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Suppose $O$ is circumcenter of triangle $ABC$ . Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$ . Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\] | \frac{^2}{\sqrt{9R^2-(^2+b^2+^2)}} | 1/8 |
Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:
The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation.
Lisa repeats this until she gets a number for the first time was on the board.
(a) Show that the sequence of moves always ends.
(b) If Lisa begins with the number $53^{2022} - 1$ , what is the last number on the board?
Example: If Lisa starts with the number $2022$ , she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$ 2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21 $$ .
Since Lisa gets $21$ for the second time, the turn order ends.
*(Stephan Pfannerer)* | 39 | 2/8 |
Given that the polar coordinate equation of curve $C$ is $ρ=2\cos θ$, and the polar coordinate equation of line $l$ is $ρ\sin (θ+ \frac {π}{6})=m$. If line $l$ and curve $C$ have exactly one common point, find the value of the real number $m$. | \frac{3}{2} | 1/8 |
Given non-negative real numbers \( a \), \( b \), and \( c \) satisfying \( a + b + c = 1 \), find the maximum value of \( a^{2}(b-c) + b^{2}(c-a) + c^{2}(a-b) \). | \frac{\sqrt{3}}{18} | 1/8 |
The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \ldots\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). What is the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when the 1994th term of the sequence is divided by 1000? | 063 | 7/8 |
In the sequence \(\{a_n\}\),
\[
a_1 = 2, \quad a_{n+1} = \frac{a_n^2}{a_n + 2}.
\]
Prove: \(\sum_{k=1}^{n} \frac{2k a_k}{a_k + 2} < 4\). | 4 | 1/8 |
Given real numbers \(x_{1}, \ldots, x_{n}\), find the maximum value of the expression:
$$
A=\left(\sin x_{1}+\ldots+\sin x_{n}\right) \cdot\left(\cos x_{1}+\ldots+\cos x_{n}\right)
$$ | \frac{n^2}{2} | 5/8 |
What is the maximum number of circles with radius 1 that can be placed in a plane such that all of them intersect a certain fixed unit circle $S$ and none of them contain the center of $S$ or the center of another circle within themselves? | 18 | 1/8 |
Each cell of a \(50 \times 50\) square contains a number equal to the count of \(1 \times 16\) rectangles (both vertical and horizontal) for which this cell is an endpoint. How many cells contain numbers that are greater than or equal to 3? | 1600 | 5/8 |
Carlinhos likes writing numbers in his notebook. One day he wrote the numbers from 1 to 999, one after the other, to form the giant number:
$$
123456789101112 \ldots 997998999
$$
Based on this number, the following questions are asked:
(a) How many digits were written?
(b) How many times does the digit 1 appear?
(c) Considering that 1 occupies position 1, 2 occupies position 2, and 0 appears for the first time occupying position 11, which digit occupies position 2016? | 8 | 5/8 |
ABCD is a convex quadrilateral with AC perpendicular to BD. M, N, R, S are the midpoints of AB, BC, CD, and DA respectively. The feet of the perpendiculars from M, N, R, and S to CD, DA, AB, and BC are W, X, Y, and Z respectively. Show that M, N, R, S, W, X, Y, and Z lie on the same circle. | M,N,R,S,W,X,Y,Z | 1/8 |
Let $A=\{a_1, a_2, \cdots, a_n\}$ be a set of numbers and let $P(A)$ denote the arithmetic mean of all elements in $A$ such that $P(A)=\frac{a_1 + a_2 + \cdots + a_n}{n}$. If $B$ is a non-empty subset of $A$ and $P(B)=P(A)$, then $B$ is called a "balanced subset" of $A$. Find the number of "balanced subsets" of the set $M=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. | 51 | 4/8 |
A positive integer $n$ is magical if $\lfloor\sqrt{\lceil\sqrt{n}\rceil}\rfloor=\lceil\sqrt{\lfloor\sqrt{n}\rfloor}\rceil$ where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ represent the floor and ceiling function respectively. Find the number of magical integers between 1 and 10,000, inclusive. | 1330 | 1/8 |
On a drying rack, there are $n$ socks hanging in a random order (as they were taken out of the washing machine). Among them are two favorite socks of the Absent-minded Scientist. The socks are blocked by a drying sheet, so the Scientist cannot see them and picks one sock at a time by feel. Find the expected number of socks the Scientist will have removed by the time he has both of his favorite socks. | \frac{2(n+1)}{3} | 5/8 |
Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four 2-by-2 subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are considered the same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs. | 1296 | 6/8 |
Given an equilateral triangle \( \triangle ABC \) with side length 1, \( B \) and \( C \) have \( n \) equally spaced points along the segment \( BC \), named \( P_1, P_2, \ldots, P_{n-1} \) in the direction from \( B \) to \( C \). Determine \( S_n = \overrightarrow{AB} \cdot \overrightarrow{AP_1} + \overrightarrow{AP_1} \cdot \overrightarrow{AP_2} + \ldots + \overrightarrow{AP_{n-1}} \cdot \overrightarrow{AC} \) in terms of \( n \). | \frac{5n^2-2}{6n} | 2/8 |
Determine the maximum value of the sum
\[S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}\]
over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying
\[\sum_{k=1}^\infty a_k = 1.\] | 2/3 | 1/8 |
The perimeter of an isosceles right triangle is $2p$. Its area is: | $(3-2\sqrt{2})p^2$ | 6/8 |
Given a right isosceles triangle $ABC$ with hypotenuse $AB$. Point $M$ is the midpoint of side $BC$. Point $K$ is chosen on the smaller arc $AC$ of the circumcircle of triangle $ABC$. Point $H$ is the foot of the perpendicular dropped from $K$ to line $AB$. Find the angle $\angle CAK$, given that $KH = BM$ and lines $MH$ and $CK$ are parallel. | 22.5 | 5/8 |
Given the function $f(x) = \cos x \cdot \sin\left(\frac{\pi}{6} - x\right)$,
(1) Find the interval where $f(x)$ is monotonically decreasing;
(2) In $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. If $f(C) = -\frac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the length of side $c$. | 2\sqrt{3} | 7/8 |
Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$ . Let $\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\frac{AK}{KD}$ .
*Proposed by Giorgi Arabidze, Georgia* | 1/2 | 5/8 |
Let $\triangle ABC$ be a right triangle with $\angle ABC = 90^\circ$, and let $AB = 10\sqrt{21}$ be the hypotenuse. Point $E$ lies on $AB$ such that $AE = 10\sqrt{7}$ and $EB = 20\sqrt{7}$. Let $F$ be the foot of the altitude from $C$ to $AB$. Find the distance $EF$. Express $EF$ in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. | 31 | 1/8 |
A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation? | \frac{3}{20} | 7/8 |
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$ , where $ p$ is a prime and $ x$ and $ y$ are natural numbers. | (5,1,1) | 3/8 |
Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal. | 1010 | 7/8 |
An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island. | 15 | 1/8 |
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$. | 504 | 2/8 |
Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$ . The median $AM$ intersects the incircle at the points $P$ and $Q$ , with $P$ between $A$ and $Q$ , such that $AP = QM$ . Find the length of $PQ$ . | \sqrt{2\sqrt{5}-4} | 1/8 |
Let \(a_i\) and \(b_i\) be real numbers such that \(a_1 b_2 \neq a_2 b_1\). What is the maximum number of possible 4-tuples \((\text{sign}(x_1), \text{sign}(x_2), \text{sign}(x_3), \text{sign}(x_4))\) for which all \(x_i\) are non-zero and \(x_i\) is a simultaneous solution of
\[
a_1 x_1 + a_2 x_2 + a_3 x_3 + a_4 x_4 = 0
\]
and
\[
b_1 x_1 + b_2 x_2 + b_3 x_3 + b_4 x_4 = 0.
\]
Find necessary and sufficient conditions on \(a_i\) and \(b_i\) for this maximum to be achieved. | 8 | 1/8 |
In $\triangle ABC$, if $\angle A=60^{\circ}$, $\angle C=45^{\circ}$, and $b=4$, then the smallest side of this triangle is $\_\_\_\_\_\_\_.$ | 4\sqrt{3}-4 | 6/8 |
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$ , $x_2y_1-x_1y_2=5$ , and $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of $y_1^2+5y_2^2$ | 23 | 7/8 |
Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$ , such that $$ af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0 $$ for all real $x, y, z$ .
(E. Barabanov) | 0 | 3/8 |
How many solutions does the equation
$$
15x + 6y + 10z = 1973
$$
have in integers that satisfy the following inequalities:
$$
x \geq 13, \quad y \geq -4, \quad z > -6
$$ | 1953 | 7/8 |
The inclination angle of the line $\sqrt {3}x-y+1=0$ is \_\_\_\_\_\_. | \frac {\pi}{3} | 2/8 |
Find the smallest real number \( m \) such that for any positive integers \( a, b, \) and \( c \) satisfying \( a + b + c = 1 \), the inequality \( m\left(a^{3}+b^{3}+c^{3}\right) \geqslant 6\left(a^{2}+b^{2}+c^{2}\right)+1 \) holds. | 27 | 1/8 |
Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Assume that each parabola is tangent to a line that forms a $45^\circ$ angle with the x-axis. Find the radius of the circle. | \frac{1}{4} | 1/8 |
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? | 12\sqrt{3} | 3/8 |
For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals
$\textbf{(A) }2x^2\qquad \textbf{(B) }2y^2\qquad \textbf{(C) }x^2+y^2\qquad \textbf{(D) }x^2-y^2\qquad \textbf{(E) }y^2-x^2$ | \textbf{(D)}x^2-y^2 | 1/8 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $\sin A + \sin B = [\cos A - \cos (π - B)] \sin C$.
1. Determine whether triangle $ABC$ is a right triangle and explain your reasoning.
2. If $a + b + c = 1 + \sqrt{2}$, find the maximum area of triangle $ABC$. | \frac{1}{4} | 5/8 |
Given that $n$ is an integer between $1$ and $60$, inclusive, determine for how many values of $n$ the expression $\frac{((n+1)^2 - 1)!}{(n!)^{n+1}}$ is an integer. | 59 | 1/8 |
There are 1996 points on a circle, and they are painted in several different colors. From each set of points of the same color, one point is selected to form an inscribed polygon whose vertices are all of different colors. If the number of points of each color is different, into how many different colors should the 1996 points be painted to maximize the number of such polygons? And how many points should each color have? | 61 | 1/8 |
A trapezoid inscribed in a circle with a radius of $13 \mathrm{~cm}$ has its diagonals located $5 \mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid? | 288 | 1/8 |
Let
$$
R=3 \times 9+4 \times 10+5 \times 11+\cdots+2003 \times 2009
$$
$$
S=1 \times 11+2 \times 12+3 \times 13+\cdots+2001 \times 2011
$$
(a) Which number is larger: $\mathrm{R}$ or $S$?
(b) Calculate the difference between the larger and the smaller number. | 32016 | 4/8 |
In a unit cube \(ABCDA_1B_1C_1D_1\), eight planes \(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\) and \(D_1AB_1\) intersect the cube. What is the volume of the part that contains the center of the cube? | 1/6 | 1/8 |
For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $215216_n$ a multiple of $5$? | 20 | 1/8 |
Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows:
- $a_0=k$ [/*]
- For $n\geq 1$ , we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*]
Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence.
*Note.* If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$ . | \lfloor{\sqrt{2k}}\rfloor | 1/8 |
Prove that for any integers \( x \) and \( y \), the value of the following expression
$$
x^{5}+3 x^{4} y-5 x^{3} y^{2}-15 x^{2} y^{3}+4 x y^{4}+12 y^{5}
$$
is never equal to 33. | 33 | 1/8 |
How many positive integers less than $1000$ are $6$ times the sum of their digits?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$ | \textbf{(B)}\1 | 1/8 |
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\), calculate the value of \(DE + DF\). | 24 | 6/8 |
Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older brothers paid 800 rubles, and the younger brothers shared this money among themselves, so that everyone ended up with an equal share. What is the value of one house? | 2000 | 2/8 |
Teams A and B each have 7 players who compete in a go tournament following a predetermined sequence. The first player from each team competes first, with the loser being eliminated. The winner then competes against the next player from the losing team, and this process continues until all players on one team are eliminated. The other team wins, thus forming a potential sequence of the match. What is the total number of possible match sequences? | 3432 | 4/8 |
Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a *bullet*. Suppose some of the bullets are coloured red. For each pair $(i j)$ , with $ 1 \le i < j \le 8$ , let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$ . | 14 | 7/8 |
Suppose that \( ABCDEF \) is a regular hexagon with sides of length 6. Each interior angle of \( ABCDEF \) is equal to \( 120^{\circ} \).
(a) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \). Determine the area of the shaded sector.
(b) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \), and a second arc with center \( A \) and radius 6 is drawn from \( B \) to \( F \). These arcs are tangent (touch) at the center of the hexagon. Line segments \( BF \) and \( CE \) are also drawn. Determine the total area of the shaded regions.
(c) Along each edge of the hexagon, a semi-circle with diameter 6 is drawn. Determine the total area of the shaded regions; that is, determine the total area of the regions that lie inside exactly two of the semi-circles. | 18\pi - 27\sqrt{3} | 1/8 |
Let $a_0 = 3,$ $b_0 = 4,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\] for all $n \ge 0.$ Calculate $b_7 = \frac{4^p}{3^q}$ for some integers $p$ and $q.$ | (1094,1093) | 1/8 |
In an $8 \times 8$ chessboard, how many ways are there to select 56 squares so that all the black squares are selected, and each row and each column has exactly seven squares selected? | 576 | 4/8 |
Real numbers \( x_{1}, x_{2}, x_{3}, x_{4} \in [0,1] \). Find \( K_{\max} \), where
$$
\begin{aligned}
K= & \left|x_{1}-x_{2}\right|\left|x_{1}-x_{3}\right|\left|x_{1}-x_{4}\right|\left|x_{2}-x_{3}\right| \cdot \\
& \left|x_{2}-x_{4}\right|\left|x_{3}-x_{4}\right|.
\end{aligned}
$$ | \frac{\sqrt{5}}{125} | 1/8 |
In the isosceles triangle \(ABC\) with the sides \(AB = BC\), the angle \(\angle ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that \(\angle OAC = 10^\circ\) and \(\angle OCA = 30^\circ\). Find the angle \(\angle AOB\). | 70 | 4/8 |
120 schools each send 20 people to form 20 teams, with each team having exactly 1 person from each school. Find the smallest positive integer \( k \) such that when \( k \) people are selected from each team, there will be at least 20 people from the same school among all the selected individuals. | 115 | 6/8 |
Let $S=\{1,2,4,8,16,32,64,128,256\}$. A subset $P$ of $S$ is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set $Q$ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets. | 5 | 1/8 |
Twelve people are carrying 12 loaves of bread. Each man carries 2 loaves, each woman carries half a loaf, and each child carries a quarter loaf, with all 12 people participating in carrying the bread. How many men, how many women, and how many children are there? | 5,1,6children | 1/8 |
In a regular quadrilateral pyramid $P-A B C D$, the dihedral angle between a lateral face and the base is $\alpha$, and the dihedral angle between two adjacent lateral faces is $\beta$. Find the value of $2 \cos \beta + \cos 2 \alpha$. | -1 | 3/8 |
Let $A_{1} A_{2} A_{3}$ be a triangle. Construct the following points:
- $B_{1}, B_{2}$, and $B_{3}$ are the midpoints of $A_{1} A_{2}, A_{2} A_{3}$, and $A_{3} A_{1}$, respectively.
- $C_{1}, C_{2}$, and $C_{3}$ are the midpoints of $A_{1} B_{1}, A_{2} B_{2}$, and $A_{3} B_{3}$, respectively.
- $D_{1}$ is the intersection of $\left(A_{1} C_{2}\right)$ and $\left(B_{1} A_{3}\right)$. Similarly, define $D_{2}$ and $D_{3}$ cyclically.
- $E_{1}$ is the intersection of $\left(A_{1} B_{2}\right)$ and $\left(C_{1} A_{3}\right)$. Similarly, define $E_{2}$ and $E_{3}$ cyclically.
Calculate the ratio of the area of $\mathrm{D}_{1} \mathrm{D}_{2} \mathrm{D}_{3}$ to the area of $\mathrm{E}_{1} \mathrm{E}_{2} \mathrm{E}_{3}$. | 25/49 | 7/8 |
For $t = 1, 2, 3, 4$, define $S_t = \sum_{i = 1}^{350}a_i^t$, where $a_i \in \{1,2,3,4\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$. | 905 | 7/8 |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $a \neq b$, $c = \sqrt{3}$, and $\sqrt{3} \cos^2 A - \sqrt{3} \cos^2 B = \sin A \cos A - \sin B \cos B$.
(I) Find the measure of angle $C$;
(II) If $\sin A = \frac{4}{5}$, find the area of $\triangle ABC$. | \frac{24\sqrt{3} + 18}{25} | 3/8 |
A wire of length $80$cm is randomly cut into three segments. The probability that each segment is no less than $20$cm is $\_\_\_\_\_\_\_.$ | \frac{1}{16} | 3/8 |
Given that signals are composed of the digits $0$ and $1$ with equal likelihood of transmission, the probabilities of error in transmission are $0.9$ and $0.1$ for signal $0$ being received as $1$ and $0$ respectively, and $0.95$ and $0.05$ for signal $1$ being received as $1$ and $0$ respectively. | 0.525 | 2/8 |
The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home. | 25 | 1/8 |
Let $S = \{1, 2, \cdots, 50\}$. Find the smallest natural number $k$ such that any subset of $S$ with $k$ elements contains at least two distinct numbers $a$ and $b$ satisfying $a+b \mid ab$. | 39 | 1/8 |
In the complex plane, non-zero complex numbers \( z_{1} \) and \( z_{2} \) lie on the circle centered at \( \mathrm{i} \) with a radius of 1. The real part of \( \overline{z_{1}} \cdot z_{2} \) is zero, and the principal argument of \( z_{1} \) is \( \frac{\pi}{6} \). Find \( z_{2} \). | -\frac{\sqrt{3}}{2}+\frac{3}{2}i | 6/8 |
What is the maximum number of different 3-term arithmetic sequences that can be selected from a sequence of real numbers \(a_{1}<a_{2}<\cdots<a_{n}\)? | \lfloor\frac{(n-1)^2}{4}\rfloor | 2/8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $c=2$, $b=\sqrt{2}a$. The maximum area of $\triangle ABC$ is ______. | 2\sqrt{2} | 7/8 |
In the decimal representation of an even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and the digits may repeat. It is known that the sum of the digits of the number \( 2M \) equals 39, and the sum of the digits of the number \( M / 2 \) equals 30. What values can the sum of the digits of the number \( M \) take? List all possible answers. | 33 | 2/8 |
Exactly at noon, a truck left the village and headed towards the city, and at the same time, a car left the city and headed towards the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$. | 8 | 4/8 |
Greedy Vovochka has 25 classmates. For his birthday, he brought 200 candies to class. Vovochka's mother, so that he does not eat everything himself, ordered him to distribute the candies in such a way that any 16 of his classmates have at least 100 candies in total. What is the maximum number of candies Vovochka can keep for himself while fulfilling his mother's request? | 37 | 3/8 |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | 2047 | 3/8 |
Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ? | \sqrt{10} | 1/8 |
Given: $$\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\in\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}$
Find the value of $a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n}a_{n}$. | 81 | 7/8 |
From points A and B, a motorcyclist and a cyclist respectively set off towards each other simultaneously and met at a distance of 4 km from B. At the moment the motorcyclist arrived in B, the cyclist was at a distance of 15 km from A. Find the distance AB. | 20 | 7/8 |
Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes. | 34 | 3/8 |
Find all reals $ k$ such that
\[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d)
\]
holds for all $ a,b,c,d\geq \minus{} 1$ .
*Edited by orl.* | \frac{3}{4} | 1/8 |
Let \( H \) be the orthocenter of \( \triangle ABC \), and \( 3 \overrightarrow{HA} + 4 \overrightarrow{HB} + 5 \overrightarrow{HC} = \mathbf{0} \). Determine \( \cos \angle AHB \). | -\frac{\sqrt{6}}{6} | 7/8 |
Consider equation $I: x+y+z=46$ where $x, y$, and $z$ are positive integers, and equation $II: x+y+z+w=46$,
where $x, y, z$, and $w$ are positive integers. Then
$\textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad \\ \textbf{(C)}\ \text{II can be solved in consecutive integers} \qquad \\ \textbf{(D)}\ \text{II can be solved in consecutive even integers} \qquad \\ \textbf{(E)}\ \text{II can be solved in consecutive odd integers}$ | \textbf{(C)} | 1/8 |
The base of a triangle is $a$, and the angles adjacent to it measure $45^{\circ}$ and $15^{\circ}$. A circle with a radius equal to the altitude dropped to this base is drawn from the vertex opposite the base. Find the area of the part of the corresponding circle that is contained within the triangle. | \frac{\pi^2(2-\sqrt{3})}{18} | 2/8 |
Given a mall with four categories of food: grains, vegetable oils, animal products, and fruits and vegetables, with 40, 10, 20, and 20 varieties, respectively, calculate the total sample size if 6 types of animal products are sampled. | 27 | 7/8 |
Given the sets of points \( A = \left\{(x, y) \left| (x-3)^2 + (y-4)^2 \leq \left(\frac{5}{2}\right)^2 \right.\right\} \) and \( B = \left\{(x, y) \mid (x-4)^2 + (y-5)^2 > \left(\frac{5}{2}\right)^2 \right\} \), find the number of integer points in the intersection \( A \cap B \). | 7 | 5/8 |
The simplest fraction \(\frac{a}{b}\) satisfies \(\frac{1}{5}<\frac{a}{b}<\frac{1}{4}\), and \(b\) does not exceed 19. What is the product of the maximum possible value and the minimum possible value of \(a + b\)? | 253 | 7/8 |
In the Cartesian coordinate plane \(xOy\), the circle \(\Omega\) and the parabola \(\Gamma: y^2 = 4x\) have exactly one common point, and the circle \(\Omega\) is tangent to the \(x\)-axis at the focus \(F\) of the parabola \(\Gamma\). Find the radius of the circle \(\Omega\). | \frac{4 \sqrt{3}}{9} | 6/8 |
The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)? | 30 | 1/8 |
Prove that \(1 \cdot 3 \cdot 5 \cdot \cdots \cdot 1983 \cdot 1985 + 2 \cdot 4 \cdot 6 \cdot \cdots \cdot 1984 \cdot 1986\) is divisible by 1987. | 1987 | 5/8 |
How many days have passed from March 19, 1990, to March 23, 1996, inclusive? | 2197 | 1/8 |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________. | \frac{2\pi}{7} | 7/8 |
In an acute angled triangle $ABC$ , let $BB' $ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$ . Similarly are defined the angles $\alpha_B$ and $\alpha_C$ . Prove that $$ \displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32} $$ (Romania) | \frac{3\sqrt{6}}{32} | 1/8 |
Find the maximum value of \( x_0 \) for which there exists a sequence \( x_0, x_1, \ldots, x_{1995} \) of positive reals with \( x_0 = x_{1995} \) such that for \( i = 1, \ldots, 1995 \):
\[ x_{i-1} + \frac{2}{x_{i-1}} = 2x_i + \frac{1}{x_i}. \] | 2^{997} | 1/8 |
Given that \(\tan \frac{\alpha+\beta}{2}=\frac{\sqrt{6}}{2}\) and \(\cot \alpha \cdot \cot \beta=\frac{7}{13}\), find the value of \(\cos (\alpha-\beta)\). | \frac{2}{3} | 7/8 |
Find the area of the triangle formed by the axis of the parabola $y^{2}=8x$ and the two asymptotes of the hyperbola $(C)$: $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$. | 2\sqrt{2} | 1/8 |
A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: | 2:3 | 1/8 |
Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\overline{B C}$ and $\overline{D A}$. Compute $P Q^{2}$. | \frac{116}{35} | 1/8 |
You are given $n \ge 2$ distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of $2$ . For every $n$ find the largest possible number of elegant pairs.
*Proposed by Oleksiy Masalitin* | n-1 | 1/8 |
In convex quadrilateral \(ABCD\) with diagonals intersecting at point \(M\), points \(P\) and \(Q\) are the centroids of \(\triangle AMD\) and \(\triangle CMB\) respectively, and points \(R\) and \(S\) are the orthocenters of \(\triangle DMC\) and \(\triangle MAB\) respectively. Prove that \(PQ \perp RS\). | PQ\perpRS | 1/8 |
Determine the number of integers $D$ such that whenever $a$ and $b$ are both real numbers with $-1 / 4<a, b<1 / 4$, then $\left|a^{2}-D b^{2}\right|<1$. | 32 | 1/8 |
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