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# RMO 2023 Problems and Solutions

#### Version 1

###### Problem 1

Given a triangle with . The point is marked on the side so that is the bisector of . The points and are marked on the sides and , respectively, so that . Prove that the triangle is equilateral.

###### Problem 2

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

###### Problem 3

Let be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers we have

Determine in terms of .

###### Problem 4

The set of four-digit numbers formed from the digits satisfies the following condition:
for any two different digits from 1, 2, 3, 4, 5, 6, 7, 8 there exists a number in which contains both of them.

Determine the smallest possible value of .

###### Problem 5

The side-lengths of a triangle are positive integers. Let

for any positive integer . If and , determine all possible perimeters of the triangle .

###### Problem 6

The diagonals and of a cyclic quadrilateral meet at . The point is chosen on the segment so that is perpendicular to . Prove that the line joining the centres of the circumcircles of triangles and is parallel to .

#### Version 2

###### Problem 1

Let be the set of all positive integers and . Find the largest positive integer such that divides for all .

###### Problem 2

Let be a semicircle with as the bounding diameter and let be a variable chord of the semicircle of constant length such that lie in the interior of the . Let be a point on the diameter such that and are equally inclined to the line . Prove that
(a) the measure of is a constant;
(b) the circumcircle of triangle passes through a fixed point.

###### Problem 3

For any natural number , expressed in base 10 , let denote the sum of all its digits. Find all natural numbers and such that and

###### Problem 4

Let be two intersecting circles with centres respectively. Let be a line that intersects at points and at points such that are collinear in that order. Let the perpendicular bisector of segment intersect at points ; and the perpendicular bisector of segment intersect at points such that are on the same side of . Prove that the midpoints of and are collinear.

###### Problem 5

Let be positive integers. Determine all positive real numbers which satisfy

###### Problem 6

Consider a set of 16 points arranged in a 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 with . The point is marked on the side so that is the bisector of . The points and are marked on the sides and , respectively, so that . Prove that the triangle is equilateral.

###### Problem 2

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

###### Problem 3

Let be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers we have

Determine in terms of .

###### Problem 4

The set of four-digit numbers formed from the digits satisfies the following condition:
for any two different digits from 1, 2, 3, 4, 5, 6, 7, 8 there exists a number in which contains both of them.

Determine the smallest possible value of .

###### Problem 5

The side-lengths of a triangle are positive integers. Let

for any positive integer . If and , determine all possible perimeters of the triangle .

###### Problem 6

The diagonals and of a cyclic quadrilateral meet at . The point is chosen on the segment so that is perpendicular to . Prove that the line joining the centres of the circumcircles of triangles and is parallel to .

#### Version 2

###### Problem 1

Let be the set of all positive integers and . Find the largest positive integer such that divides for all .

###### Problem 2

Let be a semicircle with as the bounding diameter and let be a variable chord of the semicircle of constant length such that lie in the interior of the . Let be a point on the diameter such that and are equally inclined to the line . Prove that
(a) the measure of is a constant;
(b) the circumcircle of triangle passes through a fixed point.

###### Problem 3

For any natural number , expressed in base 10 , let denote the sum of all its digits. Find all natural numbers and such that and

###### Problem 4

Let be two intersecting circles with centres respectively. Let be a line that intersects at points and at points such that are collinear in that order. Let the perpendicular bisector of segment intersect at points ; and the perpendicular bisector of segment intersect at points such that are on the same side of . Prove that the midpoints of and are collinear.

###### Problem 5

Let be positive integers. Determine all positive real numbers which satisfy

###### Problem 6

Consider a set of 16 points arranged in a 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.

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### One comment on “RMO 2023 Problems and Solutions”

1. ... says:

Where are the solutions?