English (eng), day 1
Tuesday, 15 July 2025
Problem 1. A line in the plane is called sunny if it is not parallel to any of the x-axis, the y-axis,
and the line x + y = 0.
Let n ⩾ 3 be a given integer. Determine all nonnegative integers k such that there exist n distinct
lines in the plane satisfying both of the following:
• for all positive integers a and b with a + b ⩽ n + 1, the point (a, b) is on at least one of the
lines; and
• exactly k of the n lines are sunny.
Problem 2. Let Ω and Γ be circles with centres M and N , respectively, such that the radius of
Ω is less than the radius of Γ. Suppose circles Ω and Γ intersect at two distinct points A and B.
Line M N intersects Ω at C and Γ at D, such that points C, M , N and D lie on the line in that
order. Let P be the circumcentre of triangle ACD. Line AP intersects Ω again at E ̸= A. Line AP
intersects Γ again at F ̸= A. Let H be the orthocentre of triangle P M N .
Prove that the line through H parallel to AP is tangent to the circumcircle of triangle BEF .
(The orthocentre of a triangle is the point of intersection of its altitudes.)
Problem 3. Let N denote the set of positive integers. A function f : N → N is said to be bonza if
f (a) divides ba − f (b)f (a)
for all positive integers a and b.
Determine the smallest real constant c such that f (n) ⩽ cn for all bonza functions f and all positive
integers n.
Language: English Time: 4 hours and 30 minutes.
Each problem is worth 7 points.
English (eng), day 2
Wednesday, 16 July 2025
Problem 4. A proper divisor of a positive integer N is a positive divisor of N other than N itself.
The infinite sequence a1 , a2 , . . . consists of positive integers, each of which has at least three proper
divisors. For each n ⩾ 1, the integer an+1 is the sum of the three largest proper divisors of an .
Determine all possible values of a1 .
Problem 5. Alice and Bazza are playing the inekoalaty game, a two-player game whose rules
depend on a positive real number λ which is known to both players. On the nth turn of the game
(starting with n = 1) the following happens:
• If n is odd, Alice chooses a nonnegative real number xn such that
x1 + x2 + · · · + xn ⩽ λn.
• If n is even, Bazza chooses a nonnegative real number xn such that
x21 + x22 + · · · + x2n ⩽ n.
If a player cannot choose a suitable number xn , the game ends and the other player wins. If the
game goes on forever, neither player wins. All chosen numbers are known to both players.
Determine all values of λ for which Alice has a winning strategy and all those for which Bazza has
a winning strategy.
Problem 6. Consider a 2025 × 2025 grid of unit squares. Matilda wishes to place on the grid some
rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and
every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place so that each row and each column
of the grid has exactly one unit square that is not covered by any tile.
Language: English Time: 4 hours and 30 minutes.
Each problem is worth 7 points.
