Olympiad problems

2005 Iran MO (3rd Round), 3

Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]

2017 Korea National Olympiad, problem 7

Tags: algebra , functional equation , function , inequalities

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following. For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$. Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

2005 Today's Calculation Of Integral, 75

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

A function $f(\theta)$ satisfies the following conditions $(a),(b)$. $(a)\ f(\theta)\geq 0$ $(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$ Prove the following inequality. \[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]

2002 Olympic Revenge, 1

Tags: function , algebra

Show that there is no function $f:\mathbb{N}^* \rightarrow \mathbb{N}^*$ such that $f^n(n)=n+1$ for all $n$ (when $f^n$ is the $n$th iteration of $f$)

2013 F = Ma, 5

Tags: function

A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time. At what time(s) does the student have maximum downward velocity? $\textbf{(A)}$ At all times between $2 s$ and $4 s$ $\textbf{(B)}$ At $4 s$ only $\textbf{(C)}$ At all times between $4 s$ and $22 s$ $\textbf{(D)}$ At $22 s$ only $\textbf{(E)}$ At all times between $22 s$ and $24 s$

1992 IMO Longlists, 57

Tags: function , three variable inequality , inequality , algebra

For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that \[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]

2007 Indonesia TST, 2

Tags: function , algebra

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\] for all real numbers $x$ and $y$.

2006 India Regional Mathematical Olympiad, 7

Tags: function , search , algebra

Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$

2006 Australia National Olympiad, 2

Tags: function , calculus , integration , algebra

Let $f$ be a function defined on the positive integers, taking positive integral values, such that $f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$, $f(a) < f(b)$ if $a < b$, $f(3) \geq 7$. Find the smallest possible value of $f(3)$.

1950 AMC 12/AHSME, 41

Tags: function

The least value of the function $ ax^2\plus{}bx\plus{}c$ with $a>0$ is: $\textbf{(A)}\ -\dfrac{b}{a} \qquad \textbf{(B)}\ -\dfrac{b}{2a} \qquad \textbf{(C)}\ b^2-4ac \qquad \textbf{(D)}\ \dfrac{4ac-b^2}{4a}\qquad \textbf{(E)}\ \text{None of these}$

2002 IMC, 5

Tags: real analysis , function

Prove or disprove the following statements: (a) There exists a monotone function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$. (b) There exists a continuously differentiable function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$.

1979 IMO Longlists, 61

Tags: function , inequalities

There are two non-decreasing sequences $\{a_i\}$ and $\{b_i\}$ of $n$ real numbers each, such that $a_i\le a_{i+1}$ for each $1\le i\le n-1$, and $b_i\le b_{i+1}$ for each $1\le i\le n-1$, and $\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}$ where $m\le n$ with equality for $m=n$. For a convex function $f$ defined on the real numbers, prove that $\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}$.

2015 Postal Coaching, Problem 1

Tags: trigonometry , inequalities , number theory , calculus , integration , function

Find all positive integer $n$ such that $$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$ holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$

2020 March Advanced Contest, 1

Tags: function , number theory , modular arithmetic

In terms of $a$, $b$, and a prime $p$, find an expression which gives the number of $x \in \{0, 1, \ldots, p-1\}$ such that the remainder of $ax$ upon division by $p$ is less than the remainder of $bx$ upon division by $p$.

2021 Moldova Team Selection Test, 5

Tags: function , algebra , geometry

Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.

2006 China Northern MO, 5

Tags: function , inequalities

$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]

2011 Laurențiu Duican, 3

Tags: function , find all functions , real analysis

Find the $ \mathcal{C}^1 $ class functions $ f:[0,2]\longrightarrow\mathbb{R} $ having the property that the application $ x\mapsto e^{-x} f(x) $ is nonincreasing on $ [0,1] , $ nondecreasing on $ [1,2] , $ and satisfying $$ \int_0^2 xf(x)dx=f(0)+f(2) . $$ [i]Cristinel Mortici[/i]

1998 All-Russian Olympiad, 2

Tags: trigonometry , inequalities , geometry , rectangle , function , trig identities

Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points. By the way, can two sides of a polygon intersect?

2011 Croatia Team Selection Test, 1

Tags: function , inequalities

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

1998 Romania National Olympiad, 3

Tags: calculus , derivative , function , inequalities , arithmetic sequence , real analysis

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable

2007 Kazakhstan National Olympiad, 4

Tags: function , induction , algebra

Find all functions $ f :\mathbb{R}\to\mathbb{R} $, satisfying the condition $f (xf (y) + f (x)) = 2f (x) + xy$ for any real $x$ and $y$.

1957 Czech and Slovak Olympiad III A, 3

Tags: number theory , trigonometric function , function

Find all real numbers $\alpha$ such that both values $\cot(\alpha)$ and $\cot(2\alpha)$ are integers.

2000 South africa National Olympiad, 4

Tags: geometry , perimeter , trigonometry , function , analytic geometry , angle bisector , inequalities

$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.

2024 Nepal TST, P2

Tags: function , natural number , inequalities

Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$. [i](Proposed by David Anghel, Romania)[/i]

1989 Turkey Team Selection Test, 1

Tags: function , algorithm , number theory

Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that [list=i] [*] $f(m,m)=m$ [*] $f(m,k) = f(k,m)$ [*] $f(m, m+k) = f(m,k)$[/list] , for each $m,k \in \mathbb{Z}^+$.