Found problems: 85335
2022 Rioplatense Mathematical Olympiad, 2
Let $ABC$ be an acute triangle with $AB<AC$. Let $D,E,F$ be the feet of the altitudes relatives to the vertices $A,B,C$, respectively. The circumcircle $\Gamma$ of $AEF$ cuts the circumcircle of $ABC$ at $A$ and $M$. Assume that $BM$ is tangent to $\Gamma$.
Prove that $M$, $F$ and $D$ are collinear.
2005 Uzbekistan National Olympiad, 2
Solve in integer the equation
$\frac{1}{2}(x+y)(y+z)(x+z)+(x+y+z)^{3}=1-xyz$
1995 Grosman Memorial Mathematical Olympiad, 7
For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.
2022 HMIC, 2
Does there exist a regular pentagon whose vertices lie on the edges of a cube?
2023 HMNT, 8
There are $n \ge 2$ coins, each with a different positive integer value. Call an integer $m$ [i]sticky [/i] if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value $100$.
1999 Turkey MO (2nd round), 4
Find all sequences ${{a}_{1}},{{a}_{2}},...,{{a}_{2000}}$ of real numbers such that $\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}$ and such that $\frac{1}{2}<{{a}_{n}}<1$ and ${{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})$ for all $n\ge 1$.
2006 MOP Homework, 5
Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in
the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that
$\angle AOB +\angle COD = 180$.
2015 Romania National Olympiad, 1
Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $$p^2 + pq + q^2 = r^2 .$$
2012 Princeton University Math Competition, A1 / B4
Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$
2011 QEDMO 8th, 6
A [i]synogon [/i] is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.
2018 Taiwan TST Round 3, 1
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
VMEO I 2004, 1
Let $x, y, z$ be non-negative numbers, so that $x + y + z = 1$. Prove that
$$\sqrt{x+\frac{(y-z)^2}{12}}+\sqrt{y+\frac{(x-z)^2}{12}}+\sqrt{z+\frac{(x-y)^2}{12}}\le \sqrt{3}$$
1971 Polish MO Finals, 6
A regular tetrahedron with unit edge length is given. Prove that:
(a) There exist four points on the surface $S$ of the tetrahedron, such that the distance from any point of the surface to one of these four points does not exceed $1/2$;
(b) There do not exist three points with this property.
The distance between two points on surface $S$ is defined as the length of the shortest polygonal line going over $S$ and connecting the two points.
2018 HMNT, 7
A $5\times5$ grid of squares is filled with integers. Call a rectangle [i]corner-odd[/i] if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Note: A rectangles must have four distinct corners to be considered [i]corner-odd[/i]; i.e. no $1\times k$ rectangle can be [i]corner-odd[/i] for any positive integer $k$.
2002 Tournament Of Towns, 4
Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.
1952 Polish MO Finals, 2
On the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $, the points $ M $, $ N $, $ P $ are taken, respectively, in such a way that $$\frac{BM}{MC} = \frac{CN}{NA} = \frac{AP}{PB} = k, $$
where $ k $ means a given number greater than $ 1 $, then the segments $ AM $, $ BN $, $ CP $ were drawn . Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle bounded by the lines $ AM $, $ BN $ and $ CP $.
2019 Estonia Team Selection Test, 6
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
2005 Czech And Slovak Olympiad III A, 2
Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.
1962 Miklós Schweitzer, 1
Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and
$ b$.
[i]E. Fried[/i]
Today's calculation of integrals, 854
Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.
1970 AMC 12/AHSME, 9
Points $P$ and $Q$ are on line segment $AB$, and both points are on the same side of the midpoint of $AB$. Point $P$ divides $AB$ in the ratio $2:3$ and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of segment $AB$ is
$\textbf{(A) }12\qquad\textbf{(B) }28\qquad\textbf{(C) }70\qquad\textbf{(D) }75\qquad \textbf{(E) }105$
2000 German National Olympiad, 5
(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors.
(b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.
2009 Tournament Of Towns, 4
Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is $1$ while the right digit is $0$. Let $M$ be the number of the pairs in which $1$ and $0$ are separated by an even number of digits (possibly zero), and let $N$ be the number of the pairs in which $1$ and $0$ are separated by an odd number of digits. Prove that $M \ge N$.
2019 Centers of Excellency of Suceava, 3
The circumcenter, circumradius and orthocenter of a triangle $ ABC $ satisfying $ AB<AC $ are notated with $ O,R,H, $ respectively. Prove that the middle of the segment $ OH $ belongs to the line $ BC $ if
$$ AC^2-AB^2=2R\cdot BC. $$
[i]Marius Marchitan[/i]
2018 PUMaC Geometry B, 6
Triangle $ABC$ has $\angle{A}=90^\circ$, $\angle{C}=30^\circ$, and $AC=12$. Let the circumcircle of this triangle
be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle{CAD}=60^\circ$. Define
points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.
Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$
intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$
for positive integers $a,b,c,d,$ where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal.
Find $a+b+c+d$.