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1. $2$ circles $\Gamma$ and $\sum,$ with centers $O$ and $O',$ respectively, are such that $O'$ lies on $\Gamma.$ Let $A$ be a point on $\sum,$ and let $M$ be the midpoint of $AO'.$ Let $B$ be another point on $\sum,$ such that $AB~||~OM.$ Then prove that the midpoint of $AB$ lies on $\Gamma.$
SOLUTION: Here
2. Let $P(x)=x^2+ax+b$ be a quadratic polynomial where $a,b$ are real numbers. Suppose $\langle P(-1)^2,P(0)^2,P(1)^2\rangle$ be an $AP$ of positive integers. Prove that $a,b$ are integers.
SOLUTION: Here
3. Show that there are infinitely many triples $(x,y,z)$ of positive integers, such that $x^3+y^4=z^{31}.$
SOLUTION: Here
4. Suppose $36$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite.
SOLUTION: Here
5. Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.
SOLUTION: Here
6. Show that there are infinitely many positive real numbers, which are not integers, such that $a\left(3-\{a\}\right)$ is an integer. (Here, $\{a\}$ is the fractional part of $a.$ For example$,~\{1.5\}=0.5;~\{-3.4\}=1-0.4=0.6.$)
SOLUTION: Here

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