Found problems: 85335
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 $.
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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.
2004 Bosnia and Herzegovina Team Selection Test, 4
On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.
2018 Thailand Mathematical Olympiad, 9
In $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$. Let $P$ be a point on $BC$ different from $B$ and $C$, and let $K$ and $L$ be incenters of $\vartriangle ABP$ and $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$. Prove that $AD = AQ$.
2022 Purple Comet Problems, 21
Find the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter is $A$, the last letter is $B$, and the sequence contains no three consecutive letters reading $ABA$. For example, count $AAABBABBAB$ and $ABBBBBBBAB$ but not $AABBAABABB$ or $AAAABBBBBA$.
1991 Baltic Way, 18
Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?
2024 MMATHS, 2
Grant has a box with $6$ red balls, $5$ blue balls, $4$ green balls, $3$ yellow balls, $2$ orange balls, and $1$ purple ball. Grant selects $6$ balls at random, without replacement. Let $P$ be the probability that Grant selects six balls of different colors, and let $Q$ be the probability that Grant selects six balls of the same color. What is $\tfrac{P}{Q}$?
2014 NIMO Problems, 4
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.
2003 Romania National Olympiad, 3
Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$
Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $
[i]Radu Gologan[/i]
2015 Postal Coaching, Problem 6
Let $k \in \mathbb{N}$, let $x_k$ denote the nearest integer to $\sqrt k$. Show that for each $m \in \mathbb {N}$,
$$\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}$$,
where $f(m)$ is the integer part of $\frac{\sqrt{4m-3}-1}{2}$
1969 Czech and Slovak Olympiad III A, 1
Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]
2022 German National Olympiad, 5
Let $ABC$ be an equilateral triangle with circumcircle $k$. A circle $q$ touches $k$ from outside in a point $D$, where the point $D$ on $k$ is chosen so that $D$ and $C$ lie on different sides of the line $AB$. We now draw tangent lines from $A,B$ and $C$ to the circle $q$ and denote the lengths of the respective tangent line segments by $a,b$ and $c$.
Prove that $a+b=c$.
1991 Swedish Mathematical Competition, 1
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.
2006 Estonia Math Open Junior Contests, 4
Does there exist a natural number with the sum of digits of its $ kth$ power being
equal to $ k$, if a) $ k \equal{} 2004$; b) $ k \equal{} 2006?$
2014 South East Mathematical Olympiad, 8
Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions:
$(1)$Square bar $AB$ is bisected by square bar $CD$
$(2)$At least one square of $AB$ lay on both sides of $CD$
$(3)$At least one square of $CD$ lay on both sides of $AB$
There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet