Found problems: 230
2018 Vietnam Team Selection Test, 1
Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$, resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B',\ C'$ are defined similarly.
a. Prove that if $PO=PO'$ then $O\in(A'B'C')$;
b. Point $A'$ is mirrored by $OD$, its image is $X$. $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.
1990 IMO Longlists, 60
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$
[i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers.
[i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$
2005 Brazil National Olympiad, 5
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.
Prove that these three Euler lines pass through one common point.
[i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$.
[i]Floor van Lamoen[/i]
2003 India IMO Training Camp, 4
There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.
2005 China Team Selection Test, 3
Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\]
Prove that
\[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]
2008 Turkey MO (2nd round), 1
Given an acute angled triangle $ ABC$ , $ O$ is the circumcenter and $ H$ is the orthocenter.Let $ A_1$,$ B_1$,$ C_1$ be the midpoints of the sides $ BC$,$ AC$ and $ AB$ respectively. Rays $ [HA_1$,$ [HB_1$,$ [HC_1$ cut the circumcircle of $ ABC$ at $ A_0$,$ B_0$ and $ C_0$ respectively.Prove that $ O$,$ H$ and $ H_0$ are collinear if $ H_0$ is the orthocenter of $ A_0B_0C_0$
2000 All-Russian Olympiad, 6
A perfect number, greater than $28$ is divisible by $7$. Prove that it is also divisible by $49$.
2011 Gheorghe Vranceanu, 1
Let be a triangle $ ABC $ that's not equilateral, nor right-angled. Let $ A',B',C' $ be the feet of the heights of $ A,B,C, $ respectively. Prove that the Euler's lines of the triangles $ AB'C',BC'A',CA'B' $ meet at one point on the Euler's circle of $ ABC. $
2014 USA TSTST, 1
Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f", the result is "faecdb". We say that a string $B$ is [i]reachable[/i] from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef".
Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.
2010 Postal Coaching, 2
Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively.
$(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$.
$(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .
2010 APMO, 4
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
2010 International Zhautykov Olympiad, 3
Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that
1) $\angle OIH > 90^0$(2 points)
2)$\angle OIH >135^0$(7 points)
balls for 1) and 2) not additive.
2009 Vietnam Team Selection Test, 1
Let an acute triangle $ ABC$ with curcumcircle $ (O)$. Call $ A_1,B_1,C_1$ are foots of perpendicular line from $ A,B,C$ to opposite side. $ A_2,B_2,C_2$ are reflect points of $ A_1,B_1,C_1$ over midpoints of $ BC,CA,AB$ respectively. Circle $ (AB_2C_2),(BC_2A_2),(CA_2B_2)$ cut $ (O)$ at $ A_3,B_3,C_3$ respectively.
Prove that: $ A_1A_3,B_1B_3,C_1C_3$ are concurent.
2014 Bundeswettbewerb Mathematik, 1
Show that for all positive integers $n$, the number $2^{3^n}+1$ is divisible by $3^{n+1}$.
2018 Israel Olympic Revenge, 3
Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O$. The tangent line to from $A$ to $\omega$ intersects $BC$ at $K$. The tangent line to from $B$ to $\omega$ intersects $AC$ at $L$. Let $M,N$ be the midpoints of $AK,BL$ respectively. The line $MN$ is named by $\alpha$. The feet of perpendicular from $A,B,C$ to the edges of $\triangle ABC$ are named by $D,E,F$ respectively. The perpendicular bisectors of $EF,DF,DE$ intersect $\alpha$ at $X,Y,Z$ respectively. Let $AD,BE,CF$ intersect $\omega$ again at $D',E',F'$ respectively. If $H$ is the orthocenter of $ABC$, prove that the lines $XD',YE',ZF',OH$ are concurrent.
2023 Euler Olympiad, Round 2, 3
Let $ABCD$ be a convex quadrilateral with side lengths satisfying the equality:
$$ AB \cdot CD = AD \cdot BC = AC \cdot BD.$$
Determine the sum of the acute angles of quadrilateral $ABCD$.
[i]Proposed by Zaza Meliqidze, Georgia[/i]
2008 Hong kong National Olympiad, 2
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.
2006 IMO Shortlist, 7
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces.
[i]Proposed by Kei Irei, Japan[/i]
2011 AMC 10, 23
What is the hundreds digit of $2011^{2011}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 9 $
1990 IMO Shortlist, 5
Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.
2007 China Team Selection Test, 2
After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$
2013 NIMO Problems, 8
For a finite set $X$ define \[
S(X) = \sum_{x \in X} x \text{ and }
P(x) = \prod_{x \in X} x.
\] Let $A$ and $B$ be two finite sets of positive integers such that $\left\lvert A \right\rvert = \left\lvert B \right\rvert$, $P(A) = P(B)$ and $S(A) \neq S(B)$. Suppose for any $n \in A \cup B$ and prime $p$ dividing $n$, we have $p^{36} \mid n$ and $p^{37} \nmid n$. Prove that \[ \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}. \][i]Proposed by Evan Chen[/i]
1990 Balkan MO, 3
Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$. Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.
2023 Sharygin Geometry Olympiad, 10.2
The Euler line of a scalene triangle touches its incircle. Prove that this triangle is obtuse-angled.
2005 Romania Team Selection Test, 2
On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out.
Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit.
[i]Dan Schwartz[/i]