Olympiad problems

2002 AMC 12/AHSME, 10

Tags: function , trigonometry

Let $f_n(x)=\sin^n x + \cos^n x$. For how many $x$ in $[0,\pi]$ is it true that \[6f_4(x)-4f_6(x)=2f_2(x)?\] $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{more than 8}$

2020 USMCA, 2

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Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster [i]consumes[/i] cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many minutes does it take Oscar the Grouch to bake one cookie?

2019 Czech-Polish-Slovak Junior Match, 1

Tags: integer , number theory , rational

Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.

2013 China Western Mathematical Olympiad, 8

Tags: induction , modular arithmetic , number theory

Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.

1951 AMC 12/AHSME, 16

Tags: quadratic , quadratic formula , algebra

If in applying the quadratic formula to a quadratic equation \[ f(x)\equiv ax^2 \plus{} bx \plus{} c \equal{} 0, \] it happens that $ c \equal{} \frac {b^2}{4a}$, then the graph of $ y \equal{} f(x)$ will certainly: $ \textbf{(A)}\ \text{have a maximum} \qquad\textbf{(B)}\ \text{have a minimum} \qquad\textbf{(C)}\ \text{be tangent to the x \minus{} axis} \\ \qquad\textbf{(D)}\ \text{be tangent to the y \minus{} axis} \qquad\textbf{(E)}\ \text{lie in one quadrant only}$

2001 Nordic, 2

Tags: functional equation , periodic function , bounded , function , algebra

Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.

1987 Austrian-Polish Competition, 2

Tags: prime divisor , polynomial , algebra

Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.

2014 BMO TST, 4

Tags: function , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

2018 ELMO Shortlist, 1

Tags: function , algebra , functional equation

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

2012 China Western Mathematical Olympiad, 2

Tags: geometry , geometric transformation , reflection

Show that among any $n\geq 3$ vertices of a regular $(2n-1)$-gon we can find $3$ of them forming an isosceles triangle.

2023 Polish Junior Math Olympiad Finals, 5.

Tags: number theory

Find all pairs of positive integers $m$, $n$ such that the $(m+n)$-digit number \[\underbrace{33\ldots3}_{m}\underbrace{66\ldots 6}_{n}\] is a perfect square.

2010 AMC 10, 20

Tags: 3d geometry , geometry

A fly trapped inside a cubical box with side length $ 1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path? $ \textbf{(A)}\ 4 \plus{} 4\sqrt2 \qquad \textbf{(B)}\ 2 \plus{} 4\sqrt2 \plus{} 2\sqrt3 \qquad \textbf{(C)}\ 2 \plus{} 3\sqrt2 \plus{} 3\sqrt3 \qquad \textbf{(D)}\ 4\sqrt2 \plus{} 4\sqrt3 \\ \textbf{(E)}\ 3\sqrt2 \plus{} 5\sqrt3$

1994 AMC 12/AHSME, 27

Tags: abstract algebra , probability

A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} $

2024 India IMOTC, 9

Tags: functional equation , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $a, b, c$, we have \[ f(a+b+c)f(ab+bc+ca) - f(a)f(b)f(c) = f(a+b)f(b+c)f(c+a). \] [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

1999 Czech And Slovak Olympiad IIIA, 1

Tags: min , integer polynomial , algebra , fraction

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

2023 Polish Junior Math Olympiad First Round, 4.

Tags: combinatorics

Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that: – every red number is a sum of two distinct blue numbers; and – every blue number is a difference between two red numbers. Determine the smallest possible value of $n$ for which such a coloring exists.

2018 India IMO Training Camp, 1

Tags: sorting , distance , extremal combinatorics , radius , string , combinatorics

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

1988 IMO Longlists, 50

Tags: rectangle , combinatorics , geometry

Prove that the numbers $A,B$ and $C$ are equal, where: - $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles. - $B=$ number of sequences of ones and twos that add up to $n$ - $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and - $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$

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]

2001 Polish MO Finals, 2

Tags: geometry , parallelogram , geometric transformation , reflection , ratio , analytic geometry , angle bisector

Let $ABCD$ be a parallelogram and let $K$ and $L$ be points on the segments $BC$ and $CD$, respectively, such that $BK\cdot AD=DL\cdot AB$. Let the lines $DK$ and $BL$ intersect at $P$. Show that $\measuredangle DAP=\measuredangle BAC$.

2018 Bulgaria EGMO TST, 3

Tags: induction , function , algebra

Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds: \[ f(f(n)) \leq \frac {n+f(n)} 2 . \]

2009 F = Ma, 12

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Batman, who has a mass of $\text{M = 100 kg}$, climbs to the roof of a $\text{30 m}$ building and then lowers one end of a massless rope to his sidekick Robin. Batman then pulls Robin, who has a mass of $\text{m = 75 kg}$, up the roof of the building. Approximately how much total work has Batman done after Robin is on the roof? (A) $\text{60 J}$ (B) $\text{7} \times \text{10}^3 \text{J}$ (C) $\text{5} \times \text{10}^4 \text{J}$ (D) $\text{600 J}$ (E) $\text{3} \times \text{10}^4 \text{J}$

1983 Swedish Mathematical Competition, 6

Tags: system of equations , system , algebra

Show that the only real solution to \[\left\{ \begin{array}{l} x(x+y)^2 = 9 \\ x(y^3 - x^3) = 7 \\ \end{array} \right. \] is $x = 1$, $y = 2$.

1998 Putnam, 5

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Let $\mathcal{F}$ be a finite collection of open discs in $\mathbb{R}^2$ whose union contains a set $E\subseteq \mathbb{R}^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots,D_n$ in $\mathcal{F}$ such that \[E\subseteq\cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.

2005 Germany Team Selection Test, 1

Tags: number theory , equation , divisor

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.