New problem: 100 mathemagicians are attending an important conference on mathemagics. There are 100 seats in the conference room, one assigned to each mathemagician. If a mathemagician finds their assigned seat empty, they will sit down at their assigned seat, otherwise, they will choose one empty s...

Solution: The only \(m\) that satisfies the problem conditions are \(m \in \{0, 1, 2\}\). We can easily check this by inspection. Proof: Note that \(5 \mid a_m\) for all \(m\) Assume \(m\geq 3\). Lemma: \(2m+1\) must be prime Proof: Assume \(2m+1\) is composite equal to \(uv, (u,v > 1)\), then by Zs...

New problem: In kindergarten, nurse took \(n>1\) identical cardboard rectangles and distributed them to \(n\) children; every child got one rectangle. Every child cut their rectangle into several identical squares (squares of different children could be different). Finally, the total number of squar...

Sorry for the poorly typed solution, it's late at night and I am sleepy. Solution: Let \(S\) be a partition of \(\mathbb{N}_+\) such that if \(A\in S\), we have \(\forall a \in A, |A| \mid a\). For each element \(A\) in \(S\), we define \(f\) on \(A\) such that \(f\) is an arbitrary cyclic permutati...

Ny oppgave: Let $T_1, T_2, T_3, T_4$ be pairwise distinct collinear points lying in that order. Let $\omega_1$ be a circle through $T_1$ and $T_4$; let $\omega_2$ be the circle through $T_2$ and internally tangent to $\omega_1$ at $T_1$; let $\omega_3$ be the circle through $T_3$ and externally tang...

La $C''$ være punktet på omsirkelen til $ABA'$ slik at $(A,A';B,C'') = -1$. Det er velkjent at $\triangle ABX \sim \triangle C''AX$ (EGMO kap. 4) med skaleringsfaktor $\frac{AX}{BX}$, og fra epp er $\triangle ABX \sim \triangle CAX$ med skaleringsfaktor $-\frac{AX}{BX}$. Derfor er $\triangle C''AX \...

Hype oppgave da

Følgene strategi er bra for brødrene flip. Begge roper 1 hvis de har 6er på den første plassen, og 2 hvis det ikke er 6er på første plass. Noter at sjansen på at begge roper 1 er 1/36, og at sjansen for at de begge har 6 på plass 2 og ikke 6er på plass 1 er større enn 0. Denne strategien vil derfor...

New problem: Big Flip and Lil Flip play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number oth...

I must say that I'm thoroughly disappointed by the prevalent use of Norwegian in this forum (considering many of its members must practice writing proofs in English for international contests)! Especially the use of terminology such as "omorganisering" for "permutation" makes me ...

New problem: Let $ABCD$ be a cyclic quadrilateral. Point $P$ is on line $CB$ such that $CP=CA$and $B$ lies between $C$ and $P$. Point $Q$ is on line $CD$ such that $CQ=CA$ and $D$ lies between $C$ and $Q$. Prove that the incentre of triangle $ABD$ lies on line $PQ.$

Sorry for poor formatting, am on mobile. We use induction on number of digits, and we will prove that one can find an $n$ digit number consisting of only $2$s and $5$s that is divisible by $2^n$. For $n=1$ it's trivial, as $2$ is divisible by $2^1$. Inductive step: Assume we have an n digit number c...