Olympiad problems

2013 All-Russian Olympiad, 3

Tags: geometry , circumcircle , geometric transformation , homothety , Euler , symmetry , geometry proposed

The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $. [i]L. Emelyanov, A. Polyansky[/i]

2013 China Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , homothety , trigonometry , Euler , geometry proposed

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

2011 Serbia National Math Olympiad, 2

Tags: Euler , number theory proposed , number theory

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 $n=5$.

2007 India IMO Training Camp, 1

Tags: Euler , geometry , trigonometry , perpendicular bisector , geometry unsolved

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.

1964 AMC 12/AHSME, 35

Tags: ratio , geometry , trigonometry , Euler , circumcircle , area of a triangle , Heron's formula

The sides of a triangle are of lengths $13$, $14$, and $15$. The altitudes of the triangle meet at point $H$. If $AD$ is the altitude to the side length $14$, what is the ratio $HD:HA$? $\textbf{(A) } 3 : 11\qquad \textbf{(B) } 5 : 11\qquad \textbf{(C) } 1 : 2\qquad \textbf{(D) }2 : 3\qquad \textbf{(E) }25 : 33$

2005 China Team Selection Test, 3

Tags: Euler , floor function , number theory unsolved , number theory

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 \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

2024 Euler Olympiad, Round 2, 1

Tags: Euler , Diophantine equation , number theory

Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]

1996 Balkan MO, 1

Tags: inequalities , geometry , circumcircle , incenter , Euler , inequalities proposed

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]

2014 Contests, 3

Tags: Euler , modular arithmetic , combinatorial geometry , combinatorics proposed , combinatorics , Discrete intermediate value theorem

For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$

India EGMO 2024 TST, 1

Tags: Euler , geometry , Egmo tst , circumcircle

Let $ABC$ be a triangle with circumcentre $O$ and centroid $G$. Let $M$ be the midpoint of $BC$ and $N$ the reflection of $M$ across $O$. Prove that $NO = NA$ if and only if $\angle AOG = 90^{\circ}$. [i]Proposed by Pranjal Srivastava[/i]

2023 Euler Olympiad, Round 1, 4

Tags: Euler , algebra

Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$. [i]Proposed by Gogi Khimshiashvili, Georgia[/i]

2023 Euler Olympiad, Round 2, 4

Tags: trapezoid , Euler , right angle , geometry , harmonic bundles , Georgia

Let $ABCD$ be a trapezoid, with $AD \parallel BC$, let $M$ be the midpoint of $AD$, and let $C_1$ be symmetric point to $C$ with respect to line $BD$. Segment $BM$ meets diagonal $AC$ at point $K$, and ray $C_1K$ meets line $BD$ at point $H$. Prove that $\angle{AHD}$ is a right angle. [i]Proposed by Giorgi Arabidze, Georgia[/i]

2006 IMO Shortlist, 7

Tags: geometry , 3D geometry , Euler , polyhedron , IMO Shortlist

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]

2013 JBMO Shortlist, 3

Tags: geometry , circumcircle , trigonometry , analytic geometry , graphing lines , slope , Euler

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

2003 Silk Road, 2

Tags: geometry , perimeter , circumcircle , incenter , Euler , parallelogram , trapezoid

Let $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$. Let $L$ and $N$be a point's on ray's $AB$ and $CB$, for which $AL=CN=s$. Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$. Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$. Solution for problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2012 BMT Spring, 8

Tags: modular arithmetic , Euler , probability , expected value

You are tossing an unbiased coin. The last $ 28 $ consecutive ﬂips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $.

2005 China Team Selection Test, 2

Tags: geometry , incenter , ratio , Euler , geometry unsolved

In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.

1998 South africa National Olympiad, 2

Tags: trigonometry , geometry , perimeter , inequalities , Euler , trig identities , Law of Sines

Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.

2008 National Olympiad First Round, 14

Tags: Euler , modular arithmetic

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 $

2000 239 Open Mathematical Olympiad, 7

Tags: geometry , incenter , circumcircle , Euler , angle bisector , perpendicular bisector , geometry proposed

The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC. [i]Extension:[/i] Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC. Darij

1990 IMO Shortlist, 5

Tags: geometry , incenter , vector , Euler , trigonometry , orthocenter , IMO Shortlist

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}$.

1972 Canada National Olympiad, 5

Tags: modular arithmetic , Euler

Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$.

2000 Federal Competition For Advanced Students, Part 2, 1

Tags: Euler , geometry , circumcircle , angle bisector , geometry proposed

In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

1970 IMO Longlists, 2

Tags: modular arithmetic , Euler , number theory unsolved , number theory

Prove that the two last digits of $9^{9^{9}}$ and $9^{9^{9^{9}}}$ are the same in decimal representation.

2021 Taiwan TST Round 2, G

Tags: Euler , geometry , circumcircle , Taiwan

Let $ABCD$ be a convex quadrilateral with pairwise distinct side lengths such that $AC\perp BD$. Let $O_1,O_2$ be the circumcenters of $\Delta ABD, \Delta CBD$, respectively. Show that $AO_2, CO_1$, the Euler line of $\Delta ABC$ and the Euler line of $\Delta ADC$ are concurrent. (Remark: The [i]Euler line[/i] of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.) [i]Proposed by usjl[/i]