Olympiad problems

1947 Putnam, B1

Tags: function , trigonometry

Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$ $$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$ Prove that $$\lim_{x\to \infty} f(x)$$ exists and is less than $1+ \frac{\pi}{4}.$

2011 Peru IMO TST, 1

Tags: algebra , functional equation , function

Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $

2007 Nicolae Păun, 1

Tags: function , order , group theory

Consider a finite group $ G $ and the sequence of functions $ \left( A_n \right)_{n\ge 1} :G\longrightarrow \mathcal{P} (G) $ defined as $ A_n(g) = \left\{ x\in G|x^n=g \right\} , $ where $ \mathcal{P} (G) $ is the power of $ G. $ [b]a)[/b] Prove that if $ G $ is commutative, then for any natural numbers $ n, $ either $ A_n(g) =\emptyset , $ or $ \left| A_n(g) \right| =\left| A_n(1) \right| . $ [b]b)[/b] Provide an example of what $ G $ could be in the case that there exists an element $ g_0 $ of $ G $ and a natural number $ n_0 $ such that $ \left| A_{n_0}\left( g_0 \right) \right| >\left| A_{n_0}(1) \right| . $ [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

1999 Dutch Mathematical Olympiad, 4

Tags: linear algebra , matrix , function

Consider a matrix of size $8 \times 8$, containing positive integers only. One may repeatedly transform the entries of the matrix according to the following rules: -Multiply all entries in some row by 2. -Subtract 1 from all entries in some column. Prove that one can transform the given matrix into the zero matrix.

2015 Mathematical Talent Reward Programme, MCQ: P 15

Tags: function , number of solutions

Find out the number of real solutions of $x^2e^{\sin x}=1$ [list=1] [*] 0 [*] 1 [*] 2 [*] 3 [/list]

2018 China Team Selection Test, 4

Tags: function , functional equation , fe , algebra

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

2006 Iran Team Selection Test, 5

Tags: circumcircle , trigonometry , inequalities , function , incenter , geometry

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

2011 Romanian Masters In Mathematics, 2

Tags: vector , function , geometry , geometric transformation , reflection , combinatorial geometry , perpendicular bisector

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

2000 China Team Selection Test, 1

Tags: function , polynomial , algebra

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

2001 Miklós Schweitzer, 7

Tags: function , real analysis

Let $e_1,\ldots, e_n$ be semilines on the plane starting from a common point. Prove that if there is no $u\not\equiv 0$ harmonic function on the whole plane that vanishes on the set $e_1\cup \cdots \cup e_n$, then there exists a pair $i,j$ of indices such that no $u\not\equiv 0$ harmonic function on the whole plane exists that vanishes on $e_i\cup e_j$.

2010 Harvard-MIT Mathematics Tournament, 2

Tags: calculus , function , derivative

Let $f$ be a function such that $f(0)=1$, $f^\prime (0)=2$, and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all $t$. Compute the $4$th derivative of $f$, evaluated at $0$.

2014 Miklós Schweitzer, 11

Tags: trigonometry , function , probability and stats

Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$, and let \[S_n= 2\sum_{k=1}^n \sin(2kU\pi).\] Show that, as $n\to \infty$, the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$.

2004 Hong kong National Olympiad, 4

Tags: function , number theory , prime number , combinatorics

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

2018 Pan-African Shortlist, A5

Tags: number theory , algebra , function , functional equation , multiplicative function

Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying: [list] [*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$, [*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and [*] $g(x) \neq x$ for $2 \leq x \leq 2018$. [/list] Find the minimum possible value of $g(2)$.

2006 Romania National Olympiad, 4

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

Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \] [i]Cezar Lupu, Tudorel Lupu[/i]

2006 International Zhautykov Olympiad, 1

Tags: inequalities , function , number theory

Solve in positive integers the equation \[ n \equal{} \varphi(n) \plus{} 402 , \] where $ \varphi(n)$ is the number of positive integers less than $ n$ having no common prime factors with $ n$.

Oliforum Contest II 2009, 1

Tags: function , algebra

Find all non empty subset $ S$ of $ \mathbb{N}: \equal{} \{0,1,2,\ldots\}$ such that $ 0 \in S$ and exist two function $ h(\cdot): S \times S \to S$ and $ k(\cdot): S \to S$ which respect the following rules: i) $ k(x) \equal{} h(0,x)$ for all $ x \in S$ ii) $ k(0) \equal{} 0$ iii) $ h(k(x_1),x_2) \equal{} x_1$ for all $ x_1,x_2 \in S$. [i](Pierfrancesco Carlucci)[/i]

2010 Today's Calculation Of Integral, 526

Tags: calculus , integration , function , calculus computations

For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?

2016 IFYM, Sozopol, 8

Tags: function , special condition , algebra

Prove that there exist infinitely many natural numbers $n$, for which there $\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}$, satisfying the following conditions: 1) $f(x)\neq x$; 2) $f(f(x))=x$; 3) $f(f(f(x+1)+1)+1)=x$ for $\forall x\in \{0,1…n-1\}$.

Today's calculation of integrals, 867

Tags: calculus , integration , quadratic , function , derivative , algebra , linear equation

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2012 IFYM, Sozopol, 6

Tags: function , algebra

Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f(x^2 + f(y)) = (f(x) + y^2)^ 2 \] , for all $x,y\in \Bbb{R}.$

2010 Tournament Of Towns, 3

Tags: function , trigonometry , algebra

Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way?

2005 Bulgaria Team Selection Test, 3

Tags: function , algebra

Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.

2014 Online Math Open Problems, 21

Tags: function

Let $b = \tfrac 12 (-1 + 3\sqrt{5})$. Determine the number of rational numbers which can be written in the form \[ a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a_1b + a_0 \] where $a_0, a_1, \dots, a_{2014}$ are nonnegative integers less than $b$. [i]Proposed by Michael Kural and Evan Chen[/i]

2010 Today's Calculation Of Integral, 557

Tags: calculus , integration , limit , trigonometry , floor function , function , analytic geometry

Find the folllowing limit. \[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]