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]: Prove th

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