Found problems: 85335
2020 IMO Shortlist, C6
There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:
[list]
[*]The total weights of both piles are the same.
[*] Each pile contains two pebbles of each colour.
[/list]
[i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]
2013 Thailand Mathematical Olympiad, 3
Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$.
2005 Croatia National Olympiad, 4
The circumradius $R$ of a triangle with side lengths $a, b, c$ satisfies $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.
2020 Thailand Mathematical Olympiad, 10
Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$.
2016 Finnish National High School Mathematics Comp, 2
Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.
1997 Rioplatense Mathematical Olympiad, Level 3, 1
Find all positive integers $n$ with the following property:
there exists a polynomial $P_n(x)$ of degree $n$, with integer coefficients, such that $P_n(0)=0$ and $P_n(x)=n$ for $n$ distinct integer solutions.
2018 ITAMO, 6
Let $ABC$ be a triangle with $AB=AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$. Lines $BI$ and $CI$ intersect $\Gamma$ in two new points, $M$ and $N$ respectively. Let $D$ be another point on $\Gamma$ lying on arc $BC$ not containing $A$, and let $E,F$ be the intersections of $AD$ with $BI$ and $CI$, respectively. Let $P,Q$ be the intersections of $DM$ with $CI$ and of $DN$ with $BI$ respectively.
(i) Prove that $D,I,P,Q$ lie on the same circle $\Omega$
(ii) Prove that lines $CE$ and $BF$ intersect on $\Omega$
2008 JBMO Shortlist, 12
Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$
2021 Saudi Arabia IMO TST, 10
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2003 China Western Mathematical Olympiad, 1
Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$. Find the smallest possible sum of the 4 numbers on a side.
2008 Oral Moscow Geometry Olympiad, 1
A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.
2009 Spain Mathematical Olympiad, 2
Let $ ABC$ be an acute triangle with the incircle $ C(I,r)$ and the circumcircle $ C(O,R)$ . Denote
$ D\in BC$ for which $ AD\perp BC$ and $ AD \equal{} h_a$ . Prove that $ DI^2 \equal{} (2R \minus{} h_a)(h_a \minus{} 2r)$ .
2003 Olympic Revenge, 7
Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements.
Find $X$ such that the number of subsets with the same sum is maximum.
2015 Bosnia and Herzegovina Junior BMO TST, 3
Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.
2013 Czech And Slovak Olympiad IIIA, 2
Each of the thieves in the $n$-member party ($n \ge 3$) charged a certain number of coins. All the coins were $100n$. Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers $n \ge 3$ for which after a finite number of steps each outlaw can have $100$ coins no matter how many coins each thug has charged.
2007 Hanoi Open Mathematics Competitions, 2
What is largest positive integer n satisfying the
following inequality:
$n^{2006}$ < $7^{2007}$?
2016 China Team Selection Test, 1
Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let
$$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$
If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.
1997 Croatia National Olympiad, Problem 1
Find the last four digits of each of the numbers $3^{1000}$ and $3^{1997}$.
2023 Tuymaada Olympiad, 3
Point $L$ inside triangle $ABC$ is such that $CL = AB$ and $ \angle BAC + \angle BLC = 180^{\circ}$. Point $K$ on the side $AC$ is such that $KL \parallel BC$. Prove that $AB = BK$
2006 Miklós Schweitzer, 4
let P be a finite set with at least 2 elements. P is a partially ordered and connected set. $p:P^3 \to P$ is a 3-variable, monotone function which satisfies p(x,x,y)=y. Prove that there exists a non-empty subset $I \subset P$ such that $\forall x \in P$ $\forall y \in I$, we have $p(x, y, y) \in I$.
[P is connected means that if each element is replaced by vertices and there is an edge between 2 vertices iff the 2 elements can be compared, then the graph is connected.
p is monotone means that if $x_1\leq y_1 , x_2\leq y_2 , x_3\leq y_3$ , then $p(x_1,x_2,x_3)\leq p(y_1,y_2,y_3)$.]
2016 China National Olympiad, 3
Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent:
1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$
2) For any natural $d \leq \frac{p-1}{2}$,
$$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$
where indices are taken $\pmod p$
2021 Indonesia MO, 7
Given $\triangle ABC$ with circumcircle $\ell$. Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$. Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$. Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$. Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$.
2011 Romania National Olympiad, 3
Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.
2018-2019 SDML (High School), 15
Pentagon $ABCDE$ is such that all five diagonals $AC, BD, CE, DA,$ and $EB$ lie entirely within the pentagon. If the area of each of the triangles $ABC, BCD, CDE,$ and $DEA$ is equal to $1$ and the area of triangle $EAB$ is equal to $2$, the area of the pentagon $ABCDE$ is closest to
$ \mathrm{(A) \ } 4.42 \qquad \mathrm{(B) \ } 4.44 \qquad \mathrm {(C) \ } 4.46 \qquad \mathrm{(D) \ } 4.48 \qquad \mathrm{(E) \ } 4.5$
2008 German National Olympiad, 4
Find the smallest constant $ C$ such that for all real $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds.