New posts in Contest Math

$f(x+y)=f(x)+f(y)+99$, $f(100)=101$, find $f(7)$.

In quadrilateral $ABCD$, $\angle BAC=\angle CAD=2\,\angle ACD=40^\circ$ and $\angle ACB=70^\circ$. Find $\angle ADB$.

prove $\sum_{i=1}^{n}\sqrt{a_i}\ge (n-1)\sum_{i=0}^{n}\frac{1}{\sqrt{a_i}}$

A summation of a series based on the Fibonacci sequence.

$3^{123} \mod 100$

For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$

Suppose $\angle BAC = 60^\circ$ and $\angle ABC = 20^\circ$. A point $E$ inside $ABC$ satisfies $\angle EAB=20^\circ$ and $\angle ECB=30^\circ$.

Olympiads Number Theory Problem: Show that there are no positive $a,b,c\in\Bbb Z$ such that $\frac{a^2+b^2+c^2}{3(ab+bc+ca)}\in\Bbb Z$

Numbers from $1,\frac12,\frac13,...........\frac{1}{2010}$ are written and any two $x,y$ are taken and we replace $x,y$ by just $x+y+xy$

Let $P(x)=a_0+a_1x+a_2 x^2+a_3x^3+.......+a_nx^n$ and $P(1)=4$ and $P(5)=136$