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 $k, l$.

Suppose $28$ 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?

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.

Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$.)

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 $x^2 + ax + b - st$.

Find all integers $a,b,c$ such that $a^2 = bc + 1$ and $b^2 = ac + 1$

Suppose 32 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?

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.

Find all real numbers $a$ such that $4 < a < 5$ and $a(a-3\{a\})$ is an integer. ({x} represents the fractional part of x)

2 circles Γ and Σ, with centers O and P, respectively, are such that P lies on Γ. Let A be a point on Σ, and let M be the midpoint of AP. Let B be another point on Σ, such that AB||OM. Then prove that the midpoint of AB lies on Γ.

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 $x^3 + y^4 = z^{31}$.

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.

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.

Show that there are infinitely many positive real numbers a which are not integers such that a(a-3{a}) is an integer.

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.

Find all integers \(a,b,c\) such that \(a^{2}=bc+4\) and \(b^{2}=ca+4\).

Suppose \(40\) 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?

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 \(\Gamma\).

For how many integer values of $m$, (i) $1\le m \le 5000$ (ii) $[\sqrt{m}] =[\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function

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 $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$. Click to reveal hidden textOne additional condition must be given that $P$ is non-constant, which even though is understood.

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.