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Among six different quadratic trinomials, differing by permutation of coefficients, what is the maximum number that can have two distinct roots? | 6 | 0.375 |
Given a set \( A \) of \( n \) points in the plane, no three collinear, show that we can find a set \( B \) of \( 2n - 5 \) points such that a point of \( B \) lies in the interior of every triangle whose vertices belong to \( A \). | 2n-5 | 0.625 |
Let \( a, b, c, d, e, f \) be integers selected from the set \( \{1,2, \ldots, 100\} \), uniformly and at random with replacement. Set
\[ M = a + 2b + 4c + 8d + 16e + 32f. \]
What is the expected value of the remainder when \( M \) is divided by 64? | 31.5 | 0.5 |
Suppose a sequence of positive real numbers \( x_{0}, x_{1}, \cdots, x_{1995} \) satisfies the following two conditions:
(1) \( x_{0} = x_{1995} \);
(2) \( x_{i-1} + \frac{2}{x_{i-1}} = 2 x_{i} + \frac{1}{x_{i}} \) for \( i = 1, 2, \cdots, 1995 \).
Find the maximum value of \( x_{0} \) among all sequences that satisfy the above conditions. | 2^{997} | 0.125 |
Two concentric circles have radii of 1 and 2 units, respectively. What is the minimum possible area of a cyclic quadrilateral inscribed in the larger circle that also contains the smaller circle? | 3 \sqrt{3} | 0.125 |
Compute the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}(n \sqrt{n}-\sqrt{n(n+1)(n+2)})
$$ | -\infty | 0.5 |
The pentagon $ABCDE$ is inscribed in a circle. Points $M, Q, N,$ and $P$ are the bases of the perpendiculars dropped from vertex $E$ to the sides $AB, BC, CD$ (or their extensions) and diagonal $AD$ respectively. It is known that $|EP|=d$, and the ratio of the area of triangle $MQE$ to the area of triangle $PNE$ is $k$. Find $|EM|$. | d \sqrt{k} | 0.625 |
Let \( q \) be a positive rational number. Two ants start from the same point \( X \) on a plane and each ant moves in one of the four directions (east, south, west, or north) a distance of \( q^n \) meters in the \( n \)-th minute \((n=1,2, \cdots)\). Suppose that after some integer minutes, they meet again, but their paths are not completely identical. Find all possible values of \( q \). | 1 | 0.625 |
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2033 \) and \( y = |x-a| + |x-b| + |x-c| \) has exactly one solution. Find the minimum possible value of \( c \). | 1017 | 0.125 |
Show that if the numbers \( a_{1}, a_{2}, \ldots \) are not all zeros and satisfy the relation \( a_{n+2} = \left|a_{n+1}\right| - a_{n} \), then from some point on they are periodic and the smallest period is 9. | 9 | 0.75 |
The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is 12. Point \( K \) lies on the extension of edge \( BC \) at a distance of 9 from vertex \( C \). Point \( L \) on edge \( AB \) is at a distance of 5 from \( A \). Point \( M \) divides segment \( A_1 C_1 \) in a ratio of 1:3, starting from \( A_1 \). Find the area of the cross-section of the cube by the plane passing through points \( K \), \( L \), and \( M \). | 156 | 0.375 |
On the board, there are two-digit numbers. Each number is composite, but any two numbers are coprime. What is the maximum number of such numbers that can be written? | 4 | 0.5 |
A circle is tangent to two adjacent sides \(AB\) and \(AD\) of square \(ABCD\) and cuts off a segment of length 4 cm from vertices \(B\) and \(D\) at the points of tangency. On the other two sides, the circle intersects cutting off segments from the vertices of lengths 2 cm and 1 cm, respectively. Find the radius of the circle. | 5 | 0.75 |
Find the smallest natural number divisible by 99, all of whose digits are even. | 228888 | 0.125 |
Maria ordered a certain number of televisions at $R$ \$ 1994.00 each. She noticed that in the total amount to be paid, there are no digits 0, 7, 8, or 9. What was the smallest number of televisions she ordered? | 56 | 0.25 |
From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle? | 3 : 4 : 5 | 0.375 |
Given a cyclic quadrilateral \(ABCD\). The rays \(AB\) and \(DC\) intersect at point \(K\). It turns out that points \(B\), \(D\), and the midpoints of segments \(AC\) and \(KC\) lie on the same circle. What values can the angle \(ADC\) take? | 90^\circ | 0.75 |
Suppose \( x \neq y \), and the sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{4} \) are both arithmetic sequences. What is \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \) equal to? | \frac{8}{3} | 0.375 |
For each pair of distinct natural numbers \(a\) and \(b\), not exceeding 20, Petya drew the line \( y = ax + b \) on the board. That is, he drew the lines \( y = x + 2, y = x + 3, \ldots, y = x + 20, y = 2x + 1, y = 2x + 3, \ldots, y = 2x + 20, \ldots, y = 3x + 1, y = 3x + 2, y = 3x + 4, \ldots, y = 3x + 20, \ldots, y = 20x + 1, \ldots, y = 20x + 19 \). Vasia drew a circle of radius 1 with center at the origin on the same board. How many of Petya’s lines intersect Vasia’s circle? | 190 | 0.5 |
Given 2022 lines in the plane, such that no two are parallel and no three are concurrent. We denote \( E \) as the set of their intersection points. We want to assign a color to each point in \( E \) such that any two points on the same line, whose connecting segment does not contain any other point from \( E \), have different colors. What is the minimum number of colors required to achieve this coloring? | 3 | 0.5 |
Find the largest solution to the inequality
\[
\frac{-\log _{3}(100+2 x \sqrt{2 x+25})^{3}+\left|\log _{3} \frac{100+2 x \sqrt{2 x+25}}{\left(x^{2}+2 x+4\right)^{4}}\right|}{3 \log _{6}(50+2 x \sqrt{2 x+25})-2 \log _{3}(100+2 x \sqrt{2 x+25})} \geqslant 0
\] | 12 + 4\sqrt{3} | 0.375 |
How many triangles can be formed by the vertices and the intersection point of the diagonals of a given rectangle (which is not a square), with all these triangles having a common vertex at a given fixed vertex of the rectangle? How many of these triangles are right-angled? How does the problem change if we use any interior point of the rectangle instead of the intersection of the diagonals? | 5 | 0.25 |
Given a trapezoid \(ABCD\) where \(AD \parallel BC\), \(BC = AC = 5\), and \(AD = 6\). The angle \(ACB\) is twice the measure of angle \(ADB\). Find the area of the trapezoid. | 22 | 0.375 |
$n$ trains travel in the same direction along a circular track at equal intervals. Stations $A$, $B$, and $C$ are located at the vertices of an equilateral triangle along this track (in the direction of travel). Ira boards at station $A$ and Alex boards at station $B$ to catch the nearest trains. It is known that if they arrive at the stations just as the engineer Roma is passing through the forest, then Ira will board a train before Alex; otherwise, Alex boards before Ira or at the same time. What portion of the track passes through the forest? | \frac{1}{3} | 0.75 |
Given a triangle \( \triangle ABC \) with internal angles \( A \), \( B \), and \( C \) such that \( \cos A = \sin B = 2 \tan \frac{C}{2} \), determine the value of \( \sin A + \cos A + 2 \tan A \). | 2 | 0.125 |
Given a rectangular prism with a base $A B C D$ and a top face parallel midpoint line $E F$ where the midpoint of this segment is $G$. The reflection of point $X$ on segment $E G$ across point $G$ is $Y$. For which position of point $X$ will the sum $A X + D X + X Y + Y B + Y C$ be minimized? | x = G | 0.25 |
A pair of natural numbers is called "good" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed? | 13 | 0.125 |
Four points in the order \( A, B, C, D \) lie on a circle with the extension of \( AB \) meeting the extension of \( DC \) at \( E \) and the extension of \( AD \) meeting the extension of \( BC \) at \( F \). Let \( EP \) and \( FQ \) be tangents to this circle with points of tangency \( P \) and \( Q \) respectively. Suppose \( EP = 60 \) and \( FQ = 63 \). Determine the length of \( EF \). | 87 | 0.5 |
A triangle \( EGF \) is inscribed in a circle with center \( O \), and the angle \( \angle EFG \) is obtuse. There exists a point \( L \) outside the circle such that \( \angle LEF = \angle FEG \) and \( \angle LGF = \angle FGE \). Find the radius of the circumcircle of triangle \( ELG \), given that the area of triangle \( EGO \) is \( 81 \sqrt{3} \) and \( \angle OEG = 60^\circ \). | 6\sqrt{3} | 0.125 |
Find the area of the triangle that is cut off by the line \( y = 2x + 2 \) from the figure defined by the inequality \( |x-2| + |y-3| \leq 3 \). | 3 | 0.875 |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin^{6} x \cos^{2} x \, dx
$$ | \frac{5\pi}{64} | 0.625 |
Given nine different numbers, how many different values of third-order determinants can be formed using all of them? | 10080 | 0.125 |
In triangle \( ABC \), side \( AB \) is longer than side \( BC \), and angle \( B \) is \( 40^\circ \). Point \( P \) is taken on side \( AB \) such that \( BP = BC \). The angle bisector \( BM \) intersects the circumcircle of triangle \( ABC \) at point \( T \). Find the angle \( MPT \). | 20^\circ | 0.25 |
As shown in the figure, on a rectangular table with dimensions $9 \mathrm{~cm}$ in length and $7 \mathrm{~cm}$ in width, a small ball is shot from point $A$ at a 45-degree angle. Upon reaching point $E$, it bounces off at a 45-degree angle and continues to roll forward. Throughout its motion, the ball bounces off the table edges at a 45-degree angle each time. Starting from point $A$, after how many bounces does the ball first reach point $C$? | 14 | 0.875 |
Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}
$$ | \frac{1}{7} | 0.625 |
The axial section $SAB$ of a conical frustum is an equilateral triangle with side length 2. $O$ is the center of the base, and $M$ is the midpoint of $SO$. The moving point $P$ is on the base of the conical frustum (including the circumference). If $AM \perp MP$, find the length of the locus formed by point $P$. | \frac{\sqrt{7}}{2} | 0.5 |
In a $28 \times 35$ table, some $k$ cells are colored red, some $r$ cells are colored pink, and the remaining $s$ cells are colored blue. It is known that:
- $k \geqslant r \geqslant s$
- Each boundary cell has at least 2 neighbors of the same color
- Each non-boundary cell has at least 3 neighbors of the same color
What is the smallest possible value of $k - s$?
(A cell is called a boundary cell if it is adjacent to the border of the table. Neighbors are cells that share a common side.) | 28 | 0.125 |
Let \( X \) be the set of residues modulo 17. We regard two members of \( X \) as adjacent if they differ by 1, so 0 and 16 are adjacent. We say that a permutation of \( X \) is dispersive if it never maps two adjacent values to two adjacent values, and connective if it always maps two adjacent values to two adjacent values. What is the largest \( N \) for which we can find a permutation \( p \) on \( X \) such that \( p \), \( p^2 \), ..., \( p^{N-1} \) are all dispersive and \( p^N \) is connective? | 8 | 0.5 |
Let $\{x\}$ denote the fractional part of the real number $x$. Given $a=(5 \sqrt{2}+7)^{2017}$, find the value of $a\{a\}$. | 1 | 0.75 |
A $100 \times 100$ square is divided into $2 \times 2$ squares. Then it is divided into dominos (rectangles $1 \times 2$ and $2 \times 1$). What is the smallest number of dominos that could have been inside the divided squares? | 100 | 0.125 |
At a height \( BH \) of triangle \( ABC \), a point \( D \) is marked. Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \) starting from point \( F \). Find the ratio \( FH:HE \). | 1:3 | 0.5 |
In the Chinese idioms "虚有其表", "表里如一", "一见如故", and "故弄玄虚", each Chinese character represents one of 11 consecutive non-zero natural numbers. Identical characters represent the same number, and different characters represent different numbers. The order of the numbers is such that "表" > "一" > "故" > "如" > "虚". Additionally, the sum of the numbers represented by the four characters in each idiom is 21. What is the maximum number that "弄" can represent? | 9 | 0.375 |
The diagonals of a convex quadrilateral $ABCD$, inscribed in a circle, intersect at point $E$. It is known that diagonal $BD$ is the angle bisector of $\angle ABC$ and that $BD = 25$ and $CD = 15$. Find $BE$. | 16 | 0.125 |
Using three colors $\mathbf{R}$, $\mathbf{G}$, and $\mathbf{B}$ to color a $2 \times 5$ table in a way that two squares sharing a common edge must be colored differently. How many different coloring methods are there? | 486 | 0.125 |
In the rectangular coordinate plane, find the number of integer points that satisfy the system of inequalities
\[
\left\{
\begin{array}{l}
y \leqslant 3x \\
y \geqslant \frac{x}{3} \\
x + y \leqslant 100
\end{array}
\right.
\] | 2551 | 0.875 |
Calculate the surface integrals of the first kind:
a) \(\iint_{\sigma}|x| dS\), where \(\sigma\) is defined by \(x^2 + y^2 + z^2 = 1\), \(z \geqslant 0\).
b) \(\iint_{\sigma} (x^2 + y^2) dS\), where \(\sigma\) is defined by \(x^2 + y^2 = 2z\), \(z = 1\).
c) \(\iint_{\sigma} (x^2 + y^2 + z^2) dS\), where \(\sigma\) is the part of the cone defined by \(z^2 - x^2 - y^2 = 0\), \(z \geqslant 0\), truncated by the cylinder \(x^2 + y^2 - 2x = 0\). | 3\sqrt{2} \pi | 0.125 |
Calculate the area of the common part of two rhombuses, where the lengths of the diagonals of the first one are 4 cm and 6 cm, and the second one is obtained by rotating the first one by 90 degrees around its center. | 9.6 \ \text{cm}^2 | 0.125 |
Through point \( A \) of a circle with a radius of 10, two mutually perpendicular chords \( AB \) and \( AC \) are drawn. Calculate the radius of a circle that is tangent to the given circle and the constructed chords, given \( AB = 16 \). | 8 | 0.375 |
Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number. | 520 | 0.125 |
In how many ways can the numbers \(1, 2, \ldots, 2002\) be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.) | 4004 | 0.125 |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1019 | 0.375 |
The expression \( x_{1} : x_{2} : x_{3} : \ldots : x_{n} \) has a definite value only when parentheses are used to indicate the order of divisions. The expression can be rewritten in the form
\[ \frac{x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}}{x_{j_{1}} x_{j_{2}} \ldots x_{j_{n-k}}} \]
where \( i_{1}, i_{2}, \ldots, i_{k} \) and \( j_{1}, j_{2}, \ldots, j_{n-k} \) are any permutation of the indices \( 1, 2, \ldots, n \) with \( i_{1} < i_{2} < \ldots < i_{k} \) and \( j_{1} < j_{2} < \ldots < j_{n-k} \).
For example,
\[ \left(x_{1} : \left(x_{2} : x_{3}\right)\right) : x_{4} = \frac{x_{1} x_{3}}{x_{2} x_{4}}, \quad \left(\left(x_{1} : x_{2}\right) : x_{3}\right) : x_{4} = \frac{x_{1}}{x_{2} x_{3} x_{4}} \]
How many distinct fractions of this form can be obtained by placing the parentheses in different ways? | 2^{n-2} | 0.625 |
In a tournament with 5 teams, there are no ties. In how many ways can the $\frac{5 \cdot 4}{2}=10$ games of the tournament occur such that no team wins all their games and no team loses all their games? | 544 | 0.25 |
Is it possible to divide a convex 2017-gon into black and white triangles such that any two triangles have either a common side when colored in different colors, a common vertex, or no common points, and each side of the 2017-gon is a side of one of the black triangles? | \text{No} | 0.625 |
S is a collection of subsets of {1, 2, ... , n} of size 3. Any two distinct elements of S have at most one common element. Show that S cannot have more than n(n-1)/6 elements. Find a set S with n(n-4)/6 elements. | \frac{n(n-1)}{6} | 0.125 |
8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the second smallest number? | 26 | 0.625 |
Numbers \(1, 2, \ldots, 2010\) are placed on the circumference of a circle in some order. The numbers \(i\) and \(j\), where \(i \neq j\) and \(i, j \in \{1, 2, \ldots, 2010\}\), form a friendly pair if:
(i) \(i\) and \(j\) are not neighbors to each other, and
(ii) on one or both of the arcs connecting \(i\) and \(j\) along the circle, all numbers in between them are greater than both \(i\) and \(j\).
Determine the minimal number of friendly pairs. | 2007 | 0.25 |
A permutation of \(\{1, 2, \ldots, 7\}\) is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation \((3, 4, 2, 1, 6, 5, 7)\) can be partitioned correctly into the blocks \([3, 4, 2, 1]\) and \([6, 5, 7]\), since when these blocks are sorted, the permutation becomes \((1, 2, 3, 4, 5, 6, 7)\). Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly. | \frac{151}{105} | 0.125 |
On the hyperbola \( x y = 1 \), the point with abscissa \( \frac{n}{n+1} \) is \( A_{n} \), and the point with abscissa \( \frac{n+1}{n} \) is \( B_{n}(n \in \mathbf{N}) \). Let the point with coordinates \( (1,1) \) be \( M \). Also, let \( P_{n}\left(x_{n}, y_{n}\right) \) be the circumcenter of \( \triangle A_{n} B_{n} M \). As \( n \rightarrow \infty \), find the coordinates of the limit point \( P_{n} \), denoted as \( (a, b) \), where \( a = \lim _{n \rightarrow \infty} x_{n} \) and \( b = \lim _{n \rightarrow \infty} y_{n} \). | (2, 2) | 0.75 |
Initially, there is a natural number \( N \) written on the board. At any moment, Misha can choose a number \( a > 1 \) on the board, erase it, and write down all of its natural divisors except for \( a \) itself (the same numbers can appear multiple times on the board). After some time, it turned out that there were \( N^2 \) numbers written on the board. For which \( N \) could this happen? | N = 1 | 0.625 |
Cat Matroskin, Uncle Fyodor, postman Pechkin, and Sharik sat at a round table. Each had a plate with 15 sandwiches in front of them. Every minute, three of them ate a sandwich from their own plate, and the fourth ate a sandwich from their neighbor's plate. Five minutes after the meal began, Uncle Fyodor had 8 sandwiches left on his plate. What is the minimum number of sandwiches that could remain on Cat Matroskin's plate? | 7 | 0.375 |
In a classroom, 24 light fixtures were installed, each of which can hold 4 bulbs. After screwing in 4 bulbs into some of the fixtures, it became apparent that the available stock would be insufficient. Subsequently, they screwed bulbs in groups of three, then in pairs, and finally individually into the fixtures. Unfortunately, there were still some fixtures left without any bulbs. How many bulbs were missing if twice as many fixtures received a single bulb as those that received four bulbs, and half as many fixtures received no bulbs at all as those that received three bulbs? | 48 | 0.75 |
For 155 boxes containing red, yellow, and blue balls, there are three classification methods: for each color, classify the boxes with the same number of balls of that color into one category. If every natural number from 1 to 30 is the number of boxes in some category in at least one of the classifications, then:
1. What is the sum of the numbers of categories in the three classifications?
2. Show that it is possible to find three boxes such that at least two of the colors have the same number of balls. | 30 | 0.875 |
Find the sum of all roots of the equation:
$$
\begin{gathered}
\sqrt{2 x^{2}-2024 x+1023131} + \sqrt{3 x^{2}-2025 x+1023132} + \sqrt{4 x^{2}-2026 x+1023133} = \\
= \sqrt{x^{2}-x+1} + \sqrt{2 x^{2}-2 x+2} + \sqrt{3 x^{2}-3 x+3}
\end{gathered}
$$ | 2023 | 0.5 |
Some out of 20 metal cubes, identical in size and appearance, are made of aluminum, and the rest are made of duralumin (which is heavier). How can you determine the number of duralumin cubes using no more than 11 weighings on a balance scale without weights?
Note. It is assumed that all the cubes could be made of aluminum, but they cannot all be made of duralumin (because if all the cubes turned out to be of the same weight, we wouldn't be able to determine whether they are aluminum or duralumin without this condition). | 11 | 0.625 |
Let \( s \) be the set of all rational numbers \( r \) that satisfy the following conditions:
\[
(1) \quad 0<r<1 ;
\]
(2) \( r=0.abcabcabc\cdots=0.ab\dot{c} \), where \( a, b, c \) are not necessarily distinct.
When the numbers \( r \) in \( s \) are written as irreducible fractions, how many different numerators are there? | 660 | 0.125 |
From the vertex of the obtuse angle $A$ of triangle $ABC$, a perpendicular $AD$ is drawn. A circle with center $D$ and radius $DA$ is drawn, which intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Find $AC$ if $AB = c$, $AM = m$, and $AN = n$. | \frac{mc}{n} | 0.625 |
Let $AB$, multiplied by $C$, equal $DE$. If $DE$ is subtracted from $FG$, the result is $HI$.
Each letter represents a distinct digit (1, 2, 3, 4, 5, 6, 7, 8, or 9). The digit 0 does not appear in the problem. | 93 - 68 = 25 | 0.125 |
Pentagon \( A B C D E \) is inscribed in a circle with radius \( R \). It is known that \( \angle B = 110^\circ \) and \( \angle E = 100^\circ \). Find the side \( C D \). | R | 0.5 |
a) What is the maximum number of bishops that can be placed on a standard chessboard (comprising 64 squares) such that no two bishops threaten each other? Solve the same problem for a chessboard consisting of \( n^2 \) squares.
b) What is the minimum number of bishops that can be placed on a standard chessboard (comprising 64 squares) such that they threaten all the squares on the board? Solve the same problem for a chessboard consisting of \( n^2 \) squares. | 8 | 0.125 |
In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning. | 7 | 0.25 |
Let \( \triangle ABC \) be a triangle and \( P \) a point inside the triangle such that the centers \( M_B \) and \( M_A \) of the circumcircles \( k_B \) and \( k_A \) of \( ACP \) and \( BCP \) respectively lie outside the triangle \( ABC \). Furthermore, the three points \( A, P, \) and \( M_A \) are collinear and likewise the three points \( B, P, \) and \( M_B \) are collinear. The line through \( P \) parallel to the side \( AB \) intersects the circles \( k_A \) and \( k_B \) at points \( D \) and \( E \) respectively, with \( D \neq P \) and \( E \neq P \).
Show that \( DE = AC + BC \).
(Walther Janous) | DE = AC + BC | 0.875 |
Find all values of the parameter \( c \) such that the system of equations has a unique solution:
$$
\left\{\begin{array}{l}
2|x+7|+|y-4|=c \\
|x+4|+2|y-7|=c
\end{array}\right.
$$ | c = 3 | 0.375 |
As shown in the figure, one cross section of the cube $A B C D-E F G H$ passes through the vertices $A$ and $C$ and a point $K$ on the edge $E F$. This cross section divides the cube into two parts with a volume ratio of $3:1$. Find the value of $\frac{E K}{K F}$. | \sqrt{3} | 0.25 |
Vojta wanted to add several three-digit natural numbers using a calculator. On the first attempt, he got a result of 2224. To check, he summed the numbers again and got 2198. He calculated it once more and this time got a sum of 2204. It turned out that the last three-digit number was troublesome—Vojta failed to press one of its digits hard enough each time, resulting in entering a two-digit number instead of a three-digit number into the calculator. No other errors occurred during the addition. What is the correct sum of Vojta's numbers? | 2324 | 0.25 |
Let \(a, b, c\) be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of \(\frac{ab + c}{a + b + c}\). | \frac{2}{3} | 0.875 |
In trapezoid \(ABCD\), \(\angle A = \angle B = 90^\circ\), \(AD = 2\sqrt{7}\), \(AB = \sqrt{21}\), and \(BC = 2\). What is the minimum possible value of the sum of the lengths \(XA + XB + XC + XD\), where \(X\) is an arbitrary point in the plane? | 12 | 0.75 |
From vertex $C$ of rhombus $A B C D$, with side length $a$, two segments $C E$ and $C F$ are drawn, dividing the rhombus into three equal areas. Given that $\cos C = \frac{1}{4}$, find the sum of $C E + C F$. | \frac{8a}{3} | 0.125 |
In triangle $ABC$, the median $AM$ is perpendicular to the angle bisector $BD$. Find the perimeter of the triangle given that $AB = 1$ and the lengths of all sides are integers. | 5 | 0.625 |
Determine all digits \( z \) such that for every integer \( k \geq 1 \), there exists an integer \( n \geq 1 \) with the property that the decimal representation of \( n^9 \) ends with at least \( k \) digits \( z \). | 9 | 0.125 |
A dart is thrown at a square dartboard of side length 2 so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance of 1 of a corner? | \frac{\pi - 2}{4} | 0.125 |
The circle touches the sides $AB$ and $BC$ of triangle $ABC$ at points $D$ and $E$ respectively. Find the height of triangle $ABC$ dropped from point $A$, given that $AB = 5$, $AC = 2$, and points $A$, $D$, $E$, and $C$ lie on the same circle. | \frac{4\sqrt{6}}{5} | 0.375 |
There are two-digit numbers written on a board. Each number is composite, but any two numbers are relatively prime. What is the maximum number of such numbers that can be written? | 4 | 0.5 |
We call a four-digit number with the following property a "centered four-digit number": Arrange the four digits of this four-digit number in any order, and when all the resulting four-digit numbers (at least 2) are sorted from smallest to largest, the original four-digit number is exactly in the middle position. For example, 2021 is a "centered four-digit number". How many "centered four-digit numbers" are there, including 2021? | 90 | 0.125 |
Given that \( 2^{2013} < 5^{867} < 2^{2014} \), how many pairs of integers \((m, n)\) satisfy:
\[ 5^n < 2^m < 2^{m+2} < 5^{n+1} \]
where \( 1 \leq m \leq 2012 \). | 279 | 0.5 |
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could have answered affirmatively? | 17 | 0.75 |
S is a finite set of numbers such that given any three there are two whose sum is in S. What is the largest number of elements that S can have? | 7 | 0.125 |
The diagonal \( AC \) of the inscribed quadrilateral \( ABCD \) is the diameter of the circumscribed circle \( \omega \). From point \( D \), a line is drawn perpendicular to the segment \( BC \), and it intersects the circle \( \omega \) again at point \( E \). Find the ratio of the areas of triangle \( BCD \) and quadrilateral \( ABEC \). | 1 | 0.125 |
Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{11 \ldots 11}_{2017} \underbrace{22 \ldots .22}_{2018} 5}$. | 6056 | 0.625 |
Inside triangle \(ABC\), a point \(P\) is chosen such that \(AP = BP\) and \(CP = AC\). Find \(\angle CBP\) given that \(\angle BAC = 2 \angle ABC\). | 30^\circ | 0.875 |
Let \( f(x) = \frac{1}{x^3 + 3x^2 + 2x} \). Determine the smallest positive integer \( n \) such that
\[ f(1) + f(2) + f(3) + \cdots + f(n) > \frac{503}{2014}. \] | 44 | 0.5 |
Subsets and Splits