Olympiad problems

2011 Ukraine Team Selection Test, 5

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

2008 ISI B.Math Entrance Exam, 1

Tags: function , integration , calculus , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

2018 AIME Problems, 10

Tags: function

Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: function , induction

Let $ f(x) \equal{} \frac{x}{1 \minus{} x}$ and let $ a$ be a real number. If $ x_0 \equal{} a, x_1 \equal{} f(x_0), x_2 \equal{} f(x_1), ...., x_{1996} \equal{} f(x_{1995})$ and $ x_{1996} \equal{} 1,$ what is $ a$? A. 0 B. 1/1997 C. 1995 D. 1995/1996 E. None of these

2002 Finnish National High School Mathematics Competition, 1

Tags: trigonometry , function , algebra

A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$

2022 VJIMC, 4

Tags: function , number theory

Let $g$ be the multiplicative function given by $$g(p^{\alpha}) = \alpha p^{\alpha-1},$$ for all $\alpha\in\mathbb Z^+$ and primes $p$. Prove that there exist infinitely many integers $n$ such that $$g(n+1) = g(n) + g(1).$$

2024 SG Originals, Q4

Tags: function , number theory

Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying \[f_k(x)=x+k\varphi(x)\] where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with [list] [*] $a_1=c$, and [*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$ [/list] Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if (a) $k=2025$ ? (b) $k=2065$ ? [i]Proposed by chorn[/i]

2023 AMC 12/AHSME, 22

Tags: function

Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$? $\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144$

2012 ISI Entrance Examination, 2

Tags: limit , derivative , calculus , calculus computations , function

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

2007 Ukraine Team Selection Test, 4

Tags: algebra , function

Find all functions $f: \mathbb Q \to \mathbb Q$ such that $ f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x)$ for all $x,y \in \mathbb Q$.

1986 Iran MO (2nd round), 3

Tags: function , algebra

Prove that \[\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.\]

2011 China Team Selection Test, 1

Tags: algebra , function

Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.\]

2007 Junior Balkan MO, 2

Tags: trigonometry , geometry , circumcircle , incenter , angle bisector , cyclic quadrilateral , function

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

1999 Poland - Second Round, 1

Tags: algebra , increasing function , function

Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N. Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.

2002 National Olympiad First Round, 25

Tags: trig identities , trigonometry , rhombus , geometry , law of cosines , function

Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$? $ \textbf{a)}\ 45^\circ \qquad\textbf{b)}\ 50^\circ \qquad\textbf{c)}\ 60^\circ \qquad\textbf{d)}\ 65^\circ \qquad\textbf{e)}\ 75^\circ $

2004 Romania National Olympiad, 1

Tags: inequalities , function , algebra

Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$. [i]Cristinel Mortici[/i]

1984 Balkan MO, 1

Tags: calculus , function , inequalities , n-variable inequality , algebra

Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$. Prove that: \[\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}\]

1997 Bulgaria National Olympiad, 1

Tags: function , inequalities

Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$. Prove that $ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.

2013 Albania Team Selection Test, 1

Tags: number theory , ratio , function

Find the 3-digit number whose ratio with the sum of its digits it's minimal.

2014 European Mathematical Cup, 4

Tags: algebra , function

Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds $f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$ [i]Proposed by Matija Bucić[/i]

PEN I Problems, 13

Tags: function , floor function

Suppose that $n \ge 2$. Prove that \[\sum_{k=2}^{n}\left\lfloor \frac{n^{2}}{k}\right\rfloor = \sum_{k=n+1}^{n^{2}}\left\lfloor \frac{n^{2}}{k}\right\rfloor.\]

2004 Germany Team Selection Test, 1

Tags: function , closed formula , functional equation

A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

2006 Pre-Preparation Course Examination, 7

Tags: limit , combinatorics , logarithm , function

Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$: \[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \] Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.

2018 Thailand TST, 2

Tags: algebra , function

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

2005 Regional Competition For Advanced Students, 3

Tags: algebra , analytic geometry , function

For which values of $ k$ and $ d$ has the system $ x^3\plus{}y^3\equal{}2$ and $ y\equal{}kx\plus{}d$ no real solutions $ (x,y)$?