Olympiad problems

2013 Romanian Masters In Mathematics, 1

Tags: ratio , geometry , geometric transformation , reflection , integration , function , homothety

Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?

2014 Miklós Schweitzer, 8

Tags: polynomial , function , inequalities , functional analysis , real analysis , algebra

Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.

2018 Mathematical Talent Reward Programme, SAQ: P 4

Tags: function , algebra

Suppose $S$ is a finite subset of $\mathbb{R}$. If $f: S \rightarrow S$ is a function such that, $$ \left|f\left(s_{1}\right)-f\left(s_{2}\right)\right| \leq \frac{1}{2}\left|s_{1}-s_{2}\right|, \forall s_{1}, s_{2} \in S $$ Prove that, there exists a $x \in S$ such that $f(x)=x$

2018 Korea USCM, 3

Tags: calculus , integration , function

$\Phi$ is a function defined on collection of bounded measurable subsets of $\mathbb{R}$ defined as $$\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy$$ Find the maximum value of $\Phi$.

2007 Romania Team Selection Test, 2

Tags: inequalities , function , algebra

Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that \[|f(x)-f(y)|\leq (x-y)^{2}\] for all $x,y \in\mathbb{Q}$. Prove that $f$ is constant.

2012 Iran MO (3rd Round), 1

Tags: polynomial , induction , function , modular arithmetic , algebra

Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]

2010 Princeton University Math Competition, 7

Tags: polynomial , function , algebra

Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$. $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes. Find $p_1 + p_2 + \cdots + p_n$.

2000 Polish MO Finals, 1

Tags: trigonometry , function , algebra

Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

2004 Moldova Team Selection Test, 6

Tags: function , algebra

Find all functions $f:\mathbb R \to \mathbb R$ Such that for all real $x,y$: $(x^2+xy+y^2)(f(x)-f(y))=f(x^3)-f(y^3)$

2006 CHKMO, 3

Tags: function , algebra , inequalities

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

1991 Arnold's Trivium, 98

Tags: probability , function

In the game of "Fingers", $N$ players stand in a circle and simultaneously thrust out their right hands, each with a certain number of fingers showing. The total number of fingers shown is counted out round the circle from the leader, and the player on whom the count stops is the winner. How large must $N$ be for a suitably chosen group of $N/10$ players to contain a winner with probability at least $0.9$? How does the probability that the leader wins behave as $N\to\infty$?

2015 Greece Team Selection Test, 4

Tags: function , inequalities , algebra

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$

2011 Iran MO (3rd Round), 6

Tags: function , number theory

$a$ is an integer and $p$ is a prime number and we have $p\ge 17$. Suppose that $S=\{1,2,....,p-1\}$ and $T=\{y|1\le y\le p-1,ord_p(y)<p-1\}$. Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\longrightarrow S$ satisfying $\sum_{x\in T} x^{f(x)}\equiv a$ $(mod$ $p)$. [i]proposed by Mahyar Sefidgaran[/i]

2010 Indonesia TST, 3

Tags: function , function equation , 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]

2006 All-Russian Olympiad, 1

Tags: trigonometry , function , calculus , derivative , inequalities , real analysis

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

1994 APMO, 1

Tags: function , induction , algebra

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that (i) For all $x,y \in \Bbb{R}$, \[ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) \] (ii) For all $x \in [0,1)$, $f(0) \geq f(x)$, (iii) $-f(-1) = f(1) = 1$. Find all such functions $f$.

2012 Gulf Math Olympiad, 3

Tags: ceiling function , function , combinatorics

Consider a $3\times7$ grid of squares. Each square may be coloured green or white. [list] (a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour? (b) Is it possible for a $4\times 6$ grid? [/list] [i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]

2007 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , function , limit

$g$ is a twice differentiable function over the positive reals such that \begin{align}g(x)+2x^3g^\prime(x)+x^4g^{\prime\prime}(x)&= 0 \qquad\text{ for all positive reals } x\\\lim_{x\to\infty}xg(x)&=1\end{align} Find the real number $\alpha>1$ such that $g(\alpha)=1/2$.

2011 Romania National Olympiad, 2

Tags: function , real analysis , integration , calculus

Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that $$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$ Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $

2021 Nigerian MO Round 3, Problem 5

Tags: function , calculus , derivative , degree , algebra , polynomial

Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

2010 India IMO Training Camp, 4

Tags: function , inequalities

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

2006 Turkey Team Selection Test, 2

Tags: geometry , rectangle , calculus , integration , function , combinatorics

How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?

1980 IMO, 15

Tags: ratio , function , geometry

Three points $A,B,C$ are such that $B\in AC$. On one side of $AC$, draw the three semicircles with diameters $AB,BC,CA$. The common interior tangent at $B$ to the first two semicircles meets the third circle $E$. Let $U,V$ be the points of contact of the common exterior tangent to the first two semicircles. Evaluate the ratio $R=\frac{[EUV]}{[EAC]}$ as a function of $r_{1} = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$, where $[X]$ denotes the area of polygon $X$.

1995 IMO Shortlist, 4

Tags: function , inequalities , optimization , system of equations , 133

Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

2023 Bangladesh Mathematical Olympiad, P5

Tags: algebra , integration , functional equation , function

Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration: $$ \int_{0}^{\infty} f(x) \,dx $$