Olympiad problems

2012 Vietnam National Olympiad, 3

Find all $f:\mathbb{R} \to \mathbb{R}$ such that: (a) For every real number $a$ there exist real number $b$:$f(b)=a$ (b) If $x>y$ then $f(x)>f(y)$ (c) $f(f(x))=f(x)+12x.$

2011 AMC 12/AHSME, 23

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

Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

2011 NIMO Problems, 13

Tags: trigonometry , function , smoothing

For real $\theta_i$, $i = 1, 2, \dots, 2011$, where $\theta_1 = \theta_{2012}$, find the maximum value of the expression \[ \sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}. \] [i]Proposed by Lewis Chen [/i]

2009 Romania National Olympiad, 4

Tags: discrete , number theory , algebra , function , combinatorics

We say that a natural number $ n\ge 4 $ is [i]unusual[/i] if, for any $ n\times n $ array of real numbers, the sum of the numbers from any $ 3\times 3 $ compact subarray is negative, and the sum of the numbers from any $ 4\times 4 $ compact subarray is positive. Find all unusual numbers.

STEMS 2021 Math Cat C, Q1

Tags: function , combinatorics , algebra

Let $M>1$ be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying [list] [*]$f(M) \ne M$ [/*] [*] $f(k)<2k$ for all $k \in \mathbb{N}$[/*] [*] $f^{f(n)}(n)=n$ for all $n \in \mathbb{N}$. For each $\ell>0$ we define $f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)$ and $f^0(n)=n$[/*] [/list] Tom wins otherwise. Prove that for infinitely many $M$, Tom wins, and for infinitely many $M$, Jerry wins. [i]Proposed by Anant Mudgal[/i]

1996 South africa National Olympiad, 6

Tags: function , algebra

The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given: \[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]

2010 Contests, 524

Tags: calculus , integration , function , calculus computations

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

2004 Turkey MO (2nd round), 4

Tags: function , induction , algebra

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$

2023 Bulgaria EGMO TST, 2

Tags: number theory , greatest common divisor , function

Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.

2016 Azerbaijan National Mathematical Olympiad, 4

Tags: function , algebra

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $$\sum_{i=1}^{2015} f(x_i + x_{i+1}) + f\left( \sum_{i=1}^{2016} x_i \right) \le \sum_{i=1}^{2016} f(2x_i)$$ for all real numbers $x_1, x_2, ... , x_{2016}.$

2007 France Team Selection Test, 2

Tags: function , induction , functional equation , algebra

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

2008 All-Russian Olympiad, 8

Tags: geometry , rectangle , analytic geometry , induction , vector , function , geometric transformation

On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.

Russian TST 2019, P2

Tags: function , algebra

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

2005 Moldova Team Selection Test, 3

Tags: calculus , integration , function , number theory

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

2012 ELMO Shortlist, 3

Tags: polynomial , function , algebra

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

2000 Iran MO (3rd Round), 3

Tags: function , floor function , algebra

Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and $f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$ $(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$. $(b)$ Determine$f$.

2023 Korea - Final Round, 2

Tags: algebra , function

Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition. (Condition) For each positive real number $x$, there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$, and the number of $y$ is finite. Prove $f(x) > f(y)$ for any positive real numbers $x < y$. ($\mathbb{R^+}$ is a set for all positive real numbers.)

2012 Online Math Open Problems, 23

Tags: function , geometry , 3d geometry

For reals $x\ge3$, let $f(x)$ denote the function \[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$, be the sequence satisfying $a_1 > 3$, $a_{2013} = 2013$, and for $n=1,2,\ldots,2012$, $a_{n+1} = f(a_n)$. Determine the value of \[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\] [i]Ray Li.[/i]

2023 India IMO Training Camp, 2

Tags: algebra , functional equation , function

Let $g:\mathbb{N}\to \mathbb{N}$ be a bijective function and suppose that $f:\mathbb{N}\to \mathbb{N}$ is a function such that: [list] [*] For all naturals $x$, $$\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. $$ [*] For all naturals $x,y$ such that $x|y$, we have $f(x)|g(y)$. [/list] Prove that $f(x)=x$. [i]Proposed by Pulkit Sinha[/i]

2006 Greece National Olympiad, 4

Tags: function , algebra

Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$, which satisfies both conditions : [b]a)[/b] $f( x + y + z) \leq 3(xy + yz + zx)$ for all real numbers $x , y , z$ and [b]b)[/b] there exist function $g$ and natural number $n$, such that $g(g(x)) = x ^ {2n + 1}$ and $f(g(x)) = (g(x)) ^2$ for every real number $x$ ?

2013 Today's Calculation Of Integral, 896

Tags: calculus , integration , limit , function , calculus computations , logarithm

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2001 USA Team Selection Test, 2

Tags: integration , function , calculus , algebra , binomial theorem , combinatorics

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

1998 Turkey Team Selection Test, 3

Tags: function , combinatorics

Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$

2012 Today's Calculation Of Integral, 846

Tags: calculus , integration , limit , function , absolute value , calculus computations , logarithm

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

2016 IFYM, Sozopol, 3

Tags: functional equation , function , algebra

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that $f(x,y)+f(y,z)+f(z,x)=0$. Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.