Found problems: 230
2007 Nicolae Coculescu, 4
Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient.
[i]Cristinel Mortici[/i]
2024 Euler Olympiad, Round 1, 6
On a river with a current speed of \(3 \, \text{km/h}\), there are two harbors. Every Saturday, a cruise ship departs from Harbor 1 to Harbor 2, stays overnight, and returns to Harbor 1 on Sunday. On the ship live two snails, Romeo and Juliet. One Saturday, immediately after the ship departs, both snails start moving to meet each other and do so exactly when the ship arrives at Harbor 2. On the following Sunday, as the ship departs from Harbor 2, Romeo starts moving towards Juliet's house and reaches there exactly when the ship arrives back at Harbor 1. Given that Juliet moves half as fast as Romeo, determine the speed of the ship in still water.
[i]Proposed by Demetre Gelashvili, Georgia [/i]
1990 IMO Shortlist, 17
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 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
\[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]
2023 Sharygin Geometry Olympiad, 10.2
The Euler line of a scalene triangle touches its incircle. Prove that this triangle is obtuse-angled.
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.
2024 Euler Olympiad, Round 2, 5
Consider a circle with an arc \(AB\) and a point \(C\) on this arc. Let \(D\) be the midpoint of arc \(BC\) and \(M\) the midpoint of chord \(AD\). Suppose the tangent lines to the circle at point \(D\) intersect the ray \(AC\) at point \(K\). Prove that the areas of triangle \(MBD\) and quadrilateral \(MCKD\) are equal if and only if the measure of arc \(AB\) is \(180^\circ\).
[i]Proposed by Irakli Shalibashvili, Georgia [/i]
2009 Romania Team Selection Test, 3
Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.
PEN E Problems, 7
Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.
2005 Iran MO (3rd Round), 2
Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]
1988 AIME Problems, 12
Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.
[asy]
size(200);
defaultpen(fontsize(10));
pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C);
draw(D--A--B--C--A^^B--E^^C--F);
dot(A^^B^^C^^P);
label("$a$", P--A, dir(-90)*dir(P--A));
label("$b$", P--B, dir(90)*dir(P--B));
label("$c$", P--C, dir(90)*dir(P--C));
label("$d$", P--D, dir(90)*dir(P--D));
label("$d$", P--E, dir(-90)*dir(P--E));
label("$d$", P--F, dir(-90)*dir(P--F));
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$P$", P, 1.8*dir(285));[/asy]
2014 NIMO Problems, 14
Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.
Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
2025 Kyiv City MO Round 2, Problem 3
On sides \( AB \) and \( AC \) of an acute-angled, non-isosceles triangle \( ABC \), points \( P \) and \( Q \) are chosen such that the center \( O_9 \) of the nine-point circle of \( \triangle ABC \) is the midpoint of segment \( PQ \). Let \( O \) be the circumcenter of \( \triangle ABC \). On the ray \( OP \) beyond \( P \), segment \( PX \) is marked such that \( PX = AQ \). On the ray \( OQ \) beyond \( Q \), segment \( QY \) is marked such that \( QY = AP \). Prove that the midpoint of side \( BC \), the midpoint of segment \( XY \), and the point \( O_9 \) are collinear.
[i]The nine-point circle or the Euler circle[/i] of \( \triangle ABC \) is the circle passing through nine significant points of the triangle — the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments connecting the orthocenter with the vertices of \( \triangle ABC \).
[i]Proposed by Danylo Khilko[/i]
PEN M Problems, 4
The sequence $ \{a_{n}\}_{n \ge 1}$ is defined by \[ a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).\] Show that $ a_{n}$ is an integer for all $ n$, and show that $ n|a_{n}$ for every $ n\in\mathbb{N}$.
2010 Contests, 2
Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$
2014 CHKMO, 4
Let $\triangle ABC$ be a scalene triangle, and let $D$ and $E$ be points on sides $AB$ and $AC$ respectively such that the circumcircles of triangles $\triangle ACD$ and $\triangle ABE$ are tangent to $BC$. Let $F$ be the intersection point of $BC$ and $DE$. Prove that $AF$ is perpendicular to the Euler line of $\triangle ABC$.
2009 Kazakhstan National Olympiad, 2
Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively.
Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively.
Prove, that $I, O, H$ lies on one line.
2004 239 Open Mathematical Olympiad, 5
The incircle of triangle $ABC$ touches its sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. The point $B_2$ is symmetric to $B_1$ with respect to line $A_1C_1$, lines $BB_2$ and $AC$ meet in point $B_3$. points $A_3$ and $C_3$ may be defined analogously. Prove that points $A_3, B_3$ and $C_3$ lie on a line, which passes through the circumcentre of a triangle $ABC$.
[b]
proposed by L. Emelyanov[/b]
2008 APMO, 1
Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.
2015 USA TSTST, 2
Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)
[i]Proposed by Ivan Borsenco[/i]
2005 National Olympiad First Round, 10
Which of the following does not divide $n^{2225}-n^{2005}$ for every integer value of $n$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 23
$
1974 IMO Longlists, 41
Through the circumcenter $O$ of an arbitrary acute-angled triangle, chords $A_1A_2,B_1B_2, C_1C_2$ are drawn parallel to the sides $BC,CA,AB$ of the triangle respectively. If $R$ is the radius of the circumcircle, prove that
\[A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.\]
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 Indonesia TST, 4
Given positive integer $ n > 1$ and define
\[ S \equal{} \{1,2,\dots,n\}.
\]
Suppose
\[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}.
\]
Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)
2009 Indonesia TST, 4
Given positive integer $ n > 1$ and define
\[ S \equal{} \{1,2,\dots,n\}.
\]
Suppose
\[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}.
\]
Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)