Olympiad problems

2012 Online Math Open Problems, 30

Tags: Online Math Open , algebra , polynomial , induction , Euler , modular arithmetic , number theory

Let $P(x)$ denote the polynomial \[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$. [i]Victor Wang.[/i]

2023 Euler Olympiad, Round 1, 3

Tags: Euler , combinatorics

Leonard has a hand clock with only hour and minute hands. Determine the number of minutes in a day where the angle between the clock hands is not more than 1 degree. Both clock hands move continuously and at a constant speed. [i]Proposed by Giorgi Arabidze, Georgia [/i]

2012 South africa National Olympiad, 5

Tags: geometry , circumcircle , geometric transformation , homothety , ratio , Euler , geometry unsolved

Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.

2013 Harvard-MIT Mathematics Tournament, 36

Tags: HMMT , Euler , email , MIT , college

(Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of $\{A,B,\cdots,Z\}$, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within $20$ years of each other, then your score is $0$. Otherwise, your score is $\max(3(k-3),0)$ where $k$ is the number of elements in your set. \[\begin{tabular}{cc}Niels Abel & Isaac Newton\\Etienne Bezout & Nicole Oresme \\ Augustin-Louis Cauchy & Blaise Pascal \\ Rene Descartes & Daniel Quillen \\ Leonhard Euler & Bernhard Riemann\\ Pierre Fatou & Jean-Pierre Serre \\ Alexander Grothendieck & Alan Turing \\ David Hilbert & Stanislaw Ulam \\ Kenkichi Iwasawa & John Venn \\ Carl Jacobi & Andrew Wiles \\ Andrey Kolmogorov & Leonardo Ximenes \\ Joseph-Louis Lagrange & Shing-Tung Yau \\ John Milnor & Ernst Zermelo\end{tabular}\]

2004 Iran MO (3rd Round), 19

Tags: Euler , number theory proposed , number theory

Find all integer solutions of $ p^3\equal{}p^2\plus{}q^2\plus{}r^2$ where $ p,q,r$ are primes.