Let $a,b \in \mathbb{R}$ with $a < b,$ 2 real numbers. We say that $f: [a,b] \rightarrow \mathbb{R}$ has property $(P)$ if there is an integrable function on $[a,b]$ with property that \[ f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b]. \] Show that for all real number $t$ there exist a unique function $f:[a,b] \rightarrow \mathbb{R}$ with property $(P),$ such that $\int_{a}^{b} f(x) \text{dx} = t.$

For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i]

Let be a function $ f:\mathbb{R}\rightarrow\mathbb{R} $ satisfying the following properties: $ \text{(i)} $ is continuous on the rational numbers. $ \text{(ii)} f(x)<f\left( x+\frac{1}{n}\right) , $ for any real $ x $ and natural $ n. $ Prove that $ f $ is increasing. [i]Gabriel Mârşanu, Mihai Piticari[/i]

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have \[\int_0^1f(P(x))dx=0\] Prove that $f(x)=0,\ (\forall)x\in [0,1]$. [i]Mihai Piticari[/i]

Mongolia TST 2011 Test 1 #2 Let $p$ be a prime number. Prove that: $\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem) Note: I believe they meant to say $p>2$ as well.

Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that \[8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , \] with equality if and only if triangle $ABC$ is equilateral.

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

Let be a function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that satisfies the relation $$ \sqrt{x^2-x+1}\le f(x) e^{f(x)}\le \sqrt{x^2+x+1} , $$ for any positive real number $ x. $ Prove that [b]a)[/b] $ \lim_{x\to\infty } f(x)=\infty . $ [b]b)[/b] $ \lim_{x\to\infty } (1/x)^{1/f(x)} =1/e. $

A real-valued function $ f$ on $ \mathbb{Q}$ satisfies the following conditions for arbitrary $ \alpha, \beta \in \mathbb{Q}:$ [b](i)[/b] $ f(0) \equal{} 0,$ [b](ii)[/b] $ f(\alpha) > 0 \text{ if } \alpha \neq 0,$ [b](iii)[/b] $ f(\alpha \cdot \beta) \equal{} f(\alpha)f(\beta),$ [b](iv)[/b] $ f(\alpha \plus{} \beta) \leq f(\alpha) \plus{} f(\beta),$ [b](v)[/b] $ f(m) \leq 1989$ $ \forall m \in \mathbb{Z}.$ Prove that \[ f(\alpha \plus{} \beta) \equal{} \max\{f(\alpha), f(\beta)\} \text{ if } f(\alpha) \neq f(\beta).\]

If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]