Found problems: 85335
2008 Macedonia National Olympiad, 4
We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.
2012 India Regional Mathematical Olympiad, 4
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
1980 IMO Shortlist, 12
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
2005 Hungary-Israel Binational, 1
Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$
2020 HK IMO Preliminary Selection Contest, 7
Solve the equation $\sqrt{7-x}=7-x^2$, where $x>0$.
1982 IMO Longlists, 3
Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that
\[\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.\]
1996 Italy TST, 4
4.4. Prove that there exists a set X of 1996 positive integers with the following properties:
(i) the elements of X are pairwise coprime;
(ii) all elements of X and all sums of two or more distinct elements of X are
composite numbers
1988 Romania Team Selection Test, 4
Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \]
[i]Viorel Vajaitu[/i]
2011 ELMO Shortlist, 1
Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle.
[i]Evan O'Dorney.[/i]
2024-25 IOQM India, 7
Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
2012 India IMO Training Camp, 1
Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.
2008 China Team Selection Test, 2
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$
2017 China Western Mathematical Olympiad, 6
In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.
2019 HMNT, 4
Two players play a game, starting with a pile of $N$ tokens. On each player’s turn, they must remove $2^n$ tokens from the pile for some nonnegative integer $n$. If a player cannot make a move, they lose. For how many $N$ between $ 1$ and $2019$ (inclusive) does the first player have a winning strategy?
1991 Arnold's Trivium, 47
Map the exterior of the disc conformally onto the exterior of a given ellipse.
2004 Nicolae Coculescu, 4
Let $ H $ denote the orthocenter of an acute triangle $ ABC, $ and $ A_1,A_2,A_3 $ denote the intersections of the altitudes of this triangle with its circumcircle, and $ A',B',C' $ denote the projections of the vertices of this triangle on their opposite sides.
[b]a)[/b] Prove that the sides of the triangle $ A'B'C' $ are parallel to the sides of $ A_1B_1C_1. $
[b]b)[/b] Show that $ B_1C_1\cdot\overrightarrow{HA_1} +C_1A_1\cdot\overrightarrow{HB_1} +A_1B_1\cdot\overrightarrow{HC_1} =0. $
[i]Geoghe Duță[/i]
2014 Hanoi Open Mathematics Competitions, 2
How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ?
(A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above.
Note: $C^{m}_{n}$ stands for $\binom {m}{n}$
1993 AMC 12/AHSME, 1
For integers $a, b$ and $c$, define $\boxed{a, b, c}$ to mean $a^b-b^c+c^a$. Then $\boxed{1, -1, 2}$ equals
$ \textbf{(A)}\ -4 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4 $
2019 Poland - Second Round, 5
Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that
\begin{align*}
b_k+b_{\ell}=m.
\end{align*}
1999 USAMTS Problems, 5
In a convex pentagon $ABCDE$ the sides have lengths $1,2,3,4,$ and $5$, though not necessarily in that order. Let $F,G,H,$ and $I$ be the midpoints of the sides $AB$, $BC$, $CD$, and $DE$, respectively. Let $X$ be the midpoint of segment $FH$, and $Y$ be the midpoint of segment $GI$. The length of segment $XY$ is an integer. Find all possible values for the length of side $AE$.
1990 IberoAmerican, 3
Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$.
i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$.
ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.
2016 Harvard-MIT Mathematics Tournament, 27
Find the smallest possible area of an ellipse passing through $(2,0)$, $(0,3)$, $(0,7)$, and $(6,0)$.
2017 ELMO Shortlist, 3
Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$, is equal to a power of two.
[i]Proposed by David Stoner
2007 Spain Mathematical Olympiad, Problem 3
$O$ is the circumcenter of triangle $ABC$. The bisector from $A$ intersects the opposite side in point $P$. Prove that the following is satisfied:
$$AP^2 + OA^2 - OP^2 = bc.$$
1972 IMO Longlists, 9
Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.