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68,998 | Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$. | 234 |
5,799 | 4. For a positive integer $n$, we denote by $P(n)$ the product of the positive divisors of $n$. For example, $P(20)=8000$. The positive divisors of 20 are 1, 2, 4, 5, 10, and 20, with the product $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20=8000$.
(a) Find all positive integers $n$ for which $P(n)=15 n$.
(b) Show that there are no positive integers $n$ for which $P(n)=15 n^{2}$. | 15 |
64,812 | ## Task 1 - 180811
The FDJ members Arnim, Bertram, Christian, Dieter, Ernst, and Fritz participated in a 400-meter race. No two of them finished at the same time.
Before the race, the following three predictions were made about the results (each participant is represented by the first letter of their first name):
| Place | 1. | 2. | 3. | 4. | 5. | 6. |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| 1st Prediction | A | B | C | D | E | F |
| 2nd Prediction | A | C | B | F | E | D |
| 3rd Prediction | C | E | F | A | D | B |
After the race, it turned out that in the first prediction, the places of exactly three runners were correctly given. No two of these three places were adjacent to each other. In the second prediction, no runner's place was correctly given. In the third prediction, the place of one runner was correctly given.
Give all possible sequences of the places achieved by the runners under these conditions! | C,B,E,D,A,F |
18,097 | Let \(\{f(n)\}\) be a strictly increasing sequence of positive integers: \(0 < f(1) < f(2) < f(3) < \cdots\). Of the positive integers not belonging to the sequence, the \(n\)th in order of magnitude is \(f(f(n))+1\). Determine \(f(240)\). | 388 |
34,557 | Example 1 Find the largest constant $k$, such that for $x, y, z \in \mathbf{R}_{+}$, we have
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$
where, “$\sum$” denotes the cyclic sum. | \sqrt{\frac{3}{2}} |
51,273 | 6. If from the set $S=\{1,2, \cdots, 20\}$, we take a three-element subset $A=\left\{a_{1}, a_{2}, a_{3}\right\}$, such that it simultaneously satisfies: $a_{2}-a_{1} \geqslant 5,4 \leqslant a_{3}-a_{2} \leqslant 9$, then the number of all such subsets $A$ is $\qquad$ (answer with a specific number). | 251 |
14,851 | 3. Two quadratic trinomials have a common root -3, and for one of them, it is the larger root, while for the other, it is the smaller root. The length of the segment cut off by the graphs of these trinomials on the y-axis is 12. Find the length of the segment cut off by the graphs of the trinomials on the x-axis. | 4 |
27,756 | 11. Let the sequence of positive numbers $a_{n}, a_{1}, a_{2}, \cdots$ satisfy $a_{0}=a_{1}=1$ and $\sqrt{a_{n} a_{n-2}}-\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1}, n=2,3, \cdots$, find the general term formula of the sequence. | a_{n}=\prod_{k=1}^{n}(2^{k}-1)^{2} |
22,723 | The Fibonacci numbers are defined by \( F_{0}=0, F_{1}=1 \), and \( F_{n}=F_{n-1}+F_{n-2} \) for \( n \geq 2 \). There exist unique positive integers \( n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6} \) such that
\[
\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}.
\]
Find \( n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6} \). | 1545 |
67,489 | 3. A four-digit number ABCD, consisting of different digits, is divisible by both the two-digit number CD and the two-digit number AB. Find the smallest possible value of ABCD. | 1248 |
30,850 | 3. Find the maximum value of the expression
$$
\frac{a}{x}+\frac{a+b}{x+y}+\frac{a+b+c}{x+y+z}
$$
where $a, b, c \in[2,3]$, and the triplet of numbers $x$, $y$, and $z$ is some permutation of the triplet of numbers $a, b, c$. | \frac{15}{4} |
66,490 | Problem 4. A circle with radius 4 is inscribed in trapezoid $ABCD$, touching the base $AB$ at point $M$. Find the area of the trapezoid if $BM=16$ and $CD=3$. | 108 |
24,290 | Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each running at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \mathrm{~m}$ apart. They start at the same time, but when Luíza finishes the race, Ana still has $120 \mathrm{~m}$ to reach point $B$. Yesterday, Luíza gave Ana a chance: "We will start at the same time, but I will start some meters before point $A$ so that we arrive together." How many meters before point $A$ should Luíza start? | 125 |
28,312 | 3. Problem: In a sequence of numbers, a term is called golden if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1,2,3, \ldots, 2021$ ? | 1010 |
52,960 | 7. Solve the following problem using as many methods as possible. Methods are considered different if they use different mathematical ideas, as well as various technical approaches to implementing the same idea. Indicate the place of each method used in the school mathematics curriculum.
Compare the numbers $2+\log _{2} 6$ and $2 \sqrt{5}$.
## Sixth Olympiad (2012). I (correspondence) round | 2+\log_{2}6>2\sqrt{5} |
3,309 | How many integer solutions does the equation
$$
\frac{1}{2022}=\frac{1}{x}+\frac{1}{y}
$$
have? | 53 |
10,152 | Knowing that Piglet eats a pot of honey in 5 minutes and a can of milk in 3 minutes, Rabbit calculated the minimum time in which the guests could consume his provisions. What is this time? (A can of milk and a pot of honey can be divided into any parts.)
Given:
- Pooh eats 10 pots of honey and 22 cans of condensed milk.
- Pooh takes 2 minutes per pot of honey and 1 minute per can of milk.
- Piglet takes 5 minutes per pot of honey and 3 minutes per can of milk.
Find the minimum time needed for both Pooh and Piglet to consume all the honey and milk. | 30 |
56,596 | 6. Given that the graph of the function $y=f(x)$ is centrally symmetric about the point $(1,1)$ and axially symmetric about the line $x+y=0$. If $x \in(0,1)$, $f(x)=\log _{2}(x+1)$, then the value of $f\left(\log _{2} 10\right)$ is $\qquad$ | \frac{17}{5} |
26,312 | Let's construct the projections of a cylinder, which is tangent to two given planes and the first trace (circle) of the projection axis. (How many solutions are possible?) | 4 |
29,958 | 3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 10 \\
a^{2}+b^{2} \leqslant \min (-6 a-2 b ; 10)
\end{array}\right.
$$
Find the area of the figure $M$. | 30\pi-5\sqrt{3} |
65,099 | 2. On the legs $a, b$ of a right triangle, the centers of two circles $k_{a}, k_{b}$ lie in sequence. Both circles touch the hypotenuse and pass through the vertex opposite the hypotenuse. Let the radii of the given circles be $\varrho_{a}, \varrho_{b}$. Determine the greatest positive real number $p$ such that the inequality
$$
\frac{1}{\varrho_{a}}+\frac{1}{\varrho_{b}} \geqq p\left(\frac{1}{a}+\frac{1}{b}\right)
$$
holds for all right triangles. | 1+\sqrt{2} |
3,529 | Let \( p \) be a prime number and \( a, b, c \) and \( n \) be positive integers with \( a, b, c < p \) so that the following three statements hold:
\[ p^2 \mid a + (n-1) \cdot b \]
\[ p^2 \mid b + (n-1) \cdot c \]
\[ p^2 \mid c + (n-1) \cdot a \]
Show that \( n \) is not a prime number. | n \text{ is not a prime number} |
229 | 3. Given the function $f(x)$ satisfies
$$
f^{2}(x+1)+f^{2}(x)=f(x+1)+f(x)+4 \text {. }
$$
Then the maximum value of $f(1)+f(2020)$ is $\qquad$ | 4 |
55,017 | 1. Buses from Moscow to Voronezh depart every hour, at 00 minutes. Buses from Voronezh to Moscow depart every hour, at 30 minutes. The trip between the cities takes 8 hours. How many buses from Voronezh will the bus that left from Moscow meet on its way? | 16 |
25,654 | 12.407 A line perpendicular to the chord of a segment divides the chord in the ratio 1:4, and the arc - in the ratio $1: 2$. Find the cosine of the central angle subtended by this arc. | -\frac{23}{27} |
12,218 | In the diagram, there are three isosceles right triangles $\triangle ADC$, $\triangle DPE$, and $\triangle BEC$ with unequal hypotenuses, where:
$$
\begin{array}{l}
A D = C D, \\
D P = E P, \\
B E = C E, \\
\angle A D C = \angle D P E = \angle B E C = 90^{\circ}.
\end{array}
$$
Prove that $P$ is the midpoint of line segment $AB$. | P \text{ is the midpoint of } AB |
57,637 | 1. The number of positive integer values of $n$ that satisfy $\left|\sqrt{\frac{n}{n+2009}}-1\right|>\frac{1}{1005}$ is $\qquad$ | 1008015 |
21,781 | In triangle $ABC$ , $AB = 125$ , $AC = 117$ , and $BC = 120$ . The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$ , and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$ . Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$ , respectively. Find $MN$ .
Please give the answer directly without any intermediate steps. | 56 |
28,545 | 5. A trapezoid $ABCD (AD \| BC)$ and a rectangle $A_{1}B_{1}C_{1}D_{1}$ are inscribed in a circle $\Omega$ with a radius of 10, such that $AC \| B_{1}D_{1}, BD \| A_{1}C_{1}$. Find the ratio of the areas of $ABCD$ and $A_{1}B_{1}C_{1}D_{1}$, given that $AD=16, BC=12$. | \frac{49}{50} |
24,866 | In an isosceles right triangle \( \triangle ABC \), \( CA = CB = 1 \). Let point \( P \) be any point on the boundary of \( \triangle ABC \). Find the maximum value of \( PA \cdot PB \cdot PC \). | \dfrac{\sqrt{2}}{4} |
56,262 | 7. Given
$$
(1+x)^{50}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{50} x^{50} \text {. }
$$
then $a_{1}+2 a_{2}+\cdots+25 a_{25}=$ $\qquad$ | 50\times2^{48} |
4,438 | Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\). | 2020032 |
3,382 | Find a three-digit number whose square is a six-digit number, such that each subsequent digit from left to right is greater than the previous one. | 367 |
7,859 | There are some positive integers with more than two digits, such that each pair of adjacent digits forms a perfect square. Then the sum of all positive integers satisfying the above conditions is ? | 97104 |
20,609 | On the coordinate plane, we consider squares where all vertices have non-negative integer coordinates and the center is at the point $(50 ; 30)$. Find the number of such squares. | 930 |
63,803 | Segment $A B$ is the diameter of a circle. A second circle with center at point $B$ has a radius of 2 and intersects the first circle at points $C$ and $D$. Chord $C E$ of the second circle is part of a tangent to the first circle and is equal to 3. Find the radius of the first circle. | \frac{4}{\sqrt{7}} |
21,468 | For example, $5 n$ is a positive integer, its base $b$ representation is 777, find the smallest positive integer $b$ such that $n$ is a fourth power of an integer. | 18 |
55,215 | 11. A triangle $\triangle A B C$ is inscribed in a circle of radius 1 , with $\angle B A C=60^{\circ}$. Altitudes $A D$ and $B E$ of $\triangle A B C$ intersect at $H$. Find the smallest possible value of the length of the segment $A H$. | 1 |
10,542 | Find \(\int \frac{\sin^5(3x)}{\cos^5(3x)} \, dx\). | \frac{\tan^4(3x)}{12} - \frac{\tan^2(3x)}{6} - \frac{1}{3} \ln|\cos(3x)| + C |
54,773 | 7. [5] Convex quadrilateral $A B C D$ has sides $A B=B C=7, C D=5$, and $A D=3$. Given additionally that $m \angle A B C=60^{\circ}$, find $B D$. | 8 |
64,561 | 7. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{0}=0$,
$$
a_{n+1}=\frac{8}{5} a_{n}+\frac{6}{5} \sqrt{4^{n}-a_{n}^{2}}(n \geqslant 0, n \in \mathbf{N}) \text {. }
$$
Then $a_{10}=$ . $\qquad$ | \frac{24576}{25} |
52,398 | 11. Another incorrect equation. In the equation $101-102=1$, move one digit so that it becomes correct. | 101-10^{2}=1 |
10,244 | Let the sides of a triangle be \(a, b, c\) and the semi-perimeter be \(p\). Prove that \(\sqrt{p-a} + \sqrt{p-b} + \sqrt{p-c} \leq \sqrt{3p}\). | \sqrt{p-a} + \sqrt{p-b} + \sqrt{p-c} \leq \sqrt{3p} |
61,273 | 50th Putnam 1989 Problem A1 Which members of the sequence 101, 10101, 1010101, ... are prime? Solution | 101 |
4,284 | Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?
[i]Proposed by Evan Chen[/i] | 1099 |
28,769 | \section*{Problem 4 - 261044}
Determine for each natural number \(k \geq 2\) the number of all solutions \((x, y, z, t)\) of the equation \(\overline{x y} + \overline{z t} = \overline{y z}\), where \(x, y, z, t\) are natural numbers with
\[
1 \leq x \leq k-1, \quad 1 \leq y \leq k-1 \quad 1 \leq z \leq k-1 \quad 1 \leq t \leq k-1
\]
\section*{are allowed!}
Here, \(\overline{p q}\) denotes the number that is written in the positional system of base \(k\) with the digits \(p, q\) (in this order). | \frac{1}{2}(k-3)(k-2) |
67,729 | Problem 5.3. The figure shows a plan of the road system of a certain city. In this city, there are 8 straight streets, and 11 intersections are named with Latin letters $A, B, C, \ldots, J, K$.
Three police officers need to be placed at some intersections so that at least one police officer is on each of the 8 streets. Which three intersections should the police officers be placed at? It is sufficient to provide at least one suitable arrangement.
All streets run along straight lines.
Horizontal streets: $A-B-C-D, E-F-G, H-I-J-K$.
Vertical streets: $A-E-H, B-F-I, D-G-J$.
Diagonal streets: $H-F-C, C-G-K$.
 | B,G,H |
52,729 | 25. From 15 sticks, five triangles were formed. Is there necessarily another way to form five triangles from these sticks? | no |
27,902 | 11. (15 points) Let $x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+}$, define $S_{n} = \sum_{i=1}^{n}\left(x_{i}+\frac{n-1}{n^{2}} \frac{1}{x_{i}}\right)^{2}$,
(1) Find the minimum value of $S_{n}$;
(2) Under the condition $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$, find the minimum value of $S_{n}$;
(3) Under the condition $x_{1}+x_{2}+\cdots+x_{n}=1$, find the minimum value of $S_{n}$, and provide a proof. | n |
22,080 | A quadrilateral with angles of $120^\circ, 90^\circ, 60^\circ$, and $90^\circ$ is inscribed in a circle. The area of the quadrilateral is $9\sqrt{3}$ cm$^{2}$. Find the radius of the circle if the diagonals of the quadrilateral are mutually perpendicular. | 3 |
60,886 | 25. The function $f(x)$ defined on the set of real numbers $\mathbf{R}$ satisfies the following 3 conditions:
(1) $f(x)>0$ when $x>0$; (2) $f(1)=2$; (3) For any $m, n \in \mathbf{R}$, $f(m+n)=f(m)+f(n)$. Let the sets be
$$
\begin{array}{l}
A=\left\{(x, y) \mid f\left(3 x^{2}\right)+f\left(4 y^{2}\right) \leqslant 24\right\}, \\
B=\{(x, y) \mid f(x)-f(a y)+f(3)=0\}, \\
C=\left\{(x, y) \left\lvert\, f(x)=\frac{1}{2} f\left(y^{2}\right)+f(a)\right.\right\},
\end{array}
$$
If $A \cap B \neq \varnothing$ and $A \cap C \neq \varnothing$, find the range of the real number $a$. | [-\frac{13}{6},-\frac{\sqrt{15}}{3}]\cup[\frac{\sqrt{15}}{3},2] |
52,338 | ## Task B-2.4.
A triangle $A B C$ is given. Point $D$ is on side $\overline{A B}$, and point $E$ is on side $\overline{B C}$ of triangle $A B C$ such that $|A D|=3 \mathrm{~cm},|B D|=7 \mathrm{~cm},|B E|=8 \mathrm{~cm},|D E|=5 \mathrm{~cm}$, and $\varangle B A C=\varangle D E B$. What is the area of quadrilateral $A D E C$? | \frac{45\sqrt{3}}{8} |
63,822 | Suppose $28$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite? | 2268 |
68,797 | 6. (3 points) Given the cryptarithm: ЖАЛО + ЛОЖА = ОСЕНЬ. Identical letters represent identical digits, different letters represent different digits. Find the value of the letter А. | 8 |
28,119 | Problem 8.5.1. In the cells of a $12 \times 12$ table, natural numbers are arranged such that the following condition is met: for any number in a non-corner cell, there is an adjacent cell (by side) that contains a smaller number. What is the smallest number of different numbers that can be in the table?
(Non-corner cells are those that are not in the corner of the table. There are exactly 140 of them.) | 11 |
67,253 | 5. The angle at vertex $B$ of triangle $A B C$ is $130^{\circ}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $E$ and $D$. Find the acute angle between the diagonals of the quadrilateral with vertices at points $A, C, D$ and $E$.
Problem 1 Answer: 12 students. | 80 |
65,225 | 14. Let $x, y, z$ be positive real numbers, find the minimum value of $\left(x+\frac{1}{y}+\sqrt{2}\right)\left(y+\frac{1}{z}+\sqrt{2}\right)\left(z+\frac{1}{x}+\sqrt{2}\right)$. | 20+14\sqrt{2} |
750 | What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $ | 55 |
64,026 | 4. A rectangle with dimensions $m$ and $n$, where $m$ and $n$ are natural numbers and $n = m k$, is divided into $m \cdot n$ unit squares. Each path from point $A$ to point $C$ along the divided segments (sides of the small squares) where movement to the right and movement upwards is allowed, has a length of $m+n$. Find how many times the number of paths from $A$ to $C$ that pass through point $T$ is greater than the number of paths from $A$ to $C$ that pass through point $S$. | k |
3,461 | 1. If we consider a pair of skew lines as one pair, then among the 12 lines formed by the edges of a regular hexagonal pyramid, the number of pairs of skew lines is $\qquad$ pairs. | 24 |
6,779 | In \(\triangle ABC\), \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. Given \(a+c=2b\) and \(A-C=\frac{\pi}{3}\), find the value of \(\sin B\). | \dfrac{\sqrt{39}}{8} |
34,169 | 4. Let the line $y=k x+m$ passing through any point $P$ on the ellipse $\frac{x^{2}}{4}+y^{2}=1$ intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ at points $A$ and $B$, and let the ray $P O$ intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ at point $Q$. Then the value of $S_{\triangle A B Q}: S_{\triangle A B O}$ is $\qquad$ | 3 |
62,395 | 6. (10 points) An Englishman was the owner of a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250000 rubles per hectare. It is known that 1 acre $=4840$ square yards, 1 yard $=0.9144$ meters, 1 hectare $=10000 m^{2}$. Calculate how much the Englishman will receive as a result of the sale. | 303514 |
62,276 | LX OM - I - Task 5
For each integer $ n \geqslant 1 $, determine the largest possible number of different subsets of the set $ \{1,2,3, \cdots,n\} $ with the following property: Any two of these subsets are either disjoint or one is contained in the other. | 2n |
2,708 | Given a point \( P \) inside \( \triangle ABC \), perpendiculars are drawn from \( P \) to \( BC, CA, \) and \( AB \) with feet \( D, E, \) and \( F \) respectively. Semicircles are constructed externally on diameters \( AF, BF, BD, CD, CE, \) and \( AE \). These six semicircles have areas denoted \( S_1, S_2, S_3, S_4, S_5, \) and \( S_6 \). Given that \( S_5 - S_6 = 2 \) and \( S_1 - S_2 = 1 \), find \( S_4 - S_3 \). | 3 |
15,255 | Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$. | 3\sqrt{2} |
18,502 | From a \(6 \times 6\) checkered square, gray triangles were cut out. What is the area of the remaining figure? The side length of each cell is 1 cm. Give the answer in square centimeters. | 27 |
5,653 | Let \( k_1 \) and \( k_2 \) be two circles that touch each other externally at point \( P \). Let a third circle \( k \) touch \( k_1 \) at \( B \) and \( k_2 \) at \( C \) so that \( k_1 \) and \( k_2 \) lie inside \( k \). Let \( A \) be one of the intersection points of \( k \) with the common tangent of \( k_1 \) and \( k_2 \) through \( P \). The lines \( AB \) and \( AC \) intersect \( k_1 \) and \( k_2 \) again at \( R \) and \( S \) respectively. Show that \( RS \) is a common tangent of \( k_1 \) and \( k_2 \). | RS \text{ is a common tangent of } k_1 \text{ and } k_2 |
4,885 | 5. Let $x$ be the length of the smaller segment. On the upper side of the quadrilateral, which has a length of 1, there are 3 small segments and one large segment. Therefore, the length of the large segment is $(1-3x)$. On the lower side of the quadrilateral, which has a length of 2, there are 3 large segments and one small segment.
We get the equation $3 \cdot (1-3x) + x = 2$.
Thus, $3 - 9x + x = 2$.
Therefore, $x = \frac{1}{8}$.
So, the length of the smaller segment is $\frac{1}{8}$. Therefore, the length of the larger segment is $1 - 3 \cdot \frac{1}{8} = 1 - \frac{3}{8} = \frac{5}{8}$.
Hence, the larger segment is five times longer than the smaller one. | 5 |
7,534 | Find all functions \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) such that for all \( m, n \in \mathbf{Z}_{+} \), the following conditions hold:
$$
f(mn) = f(m)f(n)
$$
and
$$
(m+n) \mid (f(m) + f(n)).
$$ | f(n) = n^k |
65,237 | (5) Divide the positive integers $1,2,3,4,5,6,7$ into two groups arbitrarily, with each group containing at least one number, then the probability that the sum of the numbers in the first group is equal to the sum of the numbers in the second group is $\qquad$ | \frac{4}{63} |
65,139 | Example 6 As shown in Figure 6, fill the ten circles in Figure 6 with the numbers $1,2, \cdots, 10$ respectively, so that the sum of the numbers in any five consecutive adjacent circles is not greater than a certain integer $M$. Find the minimum value of $M$ and complete your filling. ${ }^{[4]}$ | 28 |
28,867 | 52. Given a triangle $T$.
a) Place a centrally symmetric polygon $m$ of the largest possible area inside $T$.
What is the area of $m$ if the area of $T$ is 1?
b) Enclose $T$ in a convex centrally symmetric polygon $M$ of the smallest possible area.
What is the area of $M$ if the area of $T$ is 1? | 2 |
55,129 | 4. In a certain city, the fare scheme for traveling by metro with a card is as follows: the first trip costs 50 rubles, and each subsequent trip costs either the same as the previous one or one ruble less. Petya spent 345 rubles on several trips, and then on several subsequent trips - another 365 rubles. How many trips did he make?
 | 15 |
59,628 | Problem 8. Given an isosceles triangle $K L M(K L=L M)$ with the angle at the vertex equal to $114^{\circ}$. Point $O$ is located inside triangle $K L M$ such that $\angle O M K=30^{\circ}$, and $\angle O K M=27^{\circ}$. Find the measure of angle $\angle L O M$. | 150 |
33,818 | The eleventh question: Given a positive integer $n \geq 2$, find the smallest real number $c(n)$ such that for any positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$, the following inequality holds: $\frac{\sum_{i=1}^{n} a_{i}}{n}-\sqrt[n]{\prod_{i=1}^{n} a_{i}} \leq c(n) \cdot \max _{1 \leq i<j \leq n}\left\{\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^{2}\right\}$. | (n)=\frac{n-1}{n} |
2,841 | Given a triangle \(ABC\) with \(\angle A = 42^\circ\) and \(AB < AC\). Point \(K\) on side \(AC\) is such that \(AB = CK\). Points \(P\) and \(Q\) are the midpoints of segments \(AK\) and \(BC\), respectively. What is the measure of \(\angle ACB\) if it is known that \(\angle PQC = 110^\circ\)? | 49^\circ |
63,663 | $A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side. | n = 9 |
32,223 | 1. (AUS 3) The integer 9 can be written as a sum of two consecutive integers: $9=4+5$. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $9=4+5=2+3+4$. Is there an integer that can be written as a sum of 1990 consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly 1990 ways? | 5^{10}\cdot199^{180} |
58,243 | Find the minimum of the function
$$
f(x, y)=\sqrt{(x+1)^{2}+(2 y+1)^{2}}+\sqrt{(2 x+1)^{2}+(3 y+1)^{2}}+\sqrt{(3 x-4)^{2}+(5 y-6)^{2}} \text {, }
$$
defined for all real $x, y>0$. | 10 |
59,736 | 6. For any closed interval $I$, let $M_{I}$ denote the maximum value of the function $y=\sin x$ on $I$. If the positive number $a$ satisfies $M_{[0, a]}=2 M_{[a, 2 a]}$, then the value of $a$ is $\qquad$ . | \frac{5\pi}{6} |
58,810 | Alice, Bob, Cindy, David, and Emily sit in a circle. Alice refuses to sit to the right of Bob, and Emily sits next to Cindy. If David sits next to two girls, determine who could sit immediately to the right of Alice. | \text{Bob} |
52,001 | 9.4. Under the Christmas tree, there are 2012 cones. Winnie-the-Pooh and donkey Eeyore are playing a game: they take cones for themselves in turns. On his turn, Winnie-the-Pooh takes 1 or 4 cones, and Eeyore takes 1 or 3. Pooh goes first. The player who cannot make a move loses. Who among the players can guarantee a win, no matter how the opponent plays? | Winnie-the-Pooh |
55,008 | ## Problem Statement
Find the derivative.
$y=\frac{x^{6}+x^{3}-2}{\sqrt{1-x^{3}}}$ | \frac{9x^{5}}{2\sqrt{1-x^{3}}} |
58,341 | Let's determine how many regions the space is divided into by a) the six planes of the cube's faces?
b) the four planes of the tetrahedron's faces? | 15 |
67,365 | The bar graph below shows the numbers of boys and girls in Mrs. Kuwabara's class. The percentage of students in the class who are girls is
(A) $40 \%$
(B) $15 \%$
(C) $25 \%$
(D) $10 \%$
(E) $60 \%$
Students in Mrs. Kuwabara's Class
Number of Students | 40 |
16,979 | The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$ . Suppose that the radius of the circle is $5$ , that $BC=6$ , and that $AD$ is bisected by $BC$ . Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$ . It follows that the sine of the central angle of minor arc $AB$ is a rational number. If this number is expressed as a fraction $\frac{m}{n}$ in lowest terms, what is the product $mn$
[asy]size(100); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=1; pair O1=(0,0); pair A=(-0.91,-0.41); pair B=(-0.99,0.13); pair C=(0.688,0.728); pair D=(-0.25,0.97); path C1=Circle(O1,1); draw(C1); label("$A$",A,W); label("$B$",B,W); label("$C$",C,NE); label("$D$",D,N); draw(A--D); draw(B--C); pair F=intersectionpoint(A--D,B--C); add(pathticks(A--F,1,0.5,0,3.5)); add(pathticks(F--D,1,0.5,0,3.5)); [/asy] | 175 |
51,519 | 11.42. If five points are given on a plane, then by considering all possible triples of these points, one can form 30 angles. Let the smallest of these angles be $\alpha$. Find the maximum value of $\alpha$.
If five points are given on a plane, then by considering all possible triples of these points, one can form 30 angles. Let the smallest of these angles be $\alpha$. Find the maximum value of $\alpha$. | 36 |
60,559 | 30. In each of the following 6-digit positive integers: $555555,555333,818811$, 300388, every digit in the number appears at least twice. Find the number of such 6-digit positive integers. | 11754 |
4,665 | What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)\] is isosceles? | 380 |
65,069 | 3. Find the largest natural number $n$, for which the number 999...99 (with 999 nines) is divisible by $9^{n}$.
---
The text has been translated from Macedonian to English while preserving the original formatting and structure. | 2 |
69,154 | We wrote the numbers from 1 to 2009 on a piece of paper. In the second step, we also wrote down twice each of these numbers on the paper, then erased the numbers that appeared twice. We repeat this step in such a way that in the $i$-th step, we also write down $i$ times each of the numbers $1, 2, \ldots, 2009$ on the paper, then erase the numbers that appear twice. How many numbers will be on the paper after the 2009th step? | 2009 |
69,137 | 【3】A key can only open one lock. Now there are 10 different locks and 11 different keys. If you need to find the key for each lock, you would need to try $(\quad)$ times at most to correctly match each lock with its key. | 55 |
55,556 | 3. Given a regular hexagon $A B C D E F$, with side $10 \sqrt[4]{27}$. Find the area of the union of triangles ACE and BDF.
 | 900 |
64,373 | 3. A circle is inscribed in trapezoid $A B C D$, touching the lateral side $A D$ at point $K$. Find the area of the trapezoid if $A K=16, D K=4$ and $C D=6$. | 432 |
51,019 | 7. If the expression $\frac{1}{1 \times 2}-\frac{1}{3 \times 4}+\frac{1}{5 \times 6}-\frac{1}{7 \times 8}+\cdots-\frac{1}{2007 \times 2008}+\frac{1}{2009 \times 2010}$ is converted to a decimal, then the first digit after the decimal point is $\qquad$ . | 4 |
5,108 | In the figure, the triangle $\triangle A B C$ is isosceles, with $B \widehat{A} C=20^{\circ}$.
Knowing that $B C=B D=B E$, determine the measure of the angle $B \widehat{D} E$.
 | 60^\circ |
24,189 | [b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$ , and y. Find the greatest possible value of: $x + y$ .
<span style="color:red">NOTE: There is a high chance that this problems was copied.</span> | 761 |
65,988 | 71. What are the last two digits of the number
$$
n^{a}+(n+1)^{a}+(n+2)^{a}+\ldots+(n+99)^{a}
$$
a) $a=4$
b) $a=8$ ? | 30 |
57,902 | 1. As shown in Figure 1, it is known that rectangle $A B C D$ can be exactly divided into seven small rectangles of the same shape and size. If the area of the small rectangle is 3, then the perimeter of rectangle $A B C D$ is | 19 |
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