RMO 2023 Problems and Solutions

Version 1

Problem 1

Given a triangle $A B C$ with $\angle A C B=120^{\circ}$ . The point $L$ is marked on the side $A B$ so that $C L$ is the bisector of $\angle A C B$ . The points $N$ and $K$ are marked on the sides $A C$ and $B C$ , respectively, so that $C N+C K=C L$ . Prove that the triangle $K L N$ is equilateral.

Problem 2

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

Problem 3

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

$f(a)=b c ; f(b)=c a ; f(c)=a b .$

Determine $f(a+b+c)$ in terms of $a, b, c$ .

Problem 4

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, 5, 6, 7, 8 there exists a number in $X$ which contains both of them.

Determine the smallest possible value of $N$ .

Problem 5

The side-lengths $a, b, c$ of a triangle $A B C$ are positive integers. Let

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

for any positive integer $n$ . If $\frac{T_2}{2 T_1}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $A B C$ .

Problem 6

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at $P$ . The point $Q$ is chosen on the segment $B C$ so that $P Q$ is perpendicular to $A C$ . Prove that the line joining the centres of the circumcircles of triangles $A P D$ and $B Q D$ is parallel to $A D$ .

Version 2

Problem 1

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 $a b c d$ for all $(a, b, c, d) \in S$ .

Problem 2

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 $\operatorname{arc} A B$ . 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.

Problem 3

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

$(s(n))^2=m \quad \text { and } \quad(s(m))^2=n .$

Problem 4

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.

Problem 5

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy

$\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n$

Problem 6

Consider a set of 16 points arranged in a $4 \times 4$ 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.

Version 1

Problem 1

Given a triangle $A B C$ with $\angle A C B=120^{\circ}$ . The point $L$ is marked on the side $A B$ so that $C L$ is the bisector of $\angle A C B$ . The points $N$ and $K$ are marked on the sides $A C$ and $B C$ , respectively, so that $C N+C K=C L$ . Prove that the triangle $K L N$ is equilateral.

Problem 2

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

Problem 3

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

$f(a)=b c ; f(b)=c a ; f(c)=a b .$

Determine $f(a+b+c)$ in terms of $a, b, c$ .

Problem 4

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, 5, 6, 7, 8 there exists a number in $X$ which contains both of them.

Determine the smallest possible value of $N$ .

Problem 5

The side-lengths $a, b, c$ of a triangle $A B C$ are positive integers. Let

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

for any positive integer $n$ . If $\frac{T_2}{2 T_1}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $A B C$ .

Problem 6

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at $P$ . The point $Q$ is chosen on the segment $B C$ so that $P Q$ is perpendicular to $A C$ . Prove that the line joining the centres of the circumcircles of triangles $A P D$ and $B Q D$ is parallel to $A D$ .

Version 2

Problem 1

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 $a b c d$ for all $(a, b, c, d) \in S$ .

Problem 2

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 $\operatorname{arc} A B$ . 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.

Problem 3

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

$(s(n))^2=m \quad \text { and } \quad(s(m))^2=n .$

Problem 4

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.

Problem 5

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy

$\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n$

Problem 6

Consider a set of 16 points arranged in a $4 \times 4$ 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.