Found problems: 85335
2008 Sharygin Geometry Olympiad, 18
(A.Abdullayev, 9--11) Prove that the triangle having sides $ a$, $ b$, $ c$ and area $ S$ satisfies the inequality
\[ a^2\plus{}b^2\plus{}c^2\minus{}\frac12(|a\minus{}b|\plus{}|b\minus{}c|\plus{}|c\minus{}a|)^2\geq 4\sqrt3 S.\]
2012 EGMO, 1
Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.
[i]Netherlands (Merlijn Staps)[/i]
2010 Romania Team Selection Test, 4
Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$.
[i]Marius Cavachi[/i]
2021 Durer Math Competition Finals, 5
A torpedo set consists of $2$ pieces of $1 \times 4$, $4$ pieces of $1 \times 3$, $6$ pieces of $1 \times 2$ and $ 8$ pieces of $1 \times 1$ ships.
a) Can one put the whole set to a $10 \times 10$ table so that the ships do not even touch with corners? (The ships can be placed both horizontally and vertically.)
b) Can we solve this problem if we change $4$ pieces of $1 \times 1$ ships to $3$ pieces of $1 \times 2$ ships?
c) Can we solve the problem if we change the remaining $4$ pieces of $1 \times 1$ ships to one piece of $1 \times 3$ ship and one piece of $1 \times 2$ ship? (So the number of pieces are $2, 5, 10, 0$.)
2012 Danube Mathematical Competition, 1
a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$ and $ca+1$ are simultaneously even perfect squares ?
b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.
2019 Pan-African Shortlist, G1
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
2003 Switzerland Team Selection Test, 4
Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.
2007 IberoAmerican, 6
Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties:
i) $ H$ has parallel opposite sides.
ii) Any 3 vertices of $ H$ can be covered with a strip of width 1.
Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$.
Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.
2005 Kyiv Mathematical Festival, 5
The plane is dissected by broken lines into some regions. It is possible to paint the map formed by these regions in three colours so that any neighbouring regions will have different colours. Call by knots the points which belong to at least two segments of broken lines. One of the segments connecting two knots is erased and replaced by arbitrary broken line connecting the same knots. Prove that it is possible to paint new map in three colours so that any neighbouring regions will have different colours.
2012 Princeton University Math Competition, A3
Let $ABC$ be a triangle with incenter $I$, and let $D$ be the foot of the angle bisector from $A$ to $BC$. Let $\Gamma$ be the circumcircle of triangle $BIC$, and let $PQ$ be a chord of $\Gamma$ passing through $D$. Prove that $AD$ bisects $\angle PAQ$.
2018 Online Math Open Problems, 29
Let $q<50$ be a prime number. Call a sequence of polynomials $P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x)$ [i]tasty[/i] if it satisfies the following conditions:
[list]
[*] $P_i$ has degree $i$ for each $i$ (where we consider constant polynomials, including the $0$ polynomial, to have degree $0$)
[*] The coefficients of $P_i$ are integers between $0$ and $q-1$ for each $i$.
[*] For any $0\le i,j\le q^2$, the polynomial $P_i(P_j(x)) - P_j(P_i(x))$ has all its coefficients divisible by $q$.
[/list]
As $q$ varies over all such prime numbers, determine the total number of tasty sequences of polynomials.
[i]Proposed by Vincent Huang[/i]
2004 National Olympiad First Round, 32
If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
2019 Greece Team Selection Test, 2
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).
Kyiv City MO Seniors 2003+ geometry, 2013.11.3
The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint.
(Igor Nagel)
2002 India National Olympiad, 5
Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).
2023 239 Open Mathematical Olympiad, 7
Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?
2023 VN Math Olympiad For High School Students, Problem 5
Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Let $a=BC, b=CA,c=AB.$
Prove that: ${a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .$
2014 Online Math Open Problems, 9
Let $N = 2014! + 2015! + 2016! + \dots + 9999!$. How many zeros are at the end of the decimal representation of $N$?
[i]Proposed by Evan Chen[/i]
1996 French Mathematical Olympiad, Problem 5
Let $n$ be a positive integer. We say that a natural number $k$ has the property $C_n$ if there exist $2k$ distinct positive integers $a_1,b_1,\ldots,a_k,b_k$ such that the sums $a_1+b_1,\ldots,a_k+b_k$ are distinct and strictly smaller than $n$.
(a) Prove that if $k$ has the property $C_n$ then $k\le \frac{2n-3}{5}$.
(b) Prove that $5$ has the property $C_{14}$.
(c) If $(2n-3)/5$ is an integer, prove that it has the property $C_n$.
2008 Harvard-MIT Mathematics Tournament, 8
Compute $ \arctan\left(\tan65^\circ \minus{} 2\tan40^\circ\right)$. (Express your answer in degrees.)
PEN H Problems, 54
Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.
2007 Hanoi Open Mathematics Competitions, 3
Find the number of di erent positive integer triples (x; y; z) satsfying the equations
x+y-z=1 and $x^2+y^2-z^2=1$.
1961 All Russian Mathematical Olympiad, 004
Given a table $4\times 4$.
a) Find, how $7$ stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars.
b) Prove that if there are less than $7$ stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.
1990 AMC 12/AHSME, 22
If the six solutions of $x^6=-64$ are written in the form $a+bi$, where $a$ and $b$ are real, then the product of those solutions with $a>0$ is
$\text{(A)} \ -2 \qquad \text{(B)} \ 0 \qquad \text{(C)} \ 2i \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 16$
2023 Romania National Olympiad, 3
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.