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52,479 | 4. [3] Three brothers Abel, Banach, and Gauss each have portable music players that can share music with each other. Initially, Abel has 9 songs, Banach has 6 songs, and Gauss has 3 songs, and none of these songs are the same. One day, Abel flips a coin to randomly choose one of his brothers and he adds all of that brother's songs to his collection. The next day, Banach flips a coin to randomly choose one of his brothers and he adds all of that brother's collection of songs to his collection. Finally, each brother randomly plays a song from his collection with each song in his collection being equally likely to be chosen. What is the probability that they all play the same song? | \frac{1}{288} |
15,594 | Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the walls of the maze. After completing the maze, he noticed that it was possible to reach any unit square from any other unit square in exactly one way, excluding paths that pass through any unit square more than once. What is the side length of the initially drawn large square? | 21 |
57,329 | 60.1. Three distinct diameters are drawn on a unit circle such that chords are drawn as shown in Figure 2. If the length of one chord is $\sqrt{2}$ units and the other two chords are of equal lengths, what is the common length of these chords? | \sqrt{2-\sqrt{2}} |
21,787 |
If all the black squares are removed from a standard $8 \times 8$ chessboard, the board loses the ability to accommodate even a single $2 \times 1$ domino piece. However, this ability is restored if at least one black square is returned. A chessboard reduced in this manner is referred to as an "elegantly destroyed" board.
It is not necessary to remove all black (or all white) squares for such destruction. It is possible to remove a mix of black and white squares. Thirty-two domino pieces can obviously cover the chessboard, so if at least one square is not removed in any of the formed 32 pairs of squares, then at least one domino piece can be placed on the remaining board.
To achieve a board destroyed in the required manner, at least 32 squares must be removed.
On the other hand, it is possible to remove more than 32 squares and still get an elegantly destroyed board. The trick lies in positioning these squares so that returning any one removed square allows the placement of at least one domino piece on the board. Under this strict constraint, determine the maximum number of squares that can be removed. | 48 |
32,242 | (50 points) Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=$ $x^{3}+a x^{2}+b x+c$ have three real roots $x_{1}, x_{2}, x_{3}$, and satisfy:
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$. | \frac{3}{2}\sqrt{3} |
8,168 | ## Task 3 - 010713
During a class day on socialist production, a student saws off a piece of square steel that weighs 475 p. The next day, a piece of square steel, whose dimensions are four times larger than those of the sawn-off piece and which is made of the same material, is processed.
How heavy is the piece? Justify your answer! | 30400 |
5,260 | In a convex quadrilateral $ABCD$, $AD$ is not parallel to $BC$, and there exists a circle $\Gamma$ that is tangent to $BC$ and $AD$ at points $C$ and $D$, respectively. Let $AC$ and $BD$ intersect at point $P$, and let the circle $\Gamma$ intersect $AB$ at two distinct points $K$ and $L$. Show that the line $KP$ bisects $CD$ if and only if $LC = LD$. | LC = LD |
3,257 | $P$ and $Q$ are points on the semicircular arc $AB$ whose diameter is $AB$. $R$ is a point on the radius $OB$ such that $\angle OPR = \angle OQR = 10^{\circ}$. If $\angle POA = 40^{\circ}$, what is $\angle QOB$? | 20^\circ |
9,901 | Given \(\triangle ABC\) inscribed in circle \(\odot O\), let \(P\) be any point inside \(\triangle ABC\). Lines parallel to \(AB\), \(AC\), and \(BC\) are drawn through point \(P\), intersecting \(BC\), \(AC\), and \(AB\) at points \(F\), \(E\), \(K\), and \(I\). Lines intersecting \(AB\) and \(AC\) at points \(G\) and \(H\) are also drawn. Let \(AD\) be a chord of \(\odot O\) passing through point \(P\). Prove that \(EF^2+KI^2+GH^2 \geq 4PA \cdot PD\). | EF^2 + KI^2 + GH^2 \geq 4PA \cdot PD |
15,980 | Given a sequence \( x_{1}, x_{2}, \ldots \) consisting of positive numbers that is monotonically decreasing, the following inequality holds for every \( n \):
$$
\frac{x_{1}}{1}+\frac{x_{4}}{2}+\ldots+\frac{x_{n^{2}}}{n} \leqq 1
$$
Prove that
$$
\frac{x_{1}}{1}+\frac{x_{2}}{2}+\ldots+\frac{x_{n}}{n}<3
$$ | 3 |
53,606 | C1. Sara has 10 blocks numbered $1 \mathrm{t} / \mathrm{m}$ 10. She wants to stack all the blocks into a tower. A block can only be placed on top of a block with a higher number, or on a block with a number that is exactly one lower. An example is, from top to bottom: 2, 1, 5, 4, 3, 6, 7, 9, 8, 10. How many different towers are possible? | 512 |
56,184 | 9. The three prime numbers $a, b, c$ satisfy $a<b<c<100$, and $(b-a) \times(c-b) \times(c-a)=240$, then the maximum sum of the three prime numbers $a, b, c$ is $\qquad$ _. | 251 |
20,784 | 4. Farmer John is inside of an ellipse with reflective sides, given by the equation $x^{2} / a^{2}+$ $y^{2} / b^{2}=1$, with $a>b>0$. He is standing at the point $(3,0)$, and he shines a laser pointer in the $y$-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair $(a, b)$ ? | (5, 4) |
57,726 | In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won? | 11 |
23,261 | \section*{Exercise 3 - 061233}
All real numbers \(x\) in the intervals \(0<x<\frac{\pi}{2}\) and \(\frac{\pi}{2}<x<\pi\) are to be specified for which
\[
f(x)=\sin x+\cos x+\tan x+\cot x
\]
is positive, and all real numbers \(x\), in the same intervals, for which \(f(x)\) is negative. Is there a smallest positive value that \(f(x)\) assumes in the above intervals, and if so, what is this value? | 2 + \sqrt{2} |
64,042 | 8. Let $x, y$ be nonnegative integers such that $x+2 y$ is a multiple of $5, x+y$ is a multiple of 3 and $2 x+y \geq 99$. Find the minimum possible value of $7 x+5 y$.
設 $x 、 y$ 少非負整數, 使得 $x+2 y$ 爲 5 的倍數、 $x+y$ 絧 3 的倍數, 且 $2 x+y \geq 99$ 。求 $7 x+5 y$ 的最小可能値。 | 366 |
64,311 | 10. A key can only open one lock. Now there are 10 keys and 10 locks, but it is unknown which key opens which lock. At most how many attempts are needed to successfully match all the keys and locks. | 45 |
24,361 | Calculate the volumes of solids formed by the rotation of regions bounded by the graphs of the functions around the y-axis.
$$
y=\arcsin \frac{x}{5}, y=\arcsin x, y=\frac{\pi}{2}
$$ | 6\pi^2 |
58,299 | 4. Among the natural numbers from 2021 to 9998, the numbers with the same tens and units digits are $\qquad$ in total. | 797 |
13,089 | Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$ . Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? | 480 |
20,082 | On cards, the numbers $415, 43, 7, 8, 74, 3$ are written. Arrange the cards in a row so that the resulting ten-digit number is the smallest possible. | 3415437478 |
50,748 | 9.1. Find the smallest six-digit number that is a multiple of 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits and is equal to the sum of the third and sixth digits. | 100122 |
55,918 | $24, 2$ red balls and $11$ white balls are arranged in a row, satisfying the following conditions: the red balls are not adjacent, and in any consecutive $7$ balls, there is at least one red ball, then there are $\qquad$ ways to arrange them. | 31 |
30,000 | 10. Given that $f(x)$ is an odd function on $\mathbf{R}$, $f(1)=1$, and for any $x<0$, $f\left(\frac{x}{x-1}\right) = x f(x)$. Find the value of $f(1) f\left(\frac{1}{100}\right) + f\left(\frac{1}{2}\right) f\left(\frac{1}{99}\right) + f\left(\frac{1}{3}\right) f\left(\frac{1}{98}\right) + \cdots + f\left(\frac{1}{50}\right) f\left(\frac{1}{51}\right)$. | \frac{2^{98}}{99!} |
3,635 | 38. Did the passenger miss the train?
- Poor thing! - Quasiturtle sobbed. Think of it! If he were a little smarter, he could have left earlier and caught the train!
- I remembered the cyclist, - Quasiturtle continued. - The train left the station eleven minutes late and traveled at a speed of ten miles per hour to the next station, which was one and a half miles from the first. At the next station, it stood for fourteen and a half minutes. The passenger arrived at the first station twelve minutes after the train's scheduled departure and walked to the next station at a speed of four miles per hour, hoping to catch the train there.
Did the passenger manage to catch up with the train? | Yes |
8,468 | Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by 2 and add 1 or he may multiply the stored value by 4 and add 3. The first player to make the stored value exceed \(2^{100}\) wins. The stored value starts at 1 and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
(Optimal play means that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.) | 33 |
67,817 | Question: A student participates in military training and engages in target shooting, which must be done 10 times. In the 6th, 7th, 8th, and 9th shots, he scored 9.0 points, 8.4 points, 8.1 points, and 9.3 points, respectively. The average score of his first 9 shots is higher than the average score of his first 5 shots. If he wants the average score of 10 shots to exceed 8.8 points, how many points does he need to score at least in the 10th shot? (The points scored in each shot are accurate to 0.1 points) | 9.9 |
19,627 | In the tetrahedron \(ABCD\), \(AB\perp BC\), \(CD \perp BC\), \(BC=2\), and the angle between the skew lines \(AB\) and \(CD\) is \(60^\circ\). If the circumradius of the tetrahedron \(ABCD\) is \(\sqrt{5}\), then find the maximum volume of this tetrahedron. | 2\sqrt{3} |
495 | Find all positive integers \(a, m, n\) such that \(a^m + 1\) divides \((a + 1)^n\). | (a, m, n) \text{ with } a = 1 \text{ or } m = 1 \text{ or } (a, m, n) = (2, 3, n) \text{ for } n \geq 2 |
62,830 | 4. From $1,2, \cdots, 10$ choose 3 different numbers $a, b, c$ as the coefficients of the quadratic equation $a x^{2}+b x=c$. Then the number of equations with different solutions is $\qquad$ | 654 |
21,771 | The teacher wrote a four-digit number on a piece of paper and had 4 rounds of Q&A with Xiaowei. Xiaowei: "Is it 8765?" Teacher: "You guessed two digits correctly, but their positions are wrong." Xiaowei: "Is it 1023?" Teacher: "You guessed two digits correctly, but their positions are wrong." Xiaowei: "Is it 8642?" Teacher: "You guessed two digits correctly, and their positions are correct." Xiaowei: "Is it 5430?" Teacher: "None of the digits are correct." The four-digit number is $\qquad$ - | 7612 |
62,605 | Let two glasses, numbered $1$ and $2$, contain an equal quantity of liquid, milk in glass $1$ and coffee in glass $2$. One does the following: Take one spoon of mixture from glass $1$ and pour it into glass $2$, and then take the same spoon of the new mixture from glass $2$ and pour it back into the first glass. What happens after this operation is repeated $n$ times, and what as $n$ tends to infinity? | \frac{v}{2} |
65,431 | ## Task 4 - V00704
Which of the two numbers is the larger?
$$
\frac{35}{47} \quad \text { or } \quad \frac{23}{31}
$$
Which four-digit decimal fraction is closest to both numbers? | 0.7433 |
58,790 | 2. How many ways are there to arrange 5 identical red balls and 5 identical blue balls in a line if there cannot be three or more consecutive blue balls in the arrangement? | 126 |
26,881 | Problem 7.4. On the sides $AB$ and $AC$ of triangle $ABC$, points $X$ and $Y$ are chosen such that $\angle A Y B = \angle A X C = 134^{\circ}$. On the ray $YB$ beyond point $B$, point $M$ is marked, and on the ray $XC$ beyond point $C$, point $N$ is marked. It turns out that $MB = AC$ and $AB = CN$. Find $\angle MAN$.
Answer: $46^{\circ}$. | 46 |
24,877 | Let $P$ be a point outside a circle $\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\Gamma$ . Let segment $PO$ intersect circle $\Gamma$ at $C$ . A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$ , respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$ , compute the area of $\triangle{AOC}$ .
*2020 CCA Math Bonanza Individual Round #6* | 12\sqrt{3} |
67,908 | 9.1. (13 points) In how many ways can eight of the nine digits $1,2,3,4,5,6$, 7,8 and 9 be placed in a $4 \times 2$ table (4 rows, 2 columns) so that the sum of the digits in each row, starting from the second, is 1 more than in the previous one? | 64 |
58,104 | 11. For the equation $\sin ^{2} x+a \cos x-2 a=0$ with respect to $x$ to have real roots, the range of the real number $a$ is | [0,4-2\sqrt{3}] |
51,970 | 72. Using the digits $1,2,3$ and a decimal point, you can form $\qquad$ decimal numbers. If not all three digits need to be used, you can form $\qquad$ decimal numbers. | 18 |
65,998 | Find the largest natural number, all digits in the decimal representation of which are different and which
is reduced by 5 times if the first digit is erased. | 3750 |
4,199 | Let \( s(n) \) be the sum of the digits of \( n \) in base \( p \). Prove that the highest power of the prime \( p \) dividing \( n! \) is given by \( \frac{n-s(n)}{p-1} \).
Find the smallest positive integer \( n \) such that \( n! \) ends with exactly 2005 zeros. | 8030 |
62,905 | Shaovalov A.V.
Along the path between the houses of Nезнayka and Sineglazka, there were 15 peonies and 15 tulips growing in a row, mixed together. Setting out from home to visit Nезнayka, Sineglazka watered all the flowers in a row. After the 10th tulip, the water ran out, and 10 flowers remained unwatered. The next day, setting out from home to visit Sineglazka, Nезнayka picked all the flowers in a row for her. After picking the 6th tulip, he decided that it was enough for a bouquet. How many flowers remained growing along the path? | 19 |
11,151 | For real numbers \( x \) and \( y \), define the operation \( \star \) as follows: \( x \star y = xy + 4y - 3x \).
Compute the value of the expression
$$
((\ldots)(((2022 \star 2021) \star 2020) \star 2019) \star \ldots) \star 2) \star 1
$$ | 12 |
53,657 | 2. A six-digit number $A$ is divisible by 19. The number obtained by removing its last digit is divisible by 17, and the number obtained by removing the last two digits of $A$ is divisible by 13. Find the largest $A$ that satisfies these conditions. | 998412 |
63,046 | 2. Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$. | 4 |
56,796 | Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$ | 30787 |
22,046 | Let \( n \geq 2 \) be a fixed integer. Find the smallest constant \( C \) such that for all non-negative reals \( x_1, x_2, \ldots, x_n \):
\[ \sum_{i < j} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4. \]
Determine when equality occurs. | \dfrac{1}{8} |
67,943 | (a) You are given the expression
$$
1 \diamond 2 \diamond 3 \diamond 4 \diamond 5 \diamond 6 \diamond 7 \text {. }
$$
Determine whether it is possible to replace one of the symbols $\diamond$ with $=$ and the other symbols $\diamond$ with + or - so as to end up with a correct equation. | 2+3+4+5-6-7 |
30,526 | 21. Each of the integers $1,2,3, \ldots, 9$ is assigned to each vertex of a regular 9 -sided polygon (that is, every vertex receives exactly one integer from $\{1,2, \ldots, 9\}$, and two vertices receive different integers) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive integer $n$. What is the least possible value of $n$ for which this assignment can be done? | 16 |
1,478 | A quadrilateral \(ABCD\) is inscribed in a circle \(S\). Circles \(S_1\) and \(S_2\) of equal radius are tangent to the circle \(S\) internally at points \(A\) and \(C\) respectively. The circle \(S_1\) intersects sides \(AB\) and \(AD\) at points \(K\) and \(N\) respectively, and the circle \(S_2\) intersects sides \(BC\) and \(CD\) at points \(L\) and \(M\) respectively. Prove that \(KLMN\) is a parallelogram. | KLMN \text{ is a parallelogram} |
4,774 | 3. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. What is the smaller part of the volume separated from it by a plane passing through the midpoint of edge $A D$ and points $X$ and $Y$ on edges $A A_{1}$ and $C C_{1}$ such that $A X: A_{1} X=C Y: C_{1} Y=1: 7 ?$ | \dfrac{25}{192} |
5,357 | A set of lines in the plane is called to be in general position if no two lines are parallel, and no three lines intersect at a single point. A set of such lines partitions the plane into regions, some of which have finite area; these regions are called finite regions of the line set.
Prove that for any sufficiently large \( n \), there exists a coloring of at least \( \sqrt{n} \) of these lines blue, such that no finite region has its boundary entirely composed of blue lines.
Note: Solutions that prove the statement for \( c \sqrt{n} \) instead of \( \sqrt{n} \) will also be awarded points, depending on the value of the constant \( c \). | \sqrt{n} |
60,421 | Find the $a, n \geq 1$ such that $\left((a+1)^{n}-a^{n}\right) / n$ is an integer. | 1 |
25,045 | Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the minimum number of different lines that can be obtained?
Answer. 9. | 9 |
12,651 | [Example 3.6.6] Find all positive integers $n>1$, such that $\frac{2^{n}+1}{n^{2}}$ is an integer. | 3 |
24,401 | Determine if there exists a positive integer \( m \) such that the equation
\[
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=\frac{m}{a+b+c}
\]
has infinitely many solutions in positive integers \( (a, b, c) \). | 12 |
56,338 | 11. (25 points) Let positive numbers $a, b$ satisfy $a+b=1$. Find
$$
M=\sqrt{1+2 a^{2}}+2 \sqrt{\left(\frac{5}{12}\right)^{2}+b^{2}}
$$
the minimum value. | \frac{5\sqrt{34}}{12} |
55,563 | 6 Given $\odot O_{1}\left(r_{1}\right), \odot O_{2}\left(r_{2}\right)$ are externally tangent to $\odot O(r)$ at $A_{1}, A_{2}$. When $r_{1}=1, r_{2}=2$, $r=3, A_{1} A_{2}=4$, find the length of the external common tangent $T_{1} T_{2}$ of $\odot O_{1}$ and $\odot O_{2}$. | \frac{8}{3}\sqrt{5} |
15,187 | Let \( l \) and \( m \) be two skew lines. On line \( l \), there are points \( A, B, \) and \( C \) such that \( AB = BC \). From points \( A, B, \) and \( C \), perpendicular lines \( AD, BE, \) and \( CF \) are drawn such that they are perpendicular to \( m \), and the feet of these perpendiculars are \( D, E, \) and \( F \) respectively. Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \). | \sqrt{6} |
57,796 | A $5 \times 5$ chessboard consists of unit squares, 7 of which are red and 18 are blue. Two of the red squares are located on the edge of the board. The segments that separate two adjacent red squares are also colored red. The segments that separate two adjacent blue squares are colored blue. All other segments, including the edges of the board, are black. In this way, a total of 35 black segments are formed. How many red segments are there? | 5 |
20,434 | Given that the real numbers \( x, y \) and \( z \) satisfy the condition \( x + y + z = 3 \), find the maximum value of \( f(x, y, z) = \sqrt{2x + 13} + \sqrt[3]{3y + 5} + \sqrt[4]{8z + 12} \). | 8 |
26,021 | 4. Given a set of points on a plane $P=\left\{P_{1}, P_{2}, \cdots, P_{1994}\right\}$, where no three points in $P$ are collinear. Divide all points in $P$ into 83 groups arbitrarily, such that each group has at least 3 points, and each point belongs to exactly one group. Then connect any two points in the same group with a line segment, and do not connect points in different groups, thus obtaining a pattern $G$. Different grouping methods result in different patterns, and the number of triangles in pattern $G$ with vertices from $P$ is denoted as $m(G)$.
(1) Find the minimum value $m_{0}$ of $m(G)$;
(2) Let $G^{*}$ be a pattern such that $m\left(G^{*}\right)=m_{0}$. If the line segments in $G^{*}$ (referring to line segments with endpoints from $P$) are colored with 4 colors, and each line segment is colored with exactly one color, prove that there exists a coloring scheme such that $G^{*}$, after coloring, contains no triangle with vertices from $P$ and all three sides of the same color. | 168544 |
63,158 | ## Task B-4.4.
Grandpa Ante, looking for a way to entertain his grandchildren Iva and Mato, found three cards. On the first card, one side has the number 1, and the other side has the number 4. On the second card, one side has the number 2, and the other side has the number 4. On the third card, one side has the number 3, and the other side also has the number 4. Grandpa Ante immediately knew how to keep the children occupied for a long time. He told the younger one, Mato, to arrange the three cards in a row, and the schoolboy Iva had to write down the three-digit numbers that Mato forms by arranging the cards and then calculate the sum of all the recorded numbers. Mato randomly picks a card and its side. How many different three-digit numbers can Ivo write down at most, and what is the sum of these numbers? | 10434 |
21,113 | From village $C$, a car departs at 7:58 AM, traveling eastward on a straight road toward village $B$ at a speed of $60 \text{ km/h}$. At 8:00 AM, a cyclist leaves the same village, traveling northward on a straight road toward village $A$, which is $10 \text{ km}$ away, at a speed of $18 \text{ km/h}$. A pedestrian had already left village $A$ at 6:44 AM, walking toward village $B$ along the straight road $A B$ at a speed of $6 \text{ km/h}$. The roads $A B$ and $A C$ form an angle of $60^{\circ}$. When will the cyclist be equidistant from both the car and the pedestrian in a straight line? | 8:06 \text{ AM} |
25,968 | 19. Grandfather Frost has many identical dials in the form of regular 12-sided polygons, on which numbers from 1 to 12 are printed. He places these dials in a stack on top of each other (one by one, face up). In doing so, the vertices of the dials coincide, but the numbers in the coinciding vertices do not necessarily match. The Christmas tree will light up as soon as the sums of the numbers in all 12 columns have the same remainder when divided by 12. How many dials can be in the stack at this moment? | 12 |
66,254 | 8.4. We will call a number remarkable if it can be decomposed into the sum of 2023 addends (not necessarily distinct), each of which is a natural composite number. Find the largest integer that is not remarkable. | 8095 |
22,712 | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(7, -5, 1)$
$B(5, -1, -3)$
$C(3, 0, -4)$ | -2x + y - z + 20 = 0 |
25,075 | Through the midpoints $M$ and $N$ of the edges $A A_1$ and $C_1 D_1$ of the parallelepiped $A B C D A_1 B_1 C_1 D_1$, a plane parallel to the diagonal $B D$ of the base is drawn. Construct the section of the parallelepiped by this plane. In what ratio does it divide the diagonal $A_1 C$? | 3:7 |
53,256 | Agakhanovo $H . X$.
Different numbers $a, b$ and $c$ are such that the equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root. In addition, the equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$ have a common real root. Find the sum $a+b+c$. | -3 |
57,587 | 11. Given the set $A=\{1,2,3,4, \cdots, 101\}, B \subseteq A$, and the smallest element in set $B$ is an even number.
(1) If the smallest element in set $B$ is 2, and the largest element is 13, find the number of sets $B$ that satisfy the condition;
(2) Find the sum of the largest elements of all sets $B$ that satisfy the condition. | \frac{50}{3}(2^{102}-1) |
7,774 | Given that \( a_1, a_2, a_3, a_4 \) are 4 distinct numbers chosen from \( 1, 2, \cdots, 100 \), satisfying
$$
\left(a_1^2 + a_2^2 + a_3^2\right)\left(a_2^2 + a_3^2 + a_4^2\right) = \left(a_1 a_2 + a_2 a_3 + a_3 a_4\right)^2,
$$
determine the number of such ordered tuples \( (a_1, a_2, a_3, a_4) \). | 40 |
29,322 | 10. find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $x, y \in \mathbb{R}$ holds for all $x, y \in \mathbb{R}$:
$$
f(x+y f(x+y))=y^{2}+f(x f(y+1))
$$ | f(x)=x |
29,821 | II. (50 points)
Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$. Determine the maximum value of $P = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$. | \frac{10}{3} |
60,582 | Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$. | 32 |
55,062 | 3.1. The sum of two unequal heights of an isosceles triangle is equal to $l$, the angle at the vertex is $\alpha$. Find the lateral side. | \frac{}{\cos\frac{\alpha}{2}+\sin\alpha} |
32,194 | Task 10.5. (20 points) Find all composite natural numbers $n$ that have the following property: each natural divisor of the number $n$ (including $n$ itself), decreased by 1, is a square of an integer. | 10 |
25,752 | 12. As shown in the right figure, fill the numbers $1.2, 3.7, 6.5, 2.9, 4.6$ into five ○s respectively, then fill each $\square$ with the average of the three $\bigcirc$ numbers it is connected to, and fill the $\triangle$ with the average of the three $\square$ numbers. Find a way to fill the numbers so that the number in $\triangle$ is as small as possible. What number should be filled in $\triangle$? | 3.1 |
2,968 | ## Task 4 - 300824
Imagine the numbers 1, 2, 3, 4, ..., 9999 written consecutively to form a number $z$.
The beginning of this representation is $z=123456789101112131415 \ldots$; for example, the digit 0 appears at the 11th position, the digit 2 appears at the 2nd position, the 15th position, and at several other positions in $z$.
Which digit stands at the 206788th position of $z$? | 7 |
65,795 | At the end-of-year concert of a music school, four violinists performed. Whenever one of them was not playing, they took a seat among the audience. In at least how many pieces did the violinists perform, if each of them had the opportunity to watch any of their (violinist) colleagues from the auditorium? | 4 |
26,271 | In the triangle $ABC$, what fraction of the area of the triangle is occupied by the points $K$ that lie inside the triangle and for which the central similarity with center $K$ and ratio $-1/2$ maps $A$ and $B$ to points inside the triangle and $C$ to a point outside the triangle? | \frac{1}{9} |
19,249 |
Let \( M \) be a set composed of a finite number of positive integers,
\[
M = \bigcup_{i=1}^{20} A_i = \bigcup_{i=1}^{20} B_i, \text{ where}
\]
\[
A_i \neq \varnothing, B_i \neq \varnothing \ (i=1,2, \cdots, 20)
\]
satisfying the following conditions:
1. For any \( 1 \leqslant i < j \leqslant 20 \),
\[
A_i \cap A_j = \varnothing, \ B_i \cap B_j = \varnothing;
\]
2. For any \( 1 \leqslant i \leqslant 20, \ 1 \leqslant j \leqslant 20 \), if \( A_i \cap B_j = \varnothing \), then \( \left|A_i \cup B_j\right| \geqslant 18 \).
Find the minimum number of elements in the set \( M \) (denoted as \( |X| \) representing the number of elements in set \( X \)). | 180 |
1,861 | A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls. | 3 : 4 : 13 |
61,370 | For each positive integer $n$, define the function $f(n)=\left\{\begin{array}{ll}0 & \text { when } n \text { is a perfect square, } \\ {\left[\frac{1}{\{\sqrt{n}\}}\right]} & \text { when } n \text { is not a perfect square. }\end{array}\right.$ (where $[x]$ denotes the greatest integer not exceeding $x$, and $\{x\}=x-[x])$, find: $\sum_{k=1}^{240} f(k)$.
---
The function $f(n)$ is defined as follows:
- $f(n) = 0$ if $n$ is a perfect square.
- $f(n) = \left[\frac{1}{\{\sqrt{n}\}}\right]$ if $n$ is not a perfect square.
Here, $[x]$ represents the greatest integer less than or equal to $x$, and $\{x\}$ represents the fractional part of $x$, which is $x - [x]$.
We need to find the value of $\sum_{k=1}^{240} f(k)$. | 768 |
4,466 | Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \left|z_{1}\right| \leq 1 \), \( \left|z_{2}\right| \leq 1 \), and \( \left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leq \left|z_{1}-z_{2}\right| \). What is the maximum value of \( \left|z_{3}\right| \)? | \sqrt{2} |
14,279 | Is it possible to divide a square into 14 triangles of equal area, with a common vertex $O$ and the other vertices on the boundary of the square? | Yes |
12,505 | Task B-4.5. (20 points) A line through the origin intersects the lines given by the equations $x+y-1=0, x-y-1=0$ at points $A$ and $B$. Determine the geometric locus of the midpoints of the segments $\overline{A B}$. | x^2 - y^2 - x = 0 |
25,378 | The number $\overline{x y z t}$ is a perfect square, and so is the number $\overline{t z y x}$, and the quotient of the numbers $\overline{x y z t}$ and $\overline{t z y x}$ is also a perfect square. Determine the number $\overline{x y z t}$. (The overline indicates that the number is written in the decimal system.)
Translating the text as requested, while preserving the original line breaks and formatting. | 9801 |
25,808 | $4 \cdot 61$ Given that in the subway network, each line has at least 4 stations, of which no more than 3 are transfer stations, and at each transfer station, no more than two lines intersect. If from any station, one can reach any other station with at most two transfers, how many lines can this network have at most? | 10 |
19,272 | In the right trapezoid \( ABCD \), it is known that \(\angle A = \angle D = 90^\circ\), \(DE \perp AC\) at point \(E\), \(\angle ACD = \angle EBC = 30^\circ\), and \(AD = \sqrt{3}\). Find \(BC\). | 3 |
21,644 | Find the number of integer points that satisfy the system of inequalities:
\[
\begin{cases}
y \leqslant 3x \\
y \geqslant \frac{1}{3}x \\
x + y \leqslant 100
\end{cases}
\] | 2551 |
54,566 | 3-4. Two people \(A\) and \(B\) need to travel from point \(M\) to point \(N\), which is 15 km away from \(M\). On foot, they can travel at a speed of 6 km/h. In addition, they have a bicycle that can be ridden at a speed of 15 km/h. \(A\) sets out on foot, while \(B\) rides the bicycle until meeting pedestrian \(C\), who is walking from \(N\) to \(M\). After the meeting, \(B\) continues on foot, and \(C\) rides the bicycle until meeting \(A\) and hands over the bicycle to him, on which \(A\) then arrives at \(N\).
When should pedestrian \(C\) leave from \(N\) so that \(A\) and \(B\) arrive at point \(N\) simultaneously (if he walks at the same speed as \(A\) and \(B\))? | \frac{3}{11} |
22,397 | Suppose that a function \( M(n) \), where \( n \) is a positive integer, is defined by
\[
M(n)=\left\{
\begin{array}{ll}
n - 10 & \text{if } n > 100 \\
M(M(n + 11)) & \text{if } n \leq 100
\end{array}
\right.
\]
How many solutions does the equation \( M(n) = 91 \) have? | 101 |
62,790 | 3. Around a circle, 129 (not necessarily integer) numbers from 5 to 25 inclusive were written. From each number, the logarithm to the base of the next number in the clockwise direction was taken, after which all the obtained logarithms were added. What is the maximum value that the sum of these logarithms can take? | 161 |
24,839 | Given that \( \mathrm{a}, \mathrm{b}, \mathrm{c} \) are three natural numbers, and the least common multiple (LCM) of \( \mathrm{a} \) and \( \mathrm{b} \) is 60, and the LCM of \( \mathrm{a} \) and \( \mathrm{c} \) is 270, find the LCM of \( \mathrm{b} \) and \( \mathrm{c} \). | 540 |
54,753 | 5. Place 7 goldfish of different colors into 3 glass fish tanks numbered $1, 2, 3$. If the number of fish in each tank must be no less than its number, then the number of different ways to place the fish is $\qquad$ kinds. | 455 |
6,046 | A pile of stones has a total weight of 100 kg, with each stone weighing no more than 2 kg. By taking out some stones in various ways, find the difference between the total weight of these stones and 10 kg. Let the minimum absolute value of these differences be \( d \). Find the maximum value of \( d \). | \dfrac{10}{11} |
60,779 | 5. The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly arranged in a circle. For each such arrangement, we determine all eight sums of four consecutive numbers and denote the largest of these sums by $m$. What is the smallest possible value of the number $m$? Justify your answer. | 19 |
56,147 | [ Logic and set theory $]
A traveler visited a village where every person either always tells the truth or always lies. The villagers stood in a circle, and each one told the traveler whether the person to their right was truthful or a liar. Based on these statements, the traveler was able to uniquely determine what fraction of the village's population consists of truthful people. Determine what this fraction is.
# | \frac{1}{2} |
19,781 | Let \( M \) be a finite set of numbers. If it is known that among any three elements of \( M \), there always exist two whose sum belongs to \( M \), what is the maximum number of elements \( M \) can have? | 7 |
55,278 | Four, on a circular road, there are 4 middle schools arranged clockwise: $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$. They have 15, 8, 5, and 12 color TVs, respectively. To make the number of color TVs in each school the same, some schools are allowed to transfer color TVs to adjacent schools. How should the TVs be transferred to minimize the total number of TVs transferred? And what is the minimum total number of TVs transferred? | 10 |
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