Olympiad problems

2007 China Team Selection Test, 3

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}$.

2011 Gheorghe Vranceanu, 1

Tags: euler , euler s circle , pure geometry , euler line , geometry

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. $

2022 Taiwan TST Round 1, 1

Tags: geometry , power of a point , euler , circumcircle

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 $B'C'.$

2013 Online Math Open Problems, 19

Tags: function , number theory , euler , geometric series , relatively prime

Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i]

1950 Miklós Schweitzer, 7

Tags: logarithm , calculus , function , euler , real analysis , integration

Examine the behavior of the expression $ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$ as $ n\rightarrow \infty$

2003 India IMO Training Camp, 4

Tags: euler , geometry

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.

1966 IMO Shortlist, 15

Tags: geometry , euler , perpendicular , circles

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 $AB.$

2008 Turkey MO (2nd round), 1

Tags: symmetry , euler , ratio , homothety , circumcircle , geometric transformation , geometry

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$

2011 Serbia National Math Olympiad, 2

Tags: number theory , euler

Let $n$ be an odd positive integer such that both $\phi(n)$ and $\phi (n+1)$ are powers of two. Prove $n+1$ is power of two or $n=5$.

PEN M Problems, 4

Tags: euler , recursive sequences

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}$.

2011 India Regional Mathematical Olympiad, 1

Tags: parallelogram , trigonometry , euler , circumcircle , incenter , geometry , geometric transformation

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 $ABC.$

2005 Iran MO (3rd Round), 2

Tags: circumcircle , radical axis , geometry , euler , power of a point

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 \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]

2024 Euler Olympiad, Round 1, 8

Tags: euler , square , geometry

Let $P$ be a point inside a square $ABCD,$ such that $\angle BPC = 135^\circ $ and the area of triangle $ADP$ is twice as much as the area of triangle $PCD.$ Find $\frac {AP}{DP}.$ [i]Proposed by Andria Gvaramia, Georgia [/i]

1996 Balkan MO, 1

Tags: circumcircle , inequalities , euler , incenter , geometry

Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$. If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively, prove that \[ OG \leq \sqrt{ R ( R - 2r ) } . \] [i]Greece[/i]

2010 Malaysia National Olympiad, 9

Tags: modular arithmetic , euler , number theory

Let $m$ and $n$ be positive integers such that $2^n+3^m$ is divisible by $5$. Prove that $2^m+3^n$ is divisible by $5$.

2009 CHKMO, 2

Tags: algebra , polynomial , number theory , euler , vieta , quadratic , function

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.

2014 Bundeswettbewerb Mathematik, 1

Tags: modular arithmetic , euler

Show that for all positive integers $n$, the number $2^{3^n}+1$ is divisible by $3^{n+1}$.

2008 National Olympiad First Round, 14

Tags: modular arithmetic , euler

What is the last three digits of $49^{303}\cdot 3993^{202}\cdot 39^{606}$? $ \textbf{(A)}\ 001 \qquad\textbf{(B)}\ 081 \qquad\textbf{(C)}\ 561 \qquad\textbf{(D)}\ 721 \qquad\textbf{(E)}\ 961 $

2006 Turkey Team Selection Test, 2

Tags: power of a point , geometry , euler , angle bisector , radical axis

From a point $Q$ on a circle with diameter $AB$ different from $A$ and $B$, we draw a perpendicular to $AB$, $QH$, where $H$ lies on $AB$. The intersection points of the circle of diameter $AB$ and the circle of center $Q$ and radius $QH$ are $C$ and $D$. Prove that $CD$ bisects $QH$.

2023 Euler Olympiad, Round 1, 5

Tags: combinatorics , euler

Consider a 3 × 4 rectangular table where each cell can be colored using one of three available colors. Determine the number of different ways the table can be colored such that no two cells sharing a common side have the same color. It is not necessary to use all three colors in each coloring. [i]Proposed by Prudencio Guerrero Fernández, Cuba[/i]

2005 National Olympiad First Round, 10

Tags: euler , modular arithmetic

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 $

2004 All-Russian Olympiad, 2

Tags: circumcircle , geometric transformation , rectangle , euler , trapezoid , incenter , geometry

Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$

2008 Baltic Way, 9

Tags: number theory , euler

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.

2010 Contests, 4

Tags: geometry , euler , trapezoid , reflection , geometric transformation , circumcircle , incenter

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 $OH$.

2024 Euler Olympiad, Round 1, 5

Tags: euler , geometry , ratio

Let $ABCDEF$ be a convex hexagon. Segments $AE$ and $BF$ intersect at $X$ and segments $BD$ and $CE$ intersect in $Y.$ It's known that $$ \angle XBC = \angle XDE = \angle YAB = \angle YEF = 80^\circ \text{ and } \angle XCB = \angle XED = \angle YBA = \angle YFE = \angle 70^\circ.$$ Let $P$ and $Q$ be such points on line $XY$ that segments $PX$ and $AF$ intersect, segments $QY$ and $CD$ intersect and $\angle APF = \angle CQD = 30 ^\circ.$ Estimate the sum: \[ \frac{BX}{BF} + \frac{BY}{BD} + \frac{EX}{EA} + \frac{EY}{EC} + \frac{PX}{PY} + \frac{QY}{QX} \] [i]Proposed by Gogi Khimshiashvili, Georgia [/i]