RMO 2023

Let $\mathbb{N}$ be the set of all positive integers and $S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}$ . Find the largest positive integer $m$ such that $m$ divides abcd for all $(a, b, c, d) \in S$ .

Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the arc $A B$ . Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$ . Prove that (a) the measure of $\angle C E D$ is a constant; (b) the circumcircle of triangle $C E D$ passes through a fixed point.

For any natural number $n$ , expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$(s(n))^2=m \text { and }(s(m))^2=n .$$

Let $\Omega_1, \Omega_2$ be two intersecting circles with centres $O_1, O_2$ respectively. Let $l$ be a line that intersects $\Omega_1$ at points $A, C$ and $\Omega_2$ at points $B, D$ such that $A, B, C, D$ are collinear in that order. Let the perpendicular bisector of segment $A B$ intersect $\Omega_1$ at points $P, Q$ ; and the perpendicular bisector of segment $C D$ intersect $\Omega_2$ at points $R, S$ such that $P, R$ are on the same side of $l$ . Prove that the midpoints of $P R, Q S$ and $O_1 O_2$ are collinear.

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy $$\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n .$$

Consider a set of $16$ points arranged in $4 \times 4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

Given a triangle $ABC$ with $\angle ACB = 120^{\circ}.$ A point $L$ is marked in the side $AB$ such that $CL$ bisects $\angle ACB.$ Points $N$ and $K$ are chosen in the sides $AC$ and $BC$ such that $CK+CN=CL.$ Prove that the triangle $KLN$ is equilateral.

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:

$$f(a)=bc , f(b)=ac, f(c)=ab$$ Dertermine $f(a+b+c)$ in terms of $a,b,c$ .

The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:

for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them.

Determine the smallest possible value of $N$ .

The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:

$$T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}$$

for any positive integer $n$ . If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$ .

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$ . The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$ . Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$ .