Bangladesh Mathematical Olympiad 2025 - PROBLEMS

Problem 1

One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements:

• There are no more than three of us here. We are all liars.
• There are no more than four of us here. Not all of us are liars.
• There are five of us here. At least three of us are liars.

How many people are in the room and how many of them are liars?

Problem 2

Find all real solutions to the equation \( (x^2 - 9x + 19)^{(x^2 - 3x + 2)} = 1 \).

Problem 3

Let \( \triangle ABC \) be a given triangle with circumcenter \( O \) and orthocenter \( H \). Let \( D, E, F \) be the feet of the perpendiculars from \( A, B, C \) to the opposite sides, respectively. Let \( A' \) be the reflection of \( A \) with respect to \( EF \). Prove that \( HOA'D \) is a cyclic quadrilateral.

Problem 4

Let set \( S \) be the smallest set of positive integers satisfying the following properties:

• 2 is in set \( S \).
• If \( n^2 \) is in set \( S \), then \( n \) is also in set \( S \).
• If \( n \) is in set \( S \), then \( (n + 5)^2 \) is also in set \( S \).

Determine which positive integers are not in set \( S \).

Problem 5

In an \( N \times N \) table consisting of small unit squares, some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows, the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of \( N \)?

Problem 6

Suppose \( X \) and \( Y \) are the common points of two circles \( \omega_1 \) and \( \omega_2 \). The third circle \( \omega \) is internally tangent to \( \omega_1 \) and \( \omega_2 \) at \( P \) and \( Q \), respectively. Segment \( XY \) intersects \( \omega \) in points \( M \) and \( N \). Rays \( PM \) and \( PN \) intersect \( \omega_1 \) in points \( A \) and \( D \); rays \( QM \) and \( QN \) intersect \( \omega_2 \) in points \( B \) and \( C \), respectively. Prove that \( AB = CD \).

Problem 7

Yamin and Tamim are playing a game with subsets of \( \{1, 2, \dots, n\} \) where \( n \geq 3 \). Tamim starts the game with the empty set. On Yamin's turn, he adds a proper non-empty subset of \( \{1, 2, \dots, n\} \) to his collection \( F \) of blocked sets. On Tamim's turn, he adds or removes a positive integer less than or equal to \( n \) to or from their set but Tamim can never add or remove an element so that his set becomes one of the blocked sets in \( F \). Tamim wins if he can make his set to be \( \{1, 2, \dots, n\} \). Yamin wins if he can stop Tamim from doing so. Yamin goes first and they alternate making their moves. Does Tamim have a winning strategy?

Problem 8

Let \( a, b, m, n \) be positive integers such that \( \gcd(a, b) = 1 \) and \( a > 1 \). Prove that if \( a^m + b^m \mid a^n + b^n \), then \( m \mid n \).

Problem 9

Let \( \triangle ABC \) be an acute triangle and \( D \) be the foot of the altitude from \( A \) onto \( BC \). A semicircle with diameter \( BC \) intersects segments \( AB, AC \) and \( AD \) in the points \( F, E \) and \( X \), respectively. The circumcircles of the triangles \( DEX \) and \( DXF \) intersect \( BC \) in points \( L \) and \( N \), respectively, other than \( D \). Prove that \( BN = LC \).

Problem 10

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(x + f(y^2)) + f(xy) = f(x) + yf(x + y) \) for all \( x, y \in \mathbb{R} \).