Olympiad problems

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 $a_n = 1$?

2023 Euler Olympiad, Round 1, 6

Tags: euler , algebra

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]

2006 MOP Homework, 4

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

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.

2000 Taiwan National Olympiad, 1

Tags: euler , function , modular arithmetic , quadratic , prime number , number theory

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2009 Indonesia TST, 4

Tags: euler , function , number theory

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 Vietnam National Olympiad, 3

Tags: geometry , incenter , circumcircle , euler , trigonometry

Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB\equal{}\alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ BC$, $ CA$, $ AB$, respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.

Geometry Mathley 2011-12, 14.2

Tags: angle bisector , geometry , euler , euler circle , tangent circle

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

2009 Romania Team Selection Test, 2

Tags: euler , circumcircle , geometry

Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.

2007 Today's Calculation Of Integral, 172

Tags: calculus , integration , trigonometry , euler , calculus computations

Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$

PEN D Problems, 6

Tags: congruence , modular arithmetic , induction , euler , search

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.

2005 Germany Team Selection Test, 3

Tags: geometry , incenter , circumcircle , inequalities , inradius , euler , vector

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.

2000 IMO Shortlist, 1

Tags: number theory , modular arithmetic , euler

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

1989 AIME Problems, 9

Tags: euler , modular arithmetic , number theory

One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5.\] Find the value of $ n$.

2014 AIME Problems, 3

Tags: algorithm , euler , function , number theory , euclidean algorithm , relatively prime , totient function

Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.

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 Iran MO (3rd Round), 5

Tags: circumcircle , euler , projective geometry , geometry , conic

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 $ABC$.

1994 Irish Math Olympiad, 5

Tags: euler

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, 13

Tags: euler , modular arithmetic , inequalities , function , number theory , relatively prime

Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$. Prove that $n \leq 4m(2^m-1)$. Created by Harazi, modified by Marian Andronache.

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}$.

2024 Euler Olympiad, Round 1, 2

Tags: euler , number theory

Given the rebus: $$AB \cdot AC \cdot BC = BBBCCC $$ where different letters correspond to different digits and the same letters to the same digits, find the sum $AB + AC + BC.$ [i]Proposed by Giorgi Arabidze, Georgia [/i]

2007 Nicolae Coculescu, 4

Tags: totient , euler , number theory , divisibility

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]

2005 Iran MO (3rd Round), 2

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

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

1996 All-Russian Olympiad, 5

Tags: ratio , euler , arithmetic sequence , number theory

Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10. [i]L. Kuptsov[/i]

2007 Moldova Team Selection Test, 3

Tags: euler , circumcircle , incenter , geometry

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

2007 China Team Selection Test, 2

Tags: algebra , polynomial , euler , number theory

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}).$