Found problems: 85335
2010 Romania National Olympiad, 3
Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$.
[i]Mircea Fianu[/i]
1987 Tournament Of Towns, (147) 4
For any natural $n$ prove the inequality
$$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$
2010 Nordic, 1
A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.
2024 Brazil Cono Sur TST, 4
Let $ABC$ be a triangle, $O$ its circumcenter and $\Gamma$ its circumcircle. Let $E$ and $F$ be points on $AB$ and $AC$, respectively, such that $O$ is the midpoint of $EF$. Let $A'=AO\cap \Gamma$, with $A'\ne A$. Finally, let $P$ be the point on line $EF$ such that $A'P\perp EF$. Prove that the lines $EF,BC$ and the tangent to $\Gamma$ at $A'$ are concurrent and that $\angle BPA' = \angle CPA'$.
1974 IMO, 3
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
2012 APMO, 1
Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.
1966 IMO Longlists, 39
Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle.
[b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle.
[b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.
2007 Germany Team Selection Test, 1
Prove the inequality:
\[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\]
for positive reals $ a_{1},a_{2},\ldots,a_{n}$.
[i]Proposed by Dusan Dukic, Serbia[/i]
2020 Canadian Mathematical Olympiad Qualification, 4
Determine all graphs $G$ with the following two properties:
$\bullet$ G contains at least one Hamilton path.
$\bullet$ For any pair of vertices, $u, v \in G$, if there is a Hamilton path from $u$ to $v$ then the edge $uv$ is in the graph $G$
2022 Belarusian National Olympiad, 9.1
Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$.
Prove that points $X,Y,Z$ and $W$ lie on a circle.
1979 IMO, 1
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.
2024 Rioplatense Mathematical Olympiad, 2
Let $ABC$ be a triangle with $AB < AC$, incentre $I$, and circumcircle $\omega$. Let $D$ be the intersection of the external bisector of angle $\widehat{ BAC}$ with line $BC$. Let $E$ be the midpoint of the arc $BC$ of $\omega$ that does not contain $A$. Let $M$ be the midpoint of $DI$, and $X$ the intersection of $EM$ with $\omega$. Prove that $IX$ and $EM$ are perpendicular.
2024 Spain Mathematical Olympiad, 4
Let $a,b,c,d$ be real numbers satisfying \[abcd=1\quad \text{and}\quad a+\frac1a+b+\frac1b+c+\frac1c+d+\frac1d=0.\] Prove that at least one of the numbers $ab$, $ac$, $ad$ equals $-1$.
2012 Greece National Olympiad, 2
Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$:
\[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]
LMT Team Rounds 2010-20, 2020.S29
Let $\mathcal{F}$ be the set of polynomials $f(x)$ with integer coefficients for which there exists an integer root of the equation $f(x)=1$. For all $k>1$, let $m_k$ be the smallest integer greater than one for which there exists $f(x)\in \mathcal{F}$ such that $f(x)=m_k$ has exactly $k$ distinct integer roots. If the value of $\sqrt{m_{2021}-m_{2020}}$ can be written as $m\sqrt{n}$ for positive integers $m,n$ where $n$ is squarefree, compute the largest integer value of $k$ such that $2^k$ divides $\frac{m}{n}$.
2016 China Team Selection Test, 2
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
2023 All-Russian Olympiad, 1
Given are two monic quadratics $f(x), g(x)$ such that $f, g, f+g$ have two distinct real roots. Suppose that the difference of the roots of $f$ is equal to the difference of the roots of $g$. Prove that the difference of the roots of $f+g$ is not bigger than the above common difference.
Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4
The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that:
a) the segment $MN$ has a constant length,
b) all circles circumscribed around triangle $DMN$ have a common point
2007 F = Ma, 12
A $2$-kg rock is suspended by a massless string from one end of a uniform $1$-meter measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the $0.20$-meter mark?
[asy]
size(250);
draw((0,0)--(7.5,0)--(7.5,0.2)--(0,0.2)--cycle);
draw((1.5,0)--(1.5,0.2));
draw((3,0)--(3,0.2));
draw((4.5,0)--(4.5,0.2));
draw((6,0)--(6,0.2));
filldraw((1.5,0)--(1.2,-1.25)--(1.8,-1.25)--cycle, gray(.8));
draw((0,0)--(0,-0.4));
filldraw((0,-0.4)--(-0.05,-0.4)--(-0.1,-0.375)--(-0.2,-0.375)--(-0.3,-0.4)--(-0.3,-0.45)--(-0.4,-0.6)--(-0.35,-0.7)--(-0.15,-0.75)--(-0.1,-0.825)--(0.1,-0.84)--(0.15,-0.8)--(0.15,-0.75)--(0.25,-0.7)--(0.25,-0.55)--(0.2,-0.4)--(0.1,-0.35)--cycle, gray(.4));
[/asy]
$ \textbf {(A) } 0.20 \, \text{kg} \qquad \textbf {(B) } 1.00 \, \text{kg} \qquad \textbf {(C) } 1.33 \, \text{kg} \qquad \textbf {(D) } 2.00 \, \text{kg} \qquad \textbf {(E) } 3.00 \, \text{kg} $
2003 Junior Balkan Team Selection Tests - Moldova, 2
Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3abc.$ Prove the following inequality:
\[ \frac{a}{b^{2}c^{2}}+\frac{b}{c^{2}a^{2}}+\frac{c}{a^{2}b^{2}}\geq\frac{9}{a+b+c} \]
2007 Grigore Moisil Intercounty, 2
Le be a real number $ |a|<1, $ a natural number $ n\ge 2, $ and a $ 2\times 2 $ real matrix $ A $ that verifies
$$ \det \left( A^{2n} -aA^{2n-1} -aA+I \right)=0. $$ Show that $ \det A=1. $
2002 Iran MO (2nd round), 1
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which
\[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\]
Find the number of elements of the set $A_n$.
[i]Proposed by Vidan Govedarica, Serbia[/i]
1977 All Soviet Union Mathematical Olympiad, 242
The polynomial $$x^{10} + ?x^9 + ?x^8 + ... + ?x + 1$$ is written on the blackboard. Two players substitute (real) numbers instead of one of the question marks in turn. ($9$ turns total.) The first wins if the polynomial will have no real roots. Who wins?
1998 Putnam, 5
Let $\mathcal{F}$ be a finite collection of open discs in $\mathbb{R}^2$ whose union contains a set $E\subseteq \mathbb{R}^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots,D_n$ in $\mathcal{F}$ such that \[E\subseteq\cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.
2021 Argentina National Olympiad Level 2, 3
A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.