Olympiad problems

2007 Germany Team Selection Test, 1

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

1990 China Team Selection Test, 2

Tags: function , algebra

Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$

2008 Turkey MO (2nd round), 1

Tags: function , algebra

$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions $ a)$ $ f(0,0)\equal{}1$ , $ f(0,1)\equal{}1$ , $ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)\equal{}0$ and $ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$ find the sum $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$

2000 District Olympiad (Hunedoara), 3

Tags: real analysis , function , wilkosz , primitive

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that: $ \text{(i)}\quad f(0)=0 $ $ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $ $ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $ Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$ is primitivable.

1965 Miklós Schweitzer, 4

Tags: function , combinatorial geometry , domain , advanced fields , algebra

The plane is divided into domains by $ n$ straight lines in general position, where $ n \geq 3$. Determine the maximum and minimum possible number of angular domains among them. (We say that $ n$ lines are in general position if no two are parallel and no three are concurrent.)

1990 Romania Team Selection Test, 7

Tags: function , algebra , induction

The sequence $ (x_n)_{n \geq 1}$ is defined by: $ x_1\equal{}1$ $ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$ Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.

2024 Romania National Olympiad, 1

Tags: function , real analysis

Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$

2008 IMAR Test, 4

Tags: function , algebra

Show that for any function $ f: (0,\plus{}\infty)\to (0,\plus{}\infty)$ there exist real numbers $ x>0$ and $ y>0$ such that: $ f(x\plus{}y)<yf(f(x)).$ [b]Dan Schwarz[/b]

2009 Balkan MO, 4

Tags: function , induction , algebra

Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

2013 India National Olympiad, 4

Tags: floor function , combinatorics , function , induction

Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.

2009 Jozsef Wildt International Math Competition, W. 21

Tags: function , riemann zeta function

If $\zeta$ denote the Riemann Zeta Function, and $s>1$ then $$\sum \limits_{k=1}^{\infty} \frac{1}{1+k^s}\geq \frac{\zeta (s)}{1+\zeta (s)}$$

1999 Miklós Schweitzer, 6

Tags: real analysis , function , hilbert space

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

2005 Postal Coaching, 9

Tags: function , combinatorics

In how many ways can $n$ identical balls be distributed to nine persons $A,B,C,D,E,F,G,H,I$ so that the number of balls recieved by $A$ is the same as the total number of balls recieved by $B,C,D,E$ together,.

2010 IMO Shortlist, 6

Tags: algebra , function , positive integer , functional equation

Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$ [i]Proposed by Alex Schreiber, Germany[/i]

2013 Middle European Mathematical Olympiad, 1

Tags: function , function equation , algebra

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that \[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \] holds for all $ x, y \in \mathbb{R}$.

2009 India National Olympiad, 3

Tags: floor function , function , algebra , special factorizations

Find all real numbers $ x$ such that: $ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$ (Here $ [x]$ denotes the largest integer not exceeding $ x$.)

1961 Putnam, B6

Tags: function , differential equation , root

Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.

2001 Romania National Olympiad, 4

Tags: logarithm , real analysis , limit , integration , function

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2010 Today's Calculation Of Integral, 593

Tags: function , integration , calculus , trigonometry , calculus computations

For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

2014 Dutch BxMO/EGMO TST, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

2011 Laurențiu Duican, 3

Tags: real analysis , function , continuity , fractional part

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

1993 AMC 12/AHSME, 12

Tags: function

If $f(2x)=\frac{2}{2+x}$ for all $x>0$, then $2f(x)=$ $ \textbf{(A)}\ \frac{2}{1+x} \qquad\textbf{(B)}\ \frac{2}{2+x} \qquad\textbf{(C)}\ \frac{4}{1+x} \qquad\textbf{(D)}\ \frac{4}{2+x} \qquad\textbf{(E)}\ \frac{8}{4+x} $

2007 Nicolae Coculescu, 1

Tags: algebra , function , find all functions

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{R} $ satisfying the equation $$ f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) , $$ for any rational numbers $ x,y. $ [i]Mihai Onucu Drîmbe[/i]

2019 PUMaC Team Round, 15

Tags: algebra , function

Determine the number of functions $f : Z^+ \to Z^+$ so that for all positive integers $x$ we have $f(f(x)) = f(x + 1)$, and $\max (f(2), . . . , f(14)) \le f(1) - 2 = 12$.

1985 IMO Longlists, 97

Tags: function , trigonometry , symmetry , pythagorean theorem , analytic geometry , geometry , perpendicular bisector

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$