Olympiad problems

1992 AIME Problems, 11

Tags: geometry , geometric transformation , reflection , rotation , modular arithmetic

Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{19}{92}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.

2007 Putnam, 1

Tags: symmetry , analytic geometry , graphing lines , slope , quadratic , geometry

Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.

2017 Estonia Team Selection Test, 1

Tags: number theory , power of 5

Do there exist two positive powers of $5$ such that the number obtained by writing one after the other is also a power of $5$?

2008 JBMO Shortlist, 3

Tags: sum of digits , number theory

Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.

2013-2014 SDML (High School), 1

Tags: factorial

What is the smallest integer $m$ such that $\frac{10!}{m}$ is a perfect square? $\text{(A) }2\qquad\text{(B) }7\qquad\text{(C) }14\qquad\text{(D) }21\qquad\text{(E) }35$

1981 AMC 12/AHSME, 28

Tags:

Consider the set of all equations $ x^3 \plus{} a_2x^2 \plus{} a_1x \plus{} a_0 \equal{} 0$, where $ a_2$, $ a_1$, $ a_0$ are real constants and $ |a_i| < 2$ for $ i \equal{} 0,1,2$. Let $ r$ be the largest positive real number which satisfies at least one of these equations. Then $ \textbf{(A)}\ 1 < r < \frac{3}{2}\qquad \textbf{(B)}\ \frac{3}{2} < r < 2\qquad \textbf{(C)}\ 2 < r < \frac{5}{2}\qquad \textbf{(D)}\ \frac{5}{2} < r < 3\qquad \\ \textbf{(E)}\ 3 < r < \frac{7}{2}$

2004 South East Mathematical Olympiad, 1

Tags: inequalities

Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.

1991 Iran MO (2nd round), 1

Tags: number theory

Prove that the equation $x+x^2=y+y^2+y^3$ do not have any solutions in positive integers.

2021 Nigerian Senior MO Round 2, 1

Tags: geometry , science olympiad

If $x$,$y$ and $z$ are the lengths of a side, a shortest diagonal and a longest diagonal respectively, of a regular nonagon. Write a correct equation consisting of the three lengths

2017 Online Math Open Problems, 25

Tags:

A [i]simple hyperplane[/i] in $\mathbb{R}^4$ has the form \[k_1x_1+k_2x_2+k_3x_3+k_4x_4=0\] for some integers $k_1,k_2,k_3,k_4\in \{-1,0,1\}$ that are not all zero. Find the number of regions that the set of all simple hyperplanes divide the unit ball $x_1^2+x_2^2+x_3^2+x_4^2\leq 1$ into. [i]Proposed by Yannick Yao[/i]

2009 Singapore MO Open, 1

Tags: geometry

let $O$ be the center of the circle inscribed in a rhombus ABCD. points E,F,G,H are chosen on sides AB, BC, CD, DA respectively so that EF and GH are tangent to inscribed circle. show that EH and FG are parallel.

Kharkiv City MO Seniors - geometry, 2019.10.5

Tags: excenter , incenter , angle bisector , midpoint , geometry

In triangle $ABC$, point$ I$ is incenter , $I_a$ is the $A$-excenter. Let $K$ be the intersection point of the $BC$ with the external bisector of the angle $BAC$, and $E$ be the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$. Prove that $K$ is the orthocenter of triangle $II_aE$.

2013 Putnam, 2

Tags: geometry , 3d geometry

Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which: (i) $f(x)\ge 0$ for all real $x,$ and (ii) $a_n=0$ whenever $n$ is a multiple of $3.$ Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.

2006 AMC 10, 15

Tags:

Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other? $ \textbf{(A) } 29 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 47 \qquad \textbf{(E) } 50$

2010 Contests, 1

Tags: induction , geometry , geometric transformation , reflection , symmetry , combinatorics

For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically. Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$. (a) Find all $n$ such that $f(n)=n$. (b) Find all $n$ such that $f(n) = n+1$.

2012 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , rational , equation

Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.

Swiss NMO - geometry, 2006.7

Tags: geometry , equal segments , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$

2000 Mongolian Mathematical Olympiad, Problem 1

Tags: number theory

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

2001 Manhattan Mathematical Olympiad, 1

Tags:

Find all integer solutions to the equation \[ x^2 + y^2 + z^2 = 2xyz \]

VII Soros Olympiad 2000 - 01, 8.3

Tags: number theory , divisible , algebra

Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

1979 Spain Mathematical Olympiad, 8

Tags: real analysis , derivative , calculus

Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$ Express the numerical value of its derivative of order $325$ for $x = 0$.

2018 Purple Comet Problems, 3

Tags: algebra

Find $x$ so that the arithmetic mean of $x, 3x, 1000$, and $3000$ is $2018$.

1962 Miklós Schweitzer, 7

Tags: function , integration , trigonometry , calculus , real analysis

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

2008 Hong Kong TST, 2

Tags: modular arithmetic , system of equations , algebra

Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]

2022 AMC 10, 13

Tags: triangle , geometry

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC$. The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D$. Suppose $BP = 2$ and $PC = 3$. What is $AD$ ? $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$