Olympiad problems

2011 IMO Shortlist, 6

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies \[f(x + y) \leq yf(x) + f(f(x))\] for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$. [i]Proposed by Igor Voronovich, Belarus[/i]

2018 Greece Team Selection Test, 3

Tags: function , number theory

Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that $$xf(x)+(f(y))^2+2xf(y)$$ is perfect square for all positive integers $x,y$. **This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.

2007 Germany Team Selection Test, 2

Tags: function , functional equation , algebra

Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]

2007 IMC, 4

Tags: induction , function

Let $ G$ be a finite group. For arbitrary sets $ U, V, W \subset G$, denote by $ N_{UVW}$ the number of triples $ (x, y, z) \in U \times V \times W$ for which $ xyz$ is the unity . Suppose that $ G$ is partitioned into three sets $ A, B$ and $ C$ (i.e. sets $ A, B, C$ are pairwise disjoint and $ G = A \cup B \cup C$). Prove that $ N_{ABC}= N_{CBA}.$

2006 China Team Selection Test, 2

Tags: function , polynomial , algebra

The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

2013 India IMO Training Camp, 1

Tags: function , induction , limit , algebra , cauchy equation

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

2017 Iran Team Selection Test, 3

Tags: algebra , functional equation , function

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

2010 Today's Calculation Of Integral, 593

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

For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

2024 Brazil Team Selection Test, 4

Tags: algebra , functional equation , function

Find all pairs of positive integers $ (a, b) $ such that $ f(x) = x $ is the only function $ f : \mathbb{R} \to \mathbb{R} $ that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, $ f^n(x) $ represents the function obtained by applying $ f $ $ n $ times to $ x $. That is, $ f^1(x) = f(x) $ and $ f^{n+1}(x) = f(f^n(x))$ for all $n \geq 1$.

2015 Turkey Junior National Olympiad, 1

Tags: function

For a non-constant function $f:\mathbb{R}\to \mathbb{R}$ prove that there exist real numbers $x,y$ satisfying $f(x+y)<f(xy)$

1990 IMO Longlists, 89

Tags: function , algebra , domain , combinatorics

Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$

1974 All Soviet Union Mathematical Olympiad, 203

Tags: function , algebra , inequalities

Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$ a) Prove that for all $x$ holds $f(x) \le 2x$. b) Is the inequality $f(x) \le 1.9x$ valid?

2015 Thailand TSTST, 2

Tags: function , algebra

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(f(x - y)) = f(x)f(y) + f(x) - f(y) - xy.\]

1964 Miklós Schweitzer, 9

Tags: function , topology , real analysis

Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.

2019 Korea USCM, 6

Tags: real analysis , inequalities , integral inequality , function

A function $f:[0,\infty)\to[0,\infty)$ is integrable and $$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$ Prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$

2004 India IMO Training Camp, 3

Tags: trigonometry , function , functional equation , system of equations , algebra

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

1983 Bulgaria National Olympiad, Problem 6

Tags: algebra , floor function , function

Let $a,b,c>0$ satisfy for all integers $n$, we have $$\lfloor an\rfloor+\lfloor bn\rfloor=\lfloor cn\rfloor$$Prove that at least one of $a,b,c$ is an integer.

2013 Romania National Olympiad, 2

Tags: function , algebra

Given $f:\mathbb{R}\to \mathbb{R}$ an arbitrary function and $g:\mathbb{R}\to \mathbb{R}$ a function of the second degree, with the property: for any real numbers m and n equation $f\left( x \right)=mx+n$ has solutions if and only if the equation $g\left( x \right)=mx+n$ has solutions Show that the functions $f$ and $g$ are equal.

1964 Miklós Schweitzer, 7

Tags: invariant , function , real analysis

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

2007 Italy TST, 3

Tags: function , polynomial , functional equation , algebra

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

2014 Canadian Mathematical Olympiad Qualification, 1

Tags: function , algebra , polynomial , number theory , greatest common divisor

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.

1996 South africa National Olympiad, 2

Tags: function , trigonometry , algebra , inequalities

Find all real numbers for which $3^x+4^x=5^x$.

2020 European Mathematical Cup, 4

Tags: algebra , functional equation , function

Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$ for all $x, y \in\mathbb{R^+}.$ [i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]

2014 Contests, 1

Tags: number theory , function

Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$. Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)

1957 Putnam, B3

Tags: function , integration , inequalities

For $f(x)$ a positive , monotone decreasing function defined in $[0,1],$ prove that $$ \int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.$$