positive real numbers $x, y$ satisfy $x-2 \sqrt{y}=\sqrt{2 x-y}$, then the maximum value of $x$ is $\qquad$ . | 10 |
6,130 | 2.3. Gavriila found out that the front tires of the car last for 42,000 km, while the rear tires last for 56,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 48000 |
6,168 | Two triangles - an equilateral triangle with side \( a \) and an isosceles right triangle with legs equal to \( b \) - are positioned in space such that their centers of mass coincide. Find the sum of the squares of the distances from all vertices of one triangle to all vertices of the other. | 3a^2 + 4b^2 |
6,189 | In land of Nyemo, the unit of currency is called a *quack*. The citizens use coins that are worth $1$ , $5$ , $25$ , and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
*Proposed by Aaron Lin* | 82 |
6,194 | 1665. It is known that the density of the distribution of a random variable $X$ has the form
$$
p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{x^{2}}{2 \sigma^{2}}}
$$
Find the density of the distribution of the random variable $Y=X^{2}$. | \dfrac{1}{\sigma \sqrt{2 \pi y}} e^{-\dfrac{y}{2 \sigma^2}} |
6,198 | * Find all positive integers $n$ such that the numbers $n+1, n+3, n+7, n+9, n+13, n+15$ are all prime. | 4 |
6,202 |
Given a triangle \(ABC\) where \(AB = AC\) and \(\angle A = 80^\circ\). Inside triangle \(ABC\) is a point \(M\) such that \(\angle MBC = 30^\circ\) and \(\angle MCB = 10^\circ\). Find \(\angle AMC\). | 70^\circ |
6,236 | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{5}-8}-n \sqrt{n\left(n^{2}+5\right)}}{\sqrt{n}}$ | -\dfrac{5}{2} |
6,240 | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$f(x)=\left\{\begin{array}{c}\tan\left(2^{x^{2} \cos (1 /(8 x))}-1+x\right), x \neq 0 ; \\ 0, x=0\end{array}\right.$ | 1 |
6,261 | 3.79. Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base. | \dfrac{1}{4} |
6,294 | An irregular $n$-gon is inscribed in a circle. When the circle is rotated about its center by an angle $\alpha \neq 2 \pi$, the $n$-gon coincides with itself. Prove that $n$ is a composite number. | n \text{ is a composite number} |
6,312 | 14. Xiao Ming puts several chess pieces into the small squares of a $3 * 3$ grid. Each small square can be left empty or can contain one or more chess pieces. Now, by counting the total number of chess pieces in each row and each column, 6 numbers are obtained, and these 6 numbers are all different. What is the minimum number of chess pieces needed? | 8 |
6,320 | 4. For what least natural $\mathrm{k}$ does the quadratic trinomial
$$
y=k x^{2}-p x+q \text { with natural coefficients } p \text{ and } q \text{ have two }
$$
distinct positive roots less than 1? | 5 |
6,329 | 【Example 5】 6 boys and 4 girls are to serve as attendants on 5 buses, with two people per bus. Assuming boys and girls are separated, and the buses are distinguishable, how many ways are there to assign them? | 5400 |
6,406 | Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. What is the probability that Nyusha wins? | \dfrac{1}{2} |
6,416 | Let $x$ and $y$ be positive real numbers. Show that $x^{2}+y^{2}+1 \geqslant x \sqrt{y^{2}+1} + y \sqrt{x^{2}+1}$. Are there any cases of equality? | x^{2} + y^{2} + 1 \geqslant x \sqrt{y^{2} + 1} + y \sqrt{x^{2} + 1} |
6,424 | 1. On the board, there is a positive integer $n$. In one step, we can erase the number on the board and write either its double, or its double increased by 1. For how many initial numbers $n$ different from 2019 can we achieve that the number 2019 appears on the board after a finite number of steps?
(Josef Tkadlec) | 10 |
6,476 | A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \sqrt{15}$. | 2\sqrt{33} |
6,482 | A circle with radius $r$ is inscribed in a triangle with perimeter $p$ and area $S$. How are these three quantities related? | S = \frac{1}{2}pr |
6,489 | The inscribed circle touches side $BC$ of triangle $ABC$ at point $K$. Prove that the area of the triangle is equal to $BK \cdot KC \cdot \cot(\alpha / 2)$. | BK \cdot KC \cdot \cot\left(\frac{\alpha}{2}\right) |
6,492 | Ex. 129. In an integer-sided triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer. | 12 |
6,512 | 3. The equation of the line passing through the intersection points of the parabolas
$$
y=2 x^{2}-2 x-1 \text { and } y=-5 x^{2}+2 x+3
$$
is $\qquad$ . | 6x + 7y = 1 |
6,555 | 4. Let's calculate the number of ways to write 3 digits:
$$
10 \cdot 10 \cdot 10=1000
$$
We will derive the formula for the number $A$, which is divisible by 5 and has a remainder of 3 when divided by 7. This number has the form
$$
A=5 k=7 n+3
$$
Solving the equation $5 k-7 n=3$ in integers, we find
$$
\left\{\begin{array}{l}
k=7 t+2 \\
n=5 t+1
\end{array}\right.
$$
Therefore, $A=35 t+10, t \in Z, t \geq 0$. Since by the problem's condition $A$ is a three-digit number, we have $100 \leq 35 t+10 \leq 999$. From this, taking into account the integer nature of $t$, we get $t=2,3, \ldots, 28$. This means that the number of three-digit numbers satisfying the problem's condition is 26. Therefore, the probability
that a three-digit number divisible by 5 and having a remainder of 3 when divided by 7 can be seen on the table is $\frac{26}{1000}$. | \dfrac{13}{500} |
6,565 | The altitude \(AH\) and the angle bisector \(CL\) of triangle \(ABC\) intersect at point \(O\). Find the angle \(BAC\) if it is known that the difference between the angle \(COH\) and half of the angle \(ABC\) is \(46^\circ\). | 92^\circ |
6,578 | Example 10. Find the integral $\int \sqrt{8+2 x-x^{2}} d x$. | \frac{(x - 1)}{2} \sqrt{8 + 2x - x^2} + \frac{9}{2} \sin^{-1}\left( \frac{x - 1}{3} \right) + C |
6,601 | 11.5. For a certain quadratic trinomial, the following information is known: its leading coefficient is one, it has integer roots, and its graph (parabola) intersects the line $y=2017$ at two points with integer coordinates. Can the ordinate of the vertex of the parabola be uniquely determined from this information? | -1016064 |
6,630 | The warehouse has nails in boxes weighing 24, 23, 17, and 16 kg. Can the storekeeper release 100 kg of nails from the warehouse without opening the boxes? | Yes |
6,633 | Given \( x_{i} > 0 \) for \( i = 1, 2, \ldots, n \), \( m \in \mathbf{R}^{+} \), \( a \geqslant 0 \), and \( \sum_{i=1}^{n} x_{i} = s \leqslant n \), prove that
\[
\prod_{i=1}^{n}\left(x_{i}^{m}+\frac{1}{x_{i}^{m}}+a\right) \geqslant\left[\left(\frac{s}{n}\right)^{m}+\left(\frac{n}{s}\right)^{m}+a\right]^{n}.
\] | \prod_{i=1}^{n}\left(x_{i}^{m}+\frac{1}{x_{i}^{m}}+a\right) \geqslant\left[\left(\frac{s}{n}\right)^{m}+\left(\frac{n}{s}\right)^{m}+a\right]^{n} |
6,679 | 26. A fence 2022 meters long is used to form a triangle, and the lengths of the three sides are all integers in meters. Among them, the lengths of two sides differ by 1 meter, and the lengths of two sides differ by 2 meters. The lengths of the three sides, sorted from smallest to largest, the middle side is $\qquad$ meters. | 674 |
6,686 | A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.
How many meters of wire were there originally in the bundle? | 54 |
6,692 | 3. (48th Slovenian Mathematical Olympiad) Mathieu first wrote down all the numbers from 1 to 10000 in order, then erased those numbers that are neither divisible by 5 nor by 11. Among the remaining numbers, what is the number at the 2004th position? | 7348 |
6,694 | 7. Find the minimum value of the function $f(x, y)=\frac{2015(x+y)}{\sqrt{2015 x^{2}+2015 y^{2}}}$ and specify all pairs $(x, y)$ at which it is achieved. | -\sqrt{4030} |
6,711 | Example 3 If a store sells a certain product, which costs 100 yuan, at 120 yuan, it can sell 300 units. If the price of the product is increased by 1 yuan based on 120 yuan, it will sell 10 fewer units, and if the price is reduced by 1 yuan, it will sell 30 more units. Question: To maximize profit, what price should the store set for the product? | 115 |
6,713 | Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ is equal to the perimeter of $\triangle CNA$, and the perimeter of $\triangle ANB$ is equal to the perimeter of $\triangle CMB$. Prove that $\triangle ABC$ is isosceles. | \triangle ABC \text{ is isosceles} |
6,718 | If $\triangle ABC$ satisfies that $\cot A, \cot B, \cot C$ form an arithmetic sequence, then the maximum value of $\angle B$ is ______. | \dfrac{\pi}{3} |
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