Olympiad problems

2010 Today's Calculation Of Integral, 524

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2014 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , derivative , logarithm

Fix a positive real number $c>1$ and positive integer $n$. Initially, a blackboard contains the numbers $1,c,\ldots, c^{n-1}$. Every minute, Bob chooses two numbers $a,b$ on the board and replaces them with $ca+c^2b$. Prove that after $n-1$ minutes, the blackboard contains a single number no less than \[\left(\dfrac{c^{n/L}-1}{c^{1/L}-1}\right)^L,\] where $\phi=\tfrac{1+\sqrt 5}2$ and $L=1+\log_\phi(c)$.

2004 Baltic Way, 5

Tags: function , algebra

Determine the range of the following function defined for integer $k$, \[f(k)=(k)_3+(2k)_5+(3k)_7-6k\] where $(k)_{2n+1}$ denotes the multiple of $2n+1$ closest to $k$

2019 USA IMO Team Selection Test, 2

Tags: number theory , function

Let $\mathbb{Z}/n\mathbb{Z}$ denote the set of integers considered modulo $n$ (hence $\mathbb{Z}/n\mathbb{Z}$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, such that the 101 functions \[g(x), \quad g(x) + x, \quad g(x) + 2x, \quad \dots, \quad g(x) + 100x\] are all bijections on $\mathbb{Z}/n\mathbb{Z}$. [i]Ashwin Sah and Yang Liu[/i]

2011 Iran Team Selection Test, 12

Tags: function , induction , modular arithmetic , pigeonhole principle , number theory

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

2011 Iran MO (3rd Round), 7

Tags: algebra , function

Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting. [i]proposed by Sepehr Ghazi-Nezami[/i]

2018 PUMaC Live Round, 1.2

Tags: function

Define a function given the following $2$ rules: $\qquad$ 1) for prime $p$, $f(p)=p+1$. $\qquad$ 2) for positive integers $a$ and $b$, $f(ab)=f(a)\cdot f(b)$. For how many positive integers $n\leq 100$ is $f(n)$ divisible by $3$?

2011 IMC, 2

Tags: matrix , linear algebra , function , polynomial , algebra

Does there exist a real $3\times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I?$ ($\text{tr}(A)$ denotes the trace of $A,\ A^t$ the transpose of $A,$ and $I$ is the identity matrix.) [i]Proposed by Moubinool Omarjee, Paris[/i]

2012 District Olympiad, 2

Tags: equation , function , algebra

[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $ [b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that $$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$ then $ f(0)+f(1)=1. $

1991 Arnold's Trivium, 32

Tags: function , vector

Find the index of the singular point $0$ of the vector field \[(xy+yz+xz)\]

2005 IMAR Test, 2

Tags: function , inequalities

Let $n \geq 3$ be an integer and let $a,b\in\mathbb{R}$ such that $nb\geq a^2$. We consider the set \[ X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . \] Find the image of the function $M: X\to \mathbb{R}$ given by \[ M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . \] [i]Dan Schwarz[/i]

2013 Hong kong National Olympiad, 1

Tags: function , inequalities

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that \[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\] When does inequality hold?

2012 Postal Coaching, 3

Tags: algebra , function , floor function , integer part , logarithm

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

Kvant 2020, M413

Tags: real analysis , function , continuity

Determine the positive numbers $a{}$ for which the following statement true: for any function $f:[0,1]\to\mathbb{R}$ which is continuous at each point of this interval and for which $f(0)=f(1)=0$, the equation $f(x+a)-f(x)=0$ has at least one solution. [i]Proposed by I. Yaglom[/i]

1993 China Team Selection Test, 2

Tags: inequalities , algebra , function , quadratic

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

2012 Online Math Open Problems, 7

Tags: probability , function , trigonometry , inequalities , relatively prime , number theory

Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

2017 Greece Junior Math Olympiad, 2

Tags: function , algebra , inequalities

Let $x,y,z$ is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}\\ {y\left( {6 - z} \right) = 9}\\ {z\left( {6 - x} \right) = 9}\end{cases}$

2014 Serbia National Math Olympiad, 1

Tags: functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold: $$f(xf(y)-yf(x))=f(xy)-xy$$ [i]Proposed by Dusan Djukic[/i]

2014 Online Math Open Problems, 22

Tags: polynomial , function , algebra , modular arithmetic

Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$. [i]Proposed by Michael Kural and Evan Chen[/i]

2008 German National Olympiad, 3

Tags: function , algebra

Find all functions $ f$ defined on non-negative real numbers having the following properties: (i) For all non-negative $ x$ it is $ f(x) \geq 0$. (ii) It is $ f\left(1\right)\equal{}\frac 12$. (iii) For all non-negative numbers $ x,y$ it is $ f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y)$.

1991 Turkey Team Selection Test, 3

Tags: algebra , polynomial , function

Let $f$ be a function on defined on $|x|<1$ such that $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials with integer coefficients.

1997 Romania National Olympiad, 2

Tags: algebra , function

Find the range of the function $f: \mathbb{R} \to \mathbb{R},$ $$f(x)=\frac{3+2\sin x}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}.$$

2010 Pan African, 3

Tags: domain , function , induction , algebra

Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?

2011 Macedonia National Olympiad, 2

Tags: geometry , function , trigonometry , circumcircle , greatest common divisor , symmetry , incenter

Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$

2013 Greece Team Selection Test, 3

Tags: inequalities , function

Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$