Found problems: 85335
2012 BMT Spring, 1
Let $S$ be the set of all rational numbers $x \in [0, 1]$ with repeating base $6$ expansion $$x = 0.\overline{a_1a_2 ... a_k} = 0.a_1a_2...a_ka_1a_2...a_k...$$ for some finite sequence $\{a_i\}^{k}_{i=1}$ of distinct nonnegative integers less than $6$. What is the sum of all numbers that can be written in this form? (Put your answer in base $10$.)
2022 Romania National Olympiad, P4
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that
$$|f(A)\cap f(B)|=|A\cap B|$$
whenever $A$ and $B$ are two distinct subsets of $X$.
[i] (Sergiu Novac)[/i]
2023 Sharygin Geometry Olympiad, 9.7
Let $H$ be the orthocenter of triangle $\mathrm T$. The sidelines of triangle $\mathrm T_1$ pass through the midpoints of $\mathrm T$ and are perpendicular to the corresponding bisectors of $\mathrm T$. The vertices of triangle $\mathrm T_2$ bisect the bisectors of $\mathrm T$. Prove that the lines joining $H$ with the vertices of $\mathrm T_1$ are perpendicular to the sidelines of $\mathrm T_2$.
2003 Chile National Olympiad, 7
Juan found an easy (but wrong) way to simplify fractions. He proposes to simplify a fraction $\frac{M}{N}$ , where $M <N$ are two natural numbers, erase simultaneously the equal digits in the numerator and denominator. For instance, $\frac{12356}{5789}$ transforms after simplification of Juan in $\frac{126}{789}$. Find out if there is at least one fraction $\frac{M}{N}$, with $10 <M <N <100$ for which this method gives a correct result.
2017 China Team Selection Test, 6
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
2008 Indonesia TST, 2
Find all positive integers $1 \le n \le 2008$ so that there exist a prime number $p \ge n$ such that $$\frac{2008^p + (n -1)!}{n}$$ is a positive integer.
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.