Olympiad problems

PEN E Problems, 25

Tags: function , logarithm

Prove that $\ln n \geq k\ln 2$, where $n$ is a natural number and $k$ is the number of distinct primes that divide $n$.

2010 AIME Problems, 9

Tags: 3d geometry , quadratic , inequalities , function

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

1992 Polish MO Finals, 1

Tags: function , induction , inequalities , algebra

The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.

1998 USAMO, 3

Tags: trigonometry , function , n-variable inequality , inequalities

Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that \[ \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1. \] Prove that \[ \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}. \]

2012 Bogdan Stan, 1

Tags: function , find all functions , algebra

Find the functions $ f:\mathbb{Z}\longrightarrow\mathbb{Z}_{\ge 0} $ that satisfy the following two conditions: $ \text{(a)} f(m+n)=f(n)+f(m)+2mn,\quad\forall m,n\in\mathbb{Z} $ $ \text{(b)} f(f(1))-f(1) $ is a perfect square [i]Marin Ionescu[/i]

2019 Philippine TST, 3

Tags: quadratic , inequalities , functional inequality , function , absolute value , constant , algebra

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

2017 IMC, 2

Tags: function , calculus

Let $f:\mathbb R\to(0,\infty)$ be a differentiabe function, and suppose that there exists a constant $L>0$ such that $$|f'(x)-f'(y)|\leq L|x-y|$$ for all $x,y$. Prove that $$(f'(x))^2<2Lf(x)$$ holds for all $x$.

2010 AMC 12/AHSME, 4

Tags: function

If $ x < 0$, then which of the following must be positive? $ \textbf{(A)}\ \frac{x}{|x|}\qquad \textbf{(B)}\ \minus{}x^2\qquad \textbf{(C)}\ \minus{}2^x\qquad \textbf{(D)}\ \minus{}x^{\minus{}1}\qquad \textbf{(E)}\ \sqrt[3]{x}$

2008 Romania National Olympiad, 3

Tags: function , algebra , domain , real analysis , calculus , integration , analytic geometry

Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that \[ \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$. Prove that $ f''(c)\equal{}0$.

2009 Today's Calculation Of Integral, 434

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

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

2001 Miklós Schweitzer, 1

Tags: function , set theory , real analysis

Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.

MIPT student olimpiad spring 2022, 1

Tags: real analysis , function

Sequence of uniformly continuous functions $f_n:R \to R$ uniformly converges to a function $f:R\to R$. Can we say that $f$ is uniformly continuous?

2013 Romania National Olympiad, 4

Tags: function , algebra

Consider a nonzero integer number $n$ and the function $f:\mathbb{N}\to \mathbb{N}$ by \[ f(x) = \begin{cases} \frac{x}{2} & \text{if } x \text{ is even} \\ \frac{x-1}{2} + 2^{n-1} & \text{if } x \text{ is odd} \end{cases}. \] Determine the set: \[ A = \{ x\in \mathbb{N} \mid \underbrace{\left( f\circ f\circ ....\circ f \right)}_{n\ f\text{'s}}\left( x \right)=x \}. \]

2012 Serbia JBMO TST, 4

Tags: function , analytic geometry

In a coordinate system there are drawn the graphs of the functions $y=ax+b$ and $y=bx+a, (a\neq b)$. Their intersection is marked with red and their intersections with the $Oy$ axis are marked with blue. Everything is erased except the marked points. Using only a ruler and a compass, find the origin of the coordinate system.

2015 Korea National Olympiad, 1

Tags: function , algebra , functional equation

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y,z$, we have $$(f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz-y)$$

2010 Contests, 2

Tags: function , rhombus , vieta , geometry

Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

2011 Iran MO (3rd Round), 2

Tags: function , topology

Prove that these three statements are equivalent: (a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$. (b) There is no antipodal mapping $f:S^n \to S^{n-1}$. (c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.

2009 ISI B.Stat Entrance Exam, 6

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

Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

2003 Italy TST, 3

Tags: function , algebra

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]

2006 Pre-Preparation Course Examination, 2

Tags: function , algebra

Show that there exists a continuos function $f: [0,1]\rightarrow [0,1]$ such that it has no periodic orbit of order $3$ but it has a periodic orbit of order $5$.

2015 CIIM, Problem 5

Tags: function

There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted: 1. Each person moves to the next seat clockwise. 2. Only the ones in seats 1 and $k$ exchange their seats. Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.

2014 AIME Problems, 5

Tags: function , algebra , polynomial , quadratic , vieta

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.

1981 Canada National Olympiad, 1

Tags: function , floor function , algebra

For any real number $t$, denote by $[t]$ the greatest integer which is less than or equal to $t$. For example: $[8] = 8$, $[\pi] = 3$, and $[-5/2] = -3$. Show that the equation \[[x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345\] has no real solution.

2003 National Olympiad First Round, 32

Tags: function

The function $f$ satisfies $f(x)+3f(1-x)=x^2$ for every real $x$. If $S=\{x \mid f(x)=0 \}$, which one is true? $\textbf{(A)}$ $S$ is an infinite set. $\textbf{(B)}$ $\{0,1\} \subset S$ $\textbf{(C)}$ $S=\phi$ $\textbf{(D)}$ $S = \{(3+\sqrt 3)/2, (3-\sqrt 3)/2\}$ $\textbf{(E)}$ None of above

2004 India IMO Training Camp, 4

Tags: function , arithmetic sequence , number theory

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$