Found problems: 85335
2002 China Team Selection Test, 1
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.
(1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets.
(2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
2014 AMC 12/AHSME, 17
Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$
$ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $
2010 Tournament Of Towns, 4
A rectangle is divided into $2\times 1$ and $1\times 2$ dominoes. In each domino, a diagonal is drawn, and no two diagonals have common endpoints. Prove that exactly two corners of the rectangle are endpoints of these diagonals.
2022 CCA Math Bonanza, I15
Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\angle APB = 120^\circ$, $\angle BPC = 45^\circ$, $\angle CPD = 60^\circ$, and $\angle DPA = 135^\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$, $PDA$. $[G_1G_2G_3G_4]$ can be expressed as $a\sqrt{b} + c\sqrt{d}$. Find $a+b+c+d$.
[i]2022 CCA Math Bonanza Individual Round #15[/i]
JOM 2023, 2
Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$)
[i]Proposed by Wong Jer Ren[/i]
2021 Indonesia MO, 4
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that
\[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]
2018 Yasinsky Geometry Olympiad, 6
Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular.
(Vyacheslav Yasinsky)
2017 BMT Spring, 3
Let $ABCDEF$ be a regular hexagon with side length $ 1$. Now, construct square $AGDQ$. What is the area of the region inside the hexagon and not the square?
2022 Yasinsky Geometry Olympiad, 6
Let $AD$, $BE$ and $CF$ be the diameters of the circle circumscribed around the acute angle triangle $ABC$. Point $N$ is the midpoint of the arc $CAD$, and point $M$ is the midpoint of arc $BAD$. Prove that the lines $EN$ and $MF$ intersect at the angle bisector of $\angle BAC$.
(Matvii Kurskyi)
2020-21 KVS IOQM India, 28
For a natural number $n$, let $n'$ denote the number obtained by deleting zero digits, if any. (For example, if $n = 260$, $n' = 26$, if $n = 2020$, $n' = 22$.),Find the number of $3$-digit numbers $n$ for which $n'$ is a divisor of $n$, different from $n$.
2023 Argentina National Olympiad Level 2, 4
Initially, Igna distributes $1000$ balls into $30$ boxes. Then, Igna and Mica alternate turns, starting with Igna. Each player, on their turn, chooses a box and removes one ball. When a player removes the last ball from a box, they earn a coin. Find the maximum integer $k$ such that, regardless of how Mica plays, Igna can earn at least $k$ coins.
PEN L Problems, 11
Let the sequence $\{K_{n}\}_{n \ge 1}$ be defined by \[K_{1}=2, K_{2}=8, K_{n+2}=3K_{n+1}-K_{n}+5(-1)^{n}.\] Prove that if $K_{n}$ is prime, then $n$ must be a power of $3$.
2021 Sharygin Geometry Olympiad, 14
Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$, touching the sides $BC$, $CA$, $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$. Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$. Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$. Prove that $XY$ touches $\gamma_A$.
1979 IMO Longlists, 45
For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.
2006 All-Russian Olympiad, 4
Given a triangle $ ABC$. The angle bisectors of the angles $ ABC$ and $ BCA$ intersect the sides $ CA$ and $ AB$ at the points $ B_1$ and $ C_1$, and intersect each other at the point $ I$. The line $ B_1C_1$ intersects the circumcircle of triangle $ ABC$ at the points $ M$ and $ N$. Prove that the circumradius of triangle $ MIN$ is twice as long as the circumradius of triangle $ ABC$.
2002 Korea - Final Round, 3
The following facts are known in a mathematical contest:
[list]
(a) The number of problems tested was $n\ge 4$
(b) Each problem was solved by exactly four contestants.
(c) For each pair of problems, there is exactly one contestant who solved both problems
[/list]
Assuming the number of contestants is greater than or equal to $4n$, find the minimum value of $n$ for which there always exists a contestant who solved all the problems.
2002 China Team Selection Test, 1
Given $ n \geq 3$, $ n$ is a integer. Prove that:
\[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\]
where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.
2010 IFYM, Sozopol, 2
Is it possible to color the cells of a table 19 x 19 in yellow, blue, red, and green so that each rectangle $a$ x $b$ ($a,b\geq 2$) in the table has at least 2 cells in different color?
2013 Romania Team Selection Test, 1
Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[
\left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}.
\] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$
1993 IMO Shortlist, 6
For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane,
\[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]
1999 IMO Shortlist, 5
Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.
1985 Spain Mathematical Olympiad, 8
A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant.
Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.
2000 National High School Mathematics League, 1
In acute triangle $ABC$, $D,E$ are two points on side $BC$, satisfying that $\angle BAE=\angle CAF$. $FM\perp AB,EN\perp AC$ ($M,N$ are foot points). $AE$ intersects the circumcircle of $\triangle ABC$ at $D$. Prove that the area of $\triangle ABC$ and quadrilateral $AMDN$ are equal.
2010 China Team Selection Test, 3
An (unordered) partition $P$ of a positive integer $n$ is an $n$-tuple of nonnegative integers $P=(x_1,x_2,\cdots,x_n)$ such that $\sum_{k=1}^n kx_k=n$. For positive integer $m\leq n$, and a partition $Q=(y_1,y_2,\cdots,y_m)$ of $m$, $Q$ is called compatible to $P$ if $y_i\leq x_i$ for $i=1,2,\cdots,m$. Let $S(n)$ be the number of partitions $P$ of $n$ such that for each odd $m<n$, $m$ has exactly one partition compatible to $P$ and for each even $m<n$, $m$ has exactly two partitions compatible to $P$. Find $S(2010)$.
2014 ASDAN Math Tournament, 1
Kevin is running $1000$ meters. He wants to have an average speed of $10$ meters a second. He runs the first $100$ meters at a speed of $4$ meters a second. Compute how quickly, in meters per second, he must run the last $900$ meters to attain his desired average speed of $10$ meters a second.