Sunaina thinks absurd

Welcome back! I tried the previous year INMO problems. And yes, I did a lot of them, here are a few! Here are the problems! ( they are kept in random order) [2008 P2]:  Find all triples $ \left(p,x,y\right)$ such that $ p^x=y^4+4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers. [some INMO]: Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying $$m(4m^2+m+12)=3(p^n-1).$$ [2015 P3]  Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$. [2005 P6]:  Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]for all $x, y, z \in \mathbb{R}$. [2012 P6] Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and $(i) f(xy) + f(x)f(y) = f(x) + f(y)$ $(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $ for all $x,y \in \mathbb{Z}$, simultaneously. $(a)$ Find the set of all possible values of the function $f$. $(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find ...

Welcome back to this blog!  My blog completed it's 1-year :D i.e 29 Nov. I am so happy that this blog grew so much and it didn't die! It also crossed 10k views! Thanks a lot! Enjoy the problems. It's more of a miscellaneous problem set with the level being INMO or less. So here are $20$ INMO level problems. Problems:  Problem 1: [IMO 2009/P1]  Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$ Problem 2[ USEMO 2021 P4]:  Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$. Can line $EF$ share a point with the circumcircle of triangle $AMN?$ Problem 3 [ 2006 n5]: P...

  Welcome back! So today I will be sharing a few problems which I did last week and some ISLs. Easy ones I guess. Happy September 2021!  Problem[APMO 2018 P1]: Let $ABC$ be a triangle with orthocenter $H$ and let $M$ and $N$ denote the midpoints of ${AB}$ and ${AC}$. Assume $H$ lies inside quadrilateral $BMNC$, and the circumcircles of $\triangle BMH$ and $\triangle CNH$ are tangent. The line through $H$ parallel to ${BC}$ intersects $(BMH)$ and $(CNH)$ again at $K$, $L$ respectively. Let $F = {MK} \cap {NL}$, and let $J$ denote the incenter of $\triangle MHN$. Prove that $FJ = FA$. Proof:  By angle chase, we get $\angle FKL=\angle FLK.$    Hence $KL||MN\implies FM=FN.$   And we get $\angle MFN=2A\implies F$ is circumcentre if $(AMN)\implies FA=FM=FN.$   And we get $\angle MHN=180-2A$    Hence $MFHN$ is cyclic.    By fact 5, $ME=FJ=FN\implies FJ=FA.$ Problem[Shortlist 2007 G3]: Let $ABCD$ be a trapezoid whose diagonals meet at $P$....