Olympiad problems

2009 Harvard-MIT Mathematics Tournament, 6

Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2}n$ for any integer $n$. Suppose \[\frac{\sin\theta}{x}=\frac{\cos\theta}{y}\] and \[ \frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}. \] Compute $\frac xy+\frac yx.$

2009 Croatia Team Selection Test, 1

Tags: inequalities , LaTeX , inequalities unsolved

Prove for all positive reals a,b,c,d: $ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$

2007 AIME Problems, 4

Tags: LaTeX , AMC , AIME II

The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Find m.

2008 India Regional Mathematical Olympiad, 3

Tags: inequalities , Vieta , calculus , derivative , trigonometry , LaTeX , algebra

Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that: \[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3} \] [19 points out of 100 for the 6 problems]

1997 Finnish National High School Mathematics Competition, 2

Tags: LaTeX , geometry unsolved , geometry

Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn. This tangent and the circles bound a region inside which a circle as large as possible is drawn. What is the radius of this circle?

2005 ISI B.Stat Entrance Exam, 2

Tags: integration , calculus , function , LaTeX , derivative , algebra , polynomial

Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

2013 Romania National Olympiad, 1

Tags: LaTeX , induction , arithmetic sequence , algebra proposed , algebra

A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.

PEN H Problems, 54

Tags: calculus , integration , ratio , geometry , perimeter , quadratics , LaTeX

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.

2013 National Olympiad First Round, 23

Tags: LaTeX

If the conditions \[\begin{array}{rcl} f(2x+1)+g(3-x) &=& x \\ f((3x+5)/(x+1))+2g((2x+1)/(x+1)) &=& x/(x+1) \end{array}\] hold for all real numbers $x\neq 1$, what is $f(2013)$? $ \textbf{(A)}\ 1007 \qquad\textbf{(B)}\ \dfrac {4021}{3} \qquad\textbf{(C)}\ \dfrac {6037}7 \qquad\textbf{(D)}\ \dfrac {4029}{5} \qquad\textbf{(E)}\ \text{None of above} $

2000 China Team Selection Test, 3

Tags: function , LaTeX , algebra unsolved , algebra

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

1987 China Team Selection Test, 2

Tags: LaTeX , number theory unsolved , number theory

Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.

2006 Team Selection Test For CSMO, 1

Tags: LaTeX , modular arithmetic , quadratics , number theory , relatively prime , algebra , difference of squares

Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities , LaTeX , inequalities solved

0<a,b,c<1 ==> \sqrt (abc) + \sqrt (1-a)(1-b)(1-c) <1

PEN O Problems, 31

Tags: induction , ratio , LaTeX , geometric sequence

Prove that, for any integer $a_{1}>1$, there exist an increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, \cdots$ such that \[a_{1}+a_{2}+\cdots+a_{n}\; \vert \; a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\] for all $n \in \mathbb{N}$.

2005 Romania National Olympiad, 2

Tags: function , LaTeX , algebra proposed , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[ x(f(x+1)-f(x)) = f(x), \] for all $x\in\mathbb{R}$ and \[ | f(x) - f(y) | \leq |x-y| , \] for all $x,y\in\mathbb{R}$. [i]Mihai Piticari[/i]

1994 Polish MO Finals, 2

Tags: inequalities , vector , LaTeX , geometry , parallelogram , triangle inequality , geometry solved

A parallelopiped has vertices $A_1, A_2, ... , A_8$ and center $O$. Show that: \[ 4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2 \]

2007 AMC 12/AHSME, 17

Tags: trigonometry , AMC , AMC 12 , LaTeX , function

Suppose that $ \sin a \plus{} \sin b \equal{} \sqrt {\frac {5}{3}}$ and $ \cos a \plus{} \cos b \equal{} 1.$ What is $ \cos(a \minus{} b)?$ $ \textbf{(A)}\ \sqrt {\frac {5}{3}} \minus{} 1 \qquad \textbf{(B)}\ \frac {1}{3}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {2}{3}\qquad \textbf{(E)}\ 1$

2003 USAMO, 6

Tags: AMC , USA(J)MO , USAMO , LaTeX , invariant , induction , symmetry

At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.

1984 AMC 12/AHSME, 28

Tags: LaTeX

The number of distinct pairs of integers $(x,y)$ such that \[0 < x < y\quad \text{and}\quad \sqrt{1984} = \sqrt{x} + \sqrt{y}\] is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }7$

2011 Turkey MO (2nd round), 4

Tags: LaTeX , modular arithmetic , number theory unsolved , number theory , legendre s theorem

$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.

2014 Puerto Rico Team Selection Test, 2

Tags: LaTeX

We have shortened the usual notation indicating with a sub-index the number of times that a digit is conseutively repeated. For example, $1119900009$ is denoted $1_3 9_2 0_4 9_1$. Find $(x, y, z)$ if $2_x 3_y 5_z + 3_z 5_x 2_y = 5_3 7_2 8_3 5_1 7_3$

2015 AMC 10, 23

Tags: factorial , modular arithmetic , floor function , LaTeX , AMC

Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? $ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $

2015 AMC 12/AHSME, 5

Tags: LaTeX , AMC , no posts

The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$? $\textbf{(A) }35\qquad\textbf{(B) }37\qquad\textbf{(C) }39\qquad\textbf{(D) }41\qquad\textbf{(E) }43$

2022 AMC 8 -, 2

Tags: AMC 8 , LaTeX

Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$ $\textbf{(A) } {-}20\qquad\textbf{(B) } 4\qquad\textbf{(C) } 16\qquad\textbf{(D) } 100\qquad\textbf{(E) } 220$

2013 NIMO Problems, 2

Tags: function , modular arithmetic , LaTeX

Let $f$ be a function from positive integers to positive integers where $f(n) = \frac{n}{2}$ if $n$ is even and $f(n) = 3n+1$ if $n$ is odd. If $a$ is the smallest positive integer satisfying \[ \underbrace{f(f(\cdots f}_{2013\ f\text{'s}} (a)\cdots)) = 2013, \] find the remainder when $a$ is divided by $1000$. [i]Based on a proposal by Ivan Koswara[/i]