Olympiad problems

2023 Rioplatense Mathematical Olympiad, 5

Tags: function , algebra

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all non-negative real number $\alpha$ such that there exist a function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)$$ for any $x,y$ positive real numbers.

2003 Romania National Olympiad, 3

Tags: limit , function , continuity , real analysis

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

2006 Federal Competition For Advanced Students, Part 2, 1

Tags: relatively prime , function , modular arithmetic , floor function , number theory

Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?

1995 Vietnam Team Selection Test, 3

Tags: algebra , function

Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$. $ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$. $ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.

2004 India IMO Training Camp, 2

Tags: function , algebra

Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule: (a) $g$ is nondecrasing (b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$, Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$

2011 Today's Calculation Of Integral, 697

Tags: function , domain , calculus , geometry , integration , inequalities , algebra

Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$

1982 IMO Longlists, 22

Tags: topology , algebra , discrete intermediate value theorem , function

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

2020 Turkey EGMO TST, 2

Tags: function , number theory

$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$

2013 ELMO Shortlist, 9

Tags: function , logarithm , inequalities

Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

2001 Romania Team Selection Test, 2

Tags: combinatorics , pigeonhole principle , prime number , algebra , function , number theory

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

2023 Harvard-MIT Mathematics Tournament, 3

Tags: trigonometry , function

Suppose $x$ is a real number such that $\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}$. Compute $\cos(1 + \sin^2 x + \cos^4 x)$.

2012 Indonesia MO, 2

Tags: function , algebra

Let $\mathbb{R}^+$ be the set of all positive real numbers. Show that there is no function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfying \[f(x+y)=f(x)+f(y)+\dfrac{1}{2012}\] for all positive real numbers $x$ and $y$. [i]Proposer: Fajar Yuliawan[/i]

2009 Indonesia TST, 4

Tags: algebra , function

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

2014 Saudi Arabia BMO TST, 1

Tags: prime number , number theory , function

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

1994 Iran MO (2nd round), 3

Tags: functional equation , function , induction

Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

2007 China Western Mathematical Olympiad, 3

Tags: inequalities , three variable inequality , function

Let $ a,b,c$ be real numbers such that $ a\plus{}b\plus{}c\equal{}3$. Prove that \[\frac{1}{5a^2\minus{}4a\plus{}11}\plus{}\frac{1}{5b^2\minus{}4b\plus{}11}\plus{}\frac{1}{5c^2\minus{}4c\plus{}11}\leq\frac{1}{4}\]

1970 Miklós Schweitzer, 4

Tags: number theory , quadratic , absolute value , modular arithmetic , function

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

2010 District Olympiad, 3

Tags: function , induction , algebra

Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

2013 NIMO Problems, 2

Tags: quadratic , polynomial , functional equation , function , algebra

Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$. [i]Proposed by Ahaan S. Rungta[/i]

2019 Dutch IMO TST, 4

Tags: functional equation , find all functions , prime number , function , number theory

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2006 Switzerland Team Selection Test, 1

Tags: invariant , function , algebra

The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D

2010 Today's Calculation Of Integral, 609

Tags: integration , function , calculus computations , calculus , trigonometry

Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number. 1972 Tokyo University of Education entrance exam

1978 AMC 12/AHSME, 17

Tags: function

If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, \[\left[f(x^2+1)\right]^{\sqrt{x}}=k;\] then, for every positive number $y$, \[\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}\] is equal to $\textbf{(A) }\sqrt{k}\qquad\textbf{(B) }2k\qquad\textbf{(C) }k\sqrt{k}\qquad\textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$

2011 Today's Calculation Of Integral, 699

Tags: domain , trigonometry , integration , calculus , inequalities , function , algebra

Find the volume of the part bounded by $z=x+y,\ z=x^2+y^2$ in the $xyz$ space.

1974 USAMO, 3

Tags: sphere , 3d geometry , geometry , function

Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.