Olympiad problems

2018 Israel Olympic Revenge, 4

Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.

2010 Today's Calculation Of Integral, 617

Tags: calculus , integration , function , calculus computations

Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

2021 AMC 12/AHSME Fall, 20

Tags: function

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

1969 Miklós Schweitzer, 8

Tags: function , integration , real analysis

Let $ f$ and $ g$ be continuous positive functions defined on the interval $ [0, +\infty)$, and let $ E \subset[0,+\infty)$ be a set of positive measure. Prove that the range of the function defined on $ E \times E$ by the relation \[ F(x,y)= %Error. "dispalymath" is a bad command. \int_0^xf(t)dt+ %Error. "dispalymath" is a bad command. \int_0^y g(t)dt\] has a nonvoid interior. [i]L. Losonczi[/i]

2009 AIME Problems, 13

Tags: induction , modular arithmetic , inequalities , pigeonhole principle , function

The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.

2019 LIMIT Category B, Problem 11

Tags: floor function , function , bounding , algebra

$$\left\lfloor\left(1\cdot2+2\cdot2^2+\ldots+100\cdot2^{100}\right)\cdot9^{-901}\right\rfloor=?$$

2014 Polish MO Finals, 1

Tags: function , functional equation

Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy $$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$ for all integers $n\ge 1$ and rational numbers $q>0$.

2015 AMC 10, 1

Tags: modular arithmetic , function

What is the value of $2-(-2)^{-2}$? $ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $

2017 Romania National Olympiad, 1

Tags: limit , function , real analysis , definite integral , calculus , integration , sequence

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

2007 Baltic Way, 20

Tags: geometry , search , function , number theory

Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

2008 Harvard-MIT Mathematics Tournament, 10

Tags: floor function , function , algebra , polynomial

Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.

1989 Brazil National Olympiad, 3

Tags: function , algebra

A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$. Determine, justifying your answer, the set of all possible values for $f$.

1988 India National Olympiad, 4

Tags: function , inequalities

If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that \[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]

2019 IFYM, Sozopol, 6

Tags: function , algebra

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: $xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$

2010 Contests, A2

Tags: function , integration , real analysis , slope

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

2005 Unirea, 4

Tags: real analysis , integration , inequalities , parameterization , function

$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable suppose $f(a)=f(-a)$ $ f'(-a)=f'(a)=a^2$ Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$ Study equality case ? Radu Miculescu

2003 Kazakhstan National Olympiad, 8

Tags: function , algebra

Determine all functions $f: \mathbb R \to \mathbb R$ with the property \[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]

2009 Princeton University Math Competition, 6

Tags: function

Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.

2006 Grigore Moisil Urziceni, 3

Tags: primitive , function , rolle , lagrange , real analysis , primitivability

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that admits a primitive $ F. $ [b]a)[/b] Show that there exists a real number $ c $ such that $ f(c)-F(c)>1 $ if $ \lim_{x\to\infty } \frac{1+F(x)}{e^x} =-\infty . $ [b]b)[/b] Prove that there exists a real number $ c' $ such that $ f(c') -(F(c'))^2<1. $ [i]Cristinel Mortici[/i]

1998 AMC 12/AHSME, 17

Tags: function , algebra , functional equation , geometry , 3d geometry

Let $ f(x)$ be a function with the two properties: [list=a] [*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and [*] $ f(0) \equal{} 2$ [/list] What is the value of $ f(1998)$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 1996\qquad \textbf{(D)}\ 1998\qquad \textbf{(E)}\ 2000$

2024 CIIM, 1

Tags: integration , function , recurrence relation , calculus , sequence

Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds: \[ f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1. \] Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\ [b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.

1972 Miklós Schweitzer, 7

Tags: advanced fields , function , inequalities

Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$ [i]P. Turan[/i]

2012 Romanian Master of Mathematics, 3

Tags: function , algebra , additive combinatorics , combinatorics

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

2012 District Olympiad, 3

Tags: superior algebra , function , greatest common divisor

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

2013 Brazil National Olympiad, 3

Tags: function , induction , limit , polynomial , functional equation , algebra

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.