Olympiad problems

2007 Vietnam National Olympiad, 2

Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that: $f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$

2002 Iran MO (3rd Round), 4

Tags: algebra , geometry , geometric transformation , inequalities , polynomial , function

$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.

2014 Brazil Team Selection Test, 4

Tags: function , algebra , functional equation

Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

2016 NIMO Problems, 2

Tags: function , algebra

For real numbers $x$ and $y$, define \[\nabla(x,y)=x-\dfrac1y.\] If \[\underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \dfrac{m}{n}\] for relatively prime positive integers $m$, $n$, compute $100m + n$. [i] Proposed by David Altizio [/i]

2010 SEEMOUS, Problem 1

Tags: function , convergence , integration , calculus , sequence

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$. a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$. b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.

1992 Romania Team Selection Test, 1

Tags: functional , function , algebra

Suppose that$ f : N \to N$ is an increasing function such that $f(f(n)) = 3n$ for all $n$. Find $f(1992)$.

2017 Miklós Schweitzer, 8

Tags: algebra , binary representation , number base , function , inequalities

Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by $$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and $$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$

2004 Purple Comet Problems, 12

Tags: function

If $f(x, y) = xy + 2x + y + 1$, find $f(f(2, f(3, 4)), 5)$.

2023 Myanmar IMO Training, 5

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

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

2008 Putnam, B5

Tags: function , number theory , greatest common divisor , calculus , derivative , integration

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

2010 Indonesia TST, 3

Tags: function equation , function , algebra

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

Kvant 2025, M2837

Tags: function , geometry , parabola , algebra

On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals. [i]A. Tereshin[/i]

1989 National High School Mathematics League, 2

Tags: function

Range of function $f(x)=\arctan x+\frac{1}{2}\arcsin x$ is $\text{(A)}(-\pi,\pi)\qquad\text{(B)}[-\frac{3}{4}\pi,\frac{3}{4}\pi]\qquad\text{(C)}(-\frac{3}{4}\pi,\frac{3}{4}\pi)\qquad\text{(D)}[-\frac{1}{2}\pi,\frac{1}{2}\pi]$

2012 China Team Selection Test, 1

Tags: floor function , function , number theory , logarithm

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2010 Today's Calculation Of Integral, 558

Tags: calculus , integration , quadratic , function , inequalities , algebra , calculus computations

For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$. Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$

2008 AMC 12/AHSME, 19

Tags: function , inequalities , complex number , triangle inequality

A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$? $ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$

2013 Iran MO (3rd Round), 1

Tags: function , combinatorics

Assume that the following generating function equation is correct, prove the following statement: $\Pi_{i=1}^{\infty} (1+x^{3i})\Pi_{j=1}^{\infty} (1-x^{6j+3})=1$ Statement: The number of partitions of $n$ to numbers not of the form $6k+1$ or $6k-1$ is equal to the number of partitions of $n$ in which each summand appears at least twice. (10 points) [i]Proposed by Morteza Saghafian[/i]

2012 Federal Competition For Advanced Students, Part 1, 1

Tags: function , greatest common divisor , geometry , geometric transformation , number theory

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$. Note: If $n\in\mathbb{Z}$, $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$.

2005 South East Mathematical Olympiad, 5

Tags: trigonometry , function , search , power of a point , geometry

Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively. Find the value of $PB \cdot PC$.

2010 Today's Calculation Of Integral, 564

Tags: calculus , integration , analytic geometry , function , geometry , calculus computations , logarithm

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.

2023 Philippine MO, 8

Tags: geometry , functional equation , combinatorial geometry , function , nice

Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$, if $f(A) \leq f(B) \leq f(C)$, then $$f(A) + f(C) = f(B) + f(H).$$

1955 Miklós Schweitzer, 2

Tags: real analysis , function

[b]2.[/b] Let $f_{1}(x), \dots , f_{n}(x)$ be Lebesgue integrable functions on $[0,1]$, with $\int_{0}^{1}f_{1}(x) dx= 0$ $ (i=1,\dots ,n)$. Show that, for every $\alpha \in (0,1)$, there existis a subset $E$ of $[0,1]$ with measure $\alpha$, such that $\int_{E}f_{i}(x)dx=0$. [b](R. 17)[/b]

MIPT Undergraduate Contest 2019, 2.2

Tags: inequalities , probability and stats , expectation , calculus , integration , probability , function

Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

2000 Taiwan National Olympiad, 1

Tags: euler , function , modular arithmetic , quadratic , prime number , number theory

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

MIPT Undergraduate Contest 2019, 2.3

Tags: function , geometry , rectangle , topology , mapping

Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.