Found problems: 85335
2010 Singapore MO Open, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
2008 Sharygin Geometry Olympiad, 4
(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.
2018 Turkey Team Selection Test, 4
In a non-isosceles acute triangle $ABC$, $D$ is the midpoint of the edge $[BC]$. The points $E$ and $F$ lie on $[AC]$ and $[AB]$, respectively, and the circumcircles of $CDE$ and $AEF$ intersect in $P$ on $[AD]$. The angle bisector from $P$ in triangle $EFP$ intersects $EF$ in $Q$. Prove that the tangent line to the circumcirle of $AQP$ at $A$ is perpendicular to $BC$.
the 16th XMO, 4
Given an integer $n$ ,For a sequence of $X$ with the number of $n$ and $Y$ with the number of $100n$ , we call it a [b]spring [/b] . We have two following rules
$\blacksquare$ Choose four adjacent character , if it is $YXXY$ , than it can be changed into $XYYX$
$\blacksquare $ Choose. four adjacent character , if it is $XYYX $ , than it can be changed into $YXXY$
If [b]spring [/b] $A$ can become $B$ using the rules , than we call they are [b][color=#3D85C6]similar [/color][/b]
Thy to find the maximum of $C$ such that there exists $C$ distinct [b]springs[/b] and they are [b][color=#3D85C6]similar[/color][/b]
2014 Israel National Olympiad, 2
Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered [u]clockwise[/u]. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$.
Prove that $\Delta A_3MN$ is an equilateral triangle.
2017-2018 SDML (Middle School), 5
If the sum of the slope and the $y$-intercept of a line is $3$, then through which point is the line guaranteed to pass?
2022 Princeton University Math Competition, 11
For the function $$ g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},$$ let $b \in R$ be the input that maximizes $g$. If $\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}$ for positive integers $m, n, p, q$, find $m + n + p + q$.
2019 Miklós Schweitzer, 6
Let $d$ be a positive integer and $1 < a \le (d+2)/(d+1)$. For given $x_0, x_1,\dots, x_d \in (0, a-1)$, let $x_{k+1} = x_k (a - x_{k-d})$, $k \ge d$. Prove that $\lim_{k \to \infty} x_k = a-1$.
2012 NIMO Problems, 1
In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel?
[i]Proposed by Aaron Lin[/i]
1972 IMO, 3
Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.
2002 HKIMO Preliminary Selection Contest, 3
Find the sum of all integers from 1 to 1000 which contain at least one “7” in their digits.
1993 APMO, 5
Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane
with the following properties:
(i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$;
(ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$.
Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
2009 Grand Duchy of Lithuania, 4
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$.
Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.
1999 Miklós Schweitzer, 2
Let e>0. Prove that for a large enough natural n, there exist natural x,y,z st $n^2+x^2=y^2+z^2$ and $y,z\leq \frac{(1+e)n}{\sqrt{2}}$.
2018 IFYM, Sozopol, 5
On the extension of the heights $AH_1$ and $BH_2$ of an acute $\triangle ABC$, after points $H_1$ and $H_2$, are chosen points $M$ and $N$ in such way that
$\angle MCB = \angle NCA = 30^\circ$.
We denote with $C_1$ the intersection point of the lines $MB$ and $NA$. Analogously we define $A_1$ and $B_1$. Prove that the straight lines $AA_1$, $BB_1$, and $CC_1$ intersect in one point.
2021 AMC 12/AHSME Fall, 22
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the $3$-by-$3$ array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over?
$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 112 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\
148 \qquad\textbf{(E)}\ 160$
2024 Korea Summer Program Practice Test, 7
$2024$ people attended a party. Eunson, the host of the party, wanted to make the participant shake hands in pairs. As a professional daydreamer, Eunsun wondered which would be greater: the number of ways each person could shake hands with $4$ others or the number of ways each person could shake hands with $3$ others. Solve Eunsun's peculiar question.
2000 AMC 8, 16
In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$
2006 QEDMO 2nd, 11
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.
2024/2025 TOURNAMENT OF TOWNS, P5
A rectangular checkered board is painted black and white as a chessboard, and is tiled by dominoes $1 \times 2$. If a horizontal and a vertical dominoes have common segment, it has a door which has the color of the adjoining cell of the domino adjacent by a short side. Is it necessarily true that the number of white doors equals the number of black doors?
2012 Centers of Excellency of Suceava, 1
Function ${{f\colon \mathbb[0, +\infty)}\to\mathbb[0, +\infty)}$ satisfies the condition $f(x)+f(y){\ge}2f(x+y)$ for all $x,y{\ge}0$.
Prove that $f(x)+f(y)+f(z){\ge}3f(x+y+z)$ for all $x,y,z{\ge}0$.
Mathematical induction?
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Azerbaijan Land of the Fire :lol:
1991 Kurschak Competition, 2
A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.
2015 Princeton University Math Competition, A1/B2
What is the $22\text{nd}$ positive integer $n$ such that $22^n$ ends in a $2$? (when written in base $10$).
2025 Thailand Mathematical Olympiad, 1
For each positive integer $m$, denote by $d(m)$ the number of positive divisors of $m$. We say that a positive integer $n$ is [i]Burapha[/i] integer if it satisfy the following condition
[list]
[*] $d(n)$ is an odd integer.
[*] $d(k) \leqslant d(\ell)$ holds for every positive divisor $k, \ell$ of $n$, such that $k < \ell$
[/list]
Find all Burapha integer.
2019 BMT Spring, 4
Two real numbers $ x $ and $ y $ are both chosen at random from the closed interval $ [-10, 10] $. Find
the probability that $ x^2 + y^2 < 10 $. Express your answer as a common fraction in terms of $ \pi $.