Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that \[ g(a, b, c)\ge \sqrt{2abc}\] [i](M. Ivanov)[/i] [hide="Remarks (containing spoilers!)"] 1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$. 2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$. [/hide]

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

\[\int_0^1 \frac{\log(x)}{\sqrt x}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

Let $f:[0,1]\to\mathbb R$ be a continuous function such that $\int^1_0x^mf(x)dx=0$ for $m=0,1,\ldots,1999$. Prove that $f$ has at least $2000$ zeroes on the segment $[0,1]$.

In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("$y=x^2$",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("$y=2x^2$",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("$y=f(x)$",(24,576/10),W); label("$O(0,0)$",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("$P$",(20,400/25),E); label("$B$",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("$A$",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.

Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(2.5cm); draw((0,0)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--cycle); draw((0,1)--(1,1)--(1,0), dotted); [/asy]

Find all continuously differentiable functions $f:[0,1]\to(0,\infty)$ such that $\frac{f(1)}{f(0)}=e$ and $$\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.$$