Found problems: 85335
2012 IFYM, Sozopol, 2
Find all natural numbers, which cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$ where $a,b\in \mathbb{N}$.
1997 ITAMO, 3
The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that:
(i) In every square of side $n$ ($n \in N$) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares;
(ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?
1977 IMO Longlists, 15
Let $n$ be an integer greater than $1$. In the Cartesian coordinate system we consider all squares with integer vertices $(x,y)$ such that $1\le x,y\le n$. Denote by $p_k\ (k=0,1,2,\ldots )$ the number of pairs of points that are vertices of exactly $k$ such squares. Prove that $\sum_k(k-1)p_k=0$.
2018 Israel Olympic Revenge, 1
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
1984 IMO Longlists, 52
Construct a scalene triangle such that
\[a(\tan B - \tan C) = b(\tan A - \tan C)\]
2024 Princeton University Math Competition, 4
Consider the $100 \times 100$ grid of points with integer coordinates $S=\{(x,y) \in \mathbb{Z}^2\mid$ $1 \le x \le 100,$ $1 \le y$ $\le$ $100\}.$ A set $C$ is formed by selecting each $p \in S$ with probability $\tfrac{1}{2}$ uniformly at random. The [I]expansion[/I] of $C$ is defined as the set of points $q \in S$ such that $\min_{p \in C} d(q,p) \le 1,$ where $d(q,p)$ denotes the Euclidean distance between $q,p.$ If the expected size of the expansion of $C$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$
1996 All-Russian Olympiad Regional Round, 11.7
In triangle $ABC$, a point $O$ is taken such that $\angle COA = \angle B + 60^o$, $\angle COB = \angle A + 60^o$, $\angle AOB = \angle C + 60^o$.Prove that if a triangle can be formed from the segments $AO$, $BO$, $CO$, then a triangle can also be formed from the altitudes of triangle $ABC$ and these triangles are similar.
2002 District Olympiad, 3
Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle.
a) Show that $O$ is at equal distances from the midpoints of the three segments considered.
b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.
2001 Junior Balkan Team Selection Tests - Moldova, 7
Noah has on his ark $4$ large coffins in which to place $8$ animals.
It is known that for any animal there are at most $5$ animals with which it is incompatible (those can't live together). Show that:
a) Noah can place the animals in the cages according to their compatibility.
b) Noah can place two animals in each cage.
2001 India National Olympiad, 5
$ABC$ is a triangle. $M$ is the midpoint of $BC$. $\angle MAB = \angle C$, and $\angle MAC = 15^{\circ}$. Show that $\angle AMC$ is obtuse. If $O$ is the circumcenter of $ADC$, show that $AOD$ is equilateral.
1966 Putnam, B6
Show that all the solutions of the differential equation $y''+e^xy=0$ remain bounded as $x\to \infty.$
1968 Bulgaria National Olympiad, Problem 4
On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$. Prove that on $g$, there exists at least one pair of points $P,Q$ symmetrical with respect to $C$, which divide the segment $AB$ internally and externally in the same ratios, i.e
$$\frac{PA}{PB}=\frac{QA}{QB}\qquad(1)$$
If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$, prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$.
[i]K. Petrov[/i]
2001 National Olympiad First Round, 28
The towns $A,B,C,D,E$ are located clockwise on a circular road such that the distance between $A$ and $B$, $B$ and $C$, $C$ and $D$, $E$ and $A$ are $5$, $5$, $2$, $1$ and $4$ km respectively. A health center will be located on that road such that the maximum of the shortest distance to each town will be minimum. How many alternative locations are there for the health center?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2024 Bulgarian Autumn Math Competition, 8.3
Find all positive integers $n$, such that: $$a+b+c \mid a^{2n}+b^{2n}+c^{2n}-n(a^2b^2+b^2c^2+c^2a^2)$$ for all pairwise different positive integers $a,b$ and $c$
2017 Dutch Mathematical Olympiad, 4
If we divide the number $13$ by the three numbers $5, 7$, and $9$, then these divisions leave remainders: when dividing by $5$ the remainder is $3$, when dividing by $7$ the remainder is $6$, and when dividing by $9$ the remainder is 4. If we add these remainders, we obtain $3 + 6 + 4 = 13$, the original number.
(a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$. Prove: if you divide $n$ by $a$ and $b$, then the sum of the two remainders never equals $n$.
(b) Determine all integers $n > 229$ having the property that if you divide $n$ by $99, 132$, and $229$, the sum of the three remainders is $n$.
1956 Polish MO Finals, 6
Given a sphere of radius $ R $ and a plane $ \alpha $ having no common points with this sphere. A point $ S $ moves in the plane $ \alpha $, which is the vertex of a cone tangent to the sphere along a circle with center $ C $. Find the locus of point $ C $.
[hide=another is Polish MO 1967 p6] [url=https://artofproblemsolving.com/community/c6h3388032p31769739]here[/url][/hide]
2007 Moldova National Olympiad, 12.6
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
2000 All-Russian Olympiad Regional Round, 9.4
Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Through point $A$ of circle $S_1$, draw straight lines $AM$ and $AN$ intersecting $S_2$ at points $B$ and $C$, and through point $D$ of circle $S_2$, draw straight lines $DM$ and $DN$ intersecting $S_1$ at points $E$ and $F$, and $A$, $E$, $F$ lie along one side of line $MN$, and $D$, $B$, $C$ lie on the other side (see figure). Prove that if $AB = DE$, then points $A$, $F$, $C$ and $D$ lie on the same circle, the position of the center of which does not depend on choosing points $A$ and $D$.
[img]https://cdn.artofproblemsolving.com/attachments/7/0/d1f9c2f39352e2b39e55bd2538677073618ef9.png[/img]
1992 All Soviet Union Mathematical Olympiad, 560
A country contains $n$ cities and some towns. There is at most one road between each pair of towns and at most one road between each town and each city, but all the towns and cities are connected, directly or indirectly. We call a route between a city and a town a gold route if there is no other route between them which passes through fewer towns. Show that we can divide the towns and cities between $n$ republics, so that each belongs to just one republic, each republic has just one city, and each republic contains all the towns on at least one of the gold routes between each of its towns and its city.
2008 Serbia National Math Olympiad, 3
Let $ a$, $ b$, $ c$ be positive real numbers such that $ a \plus{} b \plus{} c \equal{} 1$. Prove inequality:
\[ \frac{1}{bc \plus{} a \plus{} \frac{1}{a}} \plus{} \frac{1}{ac \plus{} b \plus{} \frac{1}{b}} \plus{} \frac{1}{ab \plus{} c \plus{} \frac{1}{c}} \leqslant \frac{27}{31}.\]
1984 Tournament Of Towns, (063) O4
Prove that, for any natural number $n$, the graph of any increasing function $f : [0,1] \to [0, 1]$ can be covered by $n$ rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points.
(a) Assume that $f(x)$ is continuous on $[0,1]$.
(b) Do not assume that $f(x)$ is continuous on $[0,1]$.
(A Andjans, Riga)
PS. (a) for O Level, (b) for A Level
2009 Sharygin Geometry Olympiad, 23
Is it true that for each $ n$, the regular $ 2n$-gon is a projection of some polyhedron having not greater than $ n \plus{} 2$ faces?
1994 Bulgaria National Olympiad, 2
Find all functions $f : R \to R$ such that $x f(x)-y f(y) = (x-y)f(x+y)$ for all $x,y \in R$.
2003 CentroAmerican, 4
$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$. Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$).
Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.
2018 South East Mathematical Olympiad, 7
For positive integers $m,n,$ define $f(m,n)$ as the number of ordered triples $(x,y,z)$ of integers such that
$$
\begin{cases}
xyz=x+y+z+m, \\
\max\{|x|,|y|,|z|\} \leq n
\end{cases}
$$
Does there exist positive integers $m,n,$ such that $f(m,n)=2018?$ Please prove your conclusion.