RMO 2015

In a cyclic quadrilateral $ABCD$ , let the diagonals $AC$ and $BD$ intersect at $X$ . Let the circumcircles of triangles $AXD$ and $BXC$ intersect again at $Y$ . If $X$ is the incentre of triangle $ABY$ , show that $\angle CAD = 90^o$ .

Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$ . Prove that $a_1 - a_2$ is even.

Find all fractions which can be written simultaneously in the forms $\frac{7k- 5}{5k - 3}$ and $\frac{6l - 1}{4l - 3}$ , for some integers $k, l$ .

Let ABC be a right triangle with $\angle B = 90^{\circ}$ .Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.

Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.

Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $s \neq t$ such that $P(s) = t$ and $P(t) = s$ . Prove that $b-st$ is a root of $x^2 + ax + b - st$ .

Two circles X and Y in the plane intersect at two distinct points A and B such that the centre of Y lies on X. Let points C and D be on X and Y respectively, so that C, B and D are collinear. Let point E on Y be such that DE is parallel to AC. Show that AE = AB.

2.Let $P(x) = x^2 + ax + b$ be a quadratic polynomial where a, b are real numbers. Suppose $P(-1)^2$ , $P(0)^2$ , $P(1)^2$ is an Arithmetic progression of positive integers. Prove that a, b are integers.

3. Show that there are infinitely many triples (x,y,z) of integers such that $x^3 + y^4 = z^{31}$ .

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.

Let $ABC$ be a triangle. Let $B'$ denote the reflection of $b$ in the internal angle bisector $l$ of $\angle A$.Show that the circumcentre of the triangle $CB'I$ lies on the line $l$ where $I$ is the incentre of $ABC$.

Let $P(x)=x^{2}+ax+b$ be a quadratic polynomial where $a$ is real and $b \neq 2$, is rational. Suppose $P(0)^{2},P(1)^{2},P(2)^{2}$ are integers, prove that $a$ and $b$ are integers.

Two circles $\Gamma$ and $\Sigma$ intersect at two distinct points $A$ and $B$. A line through $B$ intersects $\Gamma$ and $\Sigma$ again at $C$ and $D$, respectively. Suppose that $CA=CD$. Show that the centre of $\Sigma$ lies on $\Gamma$.

Let $ABCD$ be a convex quadrilateral with $AB=a$ , $BC=b$ , $CD=c$ and $DA=d$ . Suppose $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

Determine the number of $3-$ digit numbers in base $10$ having at least one $5$ and at most one $3$ .

Let P(x) a non-constant polynomial whose coefficients are positive integers. If P(n) divides P(P(n)-2015) for every natural no. n, prove that P (-2015) =0.

Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$ , $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$ .)

Let $ABC$ be a right-angled triangle with $\angle B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$ . Draw $DE \perp AB$ and $DF \perp BC$ . Let $P, Q, R$ and $S$ be respectively the incentres of triangle $DF C, DBF, DEB$ and $DAE$ . Suppose $S, R, Q$ are collinear. Prove that $P, Q, R, D$ lie on a circle.

Let $S=\{1,2,\cdots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S$ , say $(A_1, A_2, A_3)$ , such that $A_1\cup A_2\cup A_3=S$ . Determine, in terms of $n$ , $$\sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|$$

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2-2xyz=1$ . Prove that $$(1+x)(1+y)(1+z)\le 4+4xyz.$$

The length of each side of a convex quadrilateral $ABCD$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.