Found problems: 230
2005 APMO, 5
In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$.
2005 ITAMO, 2
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and
\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\]
For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?
PEN A Problems, 10
Let $n$ be a positive integer with $n \ge 3$. Show that \[n^{n^{n^{n}}}-n^{n^{n}}\] is divisible by $1989$.
2011 Romania Team Selection Test, 2
In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.
2008 iTest Tournament of Champions, 3
For how many integers $1\leq n\leq 9999$ is there a solution to the congruence \[\phi(n)\equiv 2\,\,\,\pmod{12},\] where $\phi(n)$ is the Euler phi-function?
2018 CMIMC Number Theory, 6
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$.
2006 MOP Homework, 4
Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a$, P_b$, and $P_c$ lie on a line.
2023 Euler Olympiad, Round 1, 2
A student took a rectangular piece of paper with length equal to one meter and width equal to five centimeters. The student brought the ends together, turning one end 180 degrees and gluing the surfaces to create a figure called a Möbius strip. On one side of this strip, the student placed a flea and an ant. It is known that if the flea and the ant move in different directions on the Möbius strip, they will meet each other in 2 minutes. However, if they move in the same direction, they will meet in 7 minutes. Given that the flea is faster than the ant and both move at constant speeds, determine the speed of the flea.
[i]Proposed by Lia Chitishvili, Georgia[/i]
2023 Euler Olympiad, Round 1, 6
Given a rebus:
$$AB + BC + CA = XY + YZ + ZX = KL + LM + MK $$
where different letters correspond to different numbers, and same letters correspond to the same numbers. Determine the value of $ AXK + BYL + CZM $.
[i]Proposed by Giorgi Arabidze[/i]
2007 India IMO Training Camp, 1
Show that in a non-equilateral triangle, the following statements are equivalent:
$(a)$ The angles of the triangle are in arithmetic progression.
$(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.
PEN E Problems, 11
In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$. Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.
1991 IMO Shortlist, 10
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
[b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
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.
2004 Iran MO (3rd Round), 11
assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle at A" in the point $X_a$; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.
1994 APMO, 2
Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.
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]
1991 Brazil National Olympiad, 4
Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).
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 $
1970 IMO Longlists, 22
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
PEN N Problems, 8
An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.
2024 Euler Olympiad, Round 2, 2
Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations
\begin{align*}
f(x+y) &= f(x) f(y) + g(x) g(y) \\
g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y)
\end{align*}
holds for all rational numbers $x$ and $y.$
[i]Proposed by Gurgen Asatryan, Armenia [/i]
1994 Irish Math Olympiad, 5
If a square is partitioned into $ n$ convex polygons, determine the maximum possible number of edges in the obtained figure.
(You may wish to use the following theorem of Euler: If a polygon is partitioned into $ n$ polygons with $ v$ vertices and $ e$ edges in the resulting figure, then $ v\minus{}e\plus{}n\equal{}1$.)
2004 Romania Team Selection Test, 15
Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white.
Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$.
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality
\[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\]
and determine when equality holds.
2020 USEMO, 3
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.
Proposed by Anant Mudgal