Olympiad problems

2019 Korea Winter Program Practice Test, 1

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

1994 Brazil National Olympiad, 4

Tags: function , polynomial , algebra

Let $a, b > 0$ be reals such that \[ a^3=a+1\\ b^6=b+3a \] Show that $a>b$

2005 Moldova National Olympiad, 11.2

Tags: function , algebra , function graphing , inequalities

Let $a$ and $b$ be two real numbers. Find these numbers given that the graphs of $f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1$ and $g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1$ have exactly two points of intersection.

2003 Brazil National Olympiad, 2

Tags: function , limit , algebra

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.

2010 District Olympiad, 4

Tags: function , induction , algebra

Determine all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ f(n)\plus{}f(n\plus{}1)\plus{}f(f(n))\equal{}3n\plus{}1, \quad \forall n\in \mathbb{N}.\]

2005 Taiwan National Olympiad, 1

Tags: inequalities , function

Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]

2019 Dutch IMO TST, 2

Tags: function , algebra

Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.

1992 Romania Team Selection Test, 3

Tags: function , triangle , algebra

Let $\pi$ be the set of points in a plane and $f : \pi \to \pi$ be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that $f(\pi)$ is a square.

2013 NIMO Problems, 2

Tags: function , modular arithmetic

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]

2015 AMC 12/AHSME, 18

Tags: quadratic , function , polynomial , vieta , sfft , algebra

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2010 Bulgaria National Olympiad, 2

Tags: function , floor function , induction , algebra

Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and \[f(n)=n - f(f(n-1)), \quad \forall n \geq 2.\] Prove that $f(n+f(n))=n $ for each positive integer $n.$

2012 Pan African, 2

Tags: function , algebra

Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.

1966 Miklós Schweitzer, 6

Tags: function , real analysis

A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.) [i]A.Csaszar[/i]

1990 IMO Longlists, 53

Tags: function , algebra

Find the real solution(s) for the system of equations \[\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}\]

2012 Kazakhstan National Olympiad, 2

Tags: function , algebra

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

1972 IMO Shortlist, 1

Tags: function , algebra , functional equation , functional inequality

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

2006 Iran Team Selection Test, 1

Tags: least common multiple , function , number theory

Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

India EGMO 2024 TST, 3

Tags: function , algebra , polynomial

Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\] [i]Proposed by Sutanay Bhattacharya[/i]

2006 Taiwan TST Round 1, 1

Tags: function , algebra

Let $d,p,q$ be fixed positive integers, and $d$ is not a perfect square. $\mathbb{N}$ is the set of all positive integers, and $S=\{m+n\sqrt{d}|m,n \in \mathbb{N}\} \cup \{0\}$. Suppose the function $f: S \to S$ satisfies the following conditions for all $x,y \in S$: (i) $f((xy)^p)=(f(x)f(y))^p$ (ii)$f((x+y)^q)=(f(x)+f(y))^q$ Find the function $f$.

2010 Indonesia TST, 1

Tags: geometry , trigonometry , algebra , polynomial , induction , function , ratio

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

2003 Purple Comet Problems, 9

Tags: function

Let $f$ be a real-valued function of real and positive argument such that $f(x) + 3xf(\tfrac1x) = 2(x + 1)$ for all real numbers $x > 0$. Find $f(2003)$.

2012 Today's Calculation Of Integral, 830

Tags: calculus , integration , limit , function , calculus computations , logarithm

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

1991 French Mathematical Olympiad, Problem 2

Tags: function , limit , algebra

For each $n\in\mathbb N$, the function $f_n$ is defined on real numbers $x\ge n$ by $$f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.$$(a) If $n$ is fixed, prove that $\lim_{x\to+\infty}f_n(x)=0$. (b) Find the limit of $f_n(n)$ as $n\to+\infty$.

2010 Germany Team Selection Test, 3

Tags: function , inequalities , algebra , functional inequality

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

2014 PUMaC Algebra A, 4

Tags: floor function , ceiling function , function , algebra

There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]