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.
2010 Romania Team Selection Test, 2
Let $ABC$ be a scalene triangle. The tangents at the perpendicular foot dropped from $A$ on the line $BC$ and the midpoint of the side $BC$ to the nine-point circle meet at the point $A'$\,; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent.
[i]Gazeta Matematica[/i]
1990 IMO Longlists, 15
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}$.
2008 All-Russian Olympiad, 3
In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.
2021 Sharygin Geometry Olympiad, 20
The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)
[b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$.
[b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point.
Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.
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$
2005 Iran MO (3rd Round), 5
Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.
2008 Iran MO (3rd Round), 2
Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.
2009 CHKMO, 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.
2008 Baltic Way, 9
Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b\minus{}b^a\equal{}1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.
2004 China Western Mathematical Olympiad, 4
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$).
Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.
2013 NIMO Problems, 4
Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Lewis Chen[/i]
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.
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. $
2011 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$
2023 Euler Olympiad, Round 2, 5
Find the smallest constant M, so that for any real numbers $a_1, a_2, \dots a_{2023} \in [4, 6]$ and $b_1, b_2, \dots b_{2023} \in [9, 12] $ following inequality holds:
$$ \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots + a_{2023} b_{2023} \right) $$
[i]Proposed by Zaza Meliqidze, Georgia[/i]
1984 Balkan MO, 3
Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.
2008 Sharygin Geometry Olympiad, 14
(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this
angle (There was an error in published condition of this problem).
MathLinks Contest 7th, 1.1
Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously.
Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$.
1966 IMO Shortlist, 15
Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$
Geometry Mathley 2011-12, 14.2
The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$.
Đỗ Thanh Sơn
1976 IMO Longlists, 15
Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.
2003 Canada National Olympiad, 2
Find the last three digits of the number $2003^{{2002}^{2001}}$.
2014 Contests, 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 $A$.
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}).$