1 Algebra

A1. Quadratic solitaire is a single-player game. To start the game, the player chooses two distinct nonzero integers \(a\) and \(b\) and writes the equation \(x^2 + ax + b = 0\) on a blackboard. On each turn, if the equation currently written on the blackboard has two distinct nonzero integer solutions \(x = u\) and \(x = v\), then the player erases the equation and replaces it by one of the equations \(x^2 + ux + v = 0\) and \(x^2 + vx + u = 0\) of their choosing. Otherwise, the game ends. Determine all initial choices of \(a\) and \(b\) such that quadratic solitaire can be played forever.

A2. The sunshine cost of a sequence \(a_1, a_2, \ldots, a_{100}\) of integers is the largest possible value of
\[
|(a_1 + a_2 + \cdots + a_i) - a_j|
\]
as \(i\) and \(j\) vary over all integers \(1, 2, \ldots, 100\).

Determine the smallest possible sunshine cost over all sequences \(a_1, a_2, \ldots, a_{100}\) of pairwise distinct integers.

A3. Alice and Bazza are playing the inekoalaty game, a twoplayer game whose rules depend on a positive real number \(\lambda\) which is known to both players. On the \(n\)th turn of the game (starting with \(n = 1\)) the following happens: If \(n\) is odd, Alice chooses a nonnegative real number \(x_n\) such that
\[
x_1 + x_2 + \cdots + x_n \leq \lambda n.
\]
If \(n\) is even, Bazza chooses a nonnegative real number \(x_n\) such that
\[
x_1^2 + x_2^2 + \cdots + x_n^2 \leq n.
\]
If a player cannot choose a suitable \(x_n\), 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 \(\lambda\) for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

A4. Let \(\mathbb{Z}_{\geq 0}\) be the set of all nonnegative integers. Let \(f : \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}\) be an unbounded function such that, if \(m\) and \(n\) are nonnegative integers satisfying
\[
f(m + n) = \max\{f(0), f(1), \ldots, f(m + n)\},
\]
then
\[
f(m + n) = f(m) + f(n).
\]
Prove that there exist positive integers \(A, B, C\) and \(D\) such that for all nonnegative integers \(n\),
\[
f(An + B) = Cn + D.
\]
We say that \(f\) is unbounded if for each nonnegative integer \(N\), there exists some nonnegative integer \(n\) such that \(f(n) \geq N\).

A5. Let \(P_n(x, y, z) = x^n + y^n + z^n - xyz^{n-2} - xy^{n-2}z - x^{n-2}yz\) be a polynomial of three variables. Find all positive integers \(n \geq 2\) such that there exists non-constant polynomials \(Q_n(x, y, z)\) and \(R_n(x, y, z)\) with integer coefficients such that
\[
P_n(x, y, z) = Q_n(x, y, z) \cdot R_n(x, y, z)
\]

A6. Let \(S\) be a set of positive integers, possibly infinite, such that no positive integer greater than 1 divides all elements of \(S\). Determine all non-periodic infinite sequences \(a_1, a_2, a_3, \ldots\) of positive integers such that, for all positive integers \(n\), \(a_n \leq |a_{n+\ell} - \ell|\) for all \(\ell\) in \(S\), and
\( a_n = |a_{n+\ell} - \ell| \) for at least one \( \ell \) in \( S \). We say that an infinite sequence \( a_1, a_2, a_3, \ldots \) is periodic if there exists a positive integer \( t \) such that \( a_n = a_{n+t} \) for all positive integer \( n \).

A7. Let \( k \geq 3 \) be a positive integer. Prove that there exists a unique \( k \)-tuplet\((x_1, x_2, \ldots, x_k)\) of positive real numbers such that \( x_1 \geq x_2 \geq \cdots \geq x_k \), and

\[
\left\lfloor nx_1 + \frac{1}{2} \right\rfloor + \left\lfloor nx_2 + \frac{1}{2} \right\rfloor + \cdots + \left\lfloor nx_k + \frac{1}{2} \right\rfloor = n
\]

for every positive integer \( n \).

A8. Tim and Tam play a game. To start the game, Tim writes some nonzero real numbers not necessarily distinct, on a blackboard. In each round, the following happens. First, Tam chooses a polynomial \( P_k(x) = a_k x^k + a_{k-1} x^{k-1} + \cdots + a_1 x + a_0 \) whose coefficients \( a_k, a_{k-1}, \ldots, a_1, a_0 \) are all of the numbers currently written on the blackboard in some order. (For example, if the numbers on the blackboard, are \( 4, -3, 4 \), then \( k = 2 \) and Tam may choose the polynomial \( 4x^2 - 3x + 4 \) or \( -3x^2 + 4x + 4 \) but not \( 4x - 3 \) or \( -3x^2 - 3x + 4 \).) Then, if the equation \( P_k(x) = 0 \) has no real solutions, the game is stopped. Otherwise, Tim chooses a real number \( r \) such that \( P_k(r) = 0 \) and writes it on the blackboard, so there is now one more number on the blackboard. Determine whether can ensure that the game is never stopped, no matter what Tam does

2 Combinatorics

C1. A line in the plane is called sunny if it is not parallel to any of the xaxis, the yaxis, or the line \( x + y = 0 \).

Let \( n \geq 3 \) 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 \( a + b \leq n + 1 \), the point \( (a, b) \) lies on at least one of the lines; and exactly \( k \) of the \( n \) lines are sunny.

C2. There is a row of \( n \) paddocks, labelled from left to right with the integers 1 to \( n \), where \( n \geq 3 \). Skippy the Kangaroo grazes in the paddocks according to the following rule:

If Skippy is grazing in paddock \( k \), then she makes a sequence of \( k \) hops from a paddock to an adjacent paddock. The first of the \( k \) hops is always to the right, unless Skippy is in paddock \( n \), in which case it is to the left. Each of the following \( k - 1 \) hops is in the same direction as the previous hop, unless Skippy is in paddock 1 or \( n \). Skippy grazes in the paddock that she is in after the \( k \)th hop.

For example, if \( n = 8 \) and Skippy is grazing in paddock 3, then the next four paddocks she grazes in are 6, 4, 8 and 2, in this order.

Skippy continues grazing in this way indefinitely. A paddock is overgrazed if Skippy eventually grazes in it, regardless of the paddock that she starts in. Determine all \( n \geq 3 \) for which there is exactly one overgrazed paddock.

C3. There are 2025 white balls and 2025 black balls arranged in a row. A balanced segment is a contiguous nonempty segment of balls that contains the same number of white balls as black balls. A balanced removal is the operation of selecting a balanced segment \( S \), removing all the balls in \( S \), and shifting left the balls that were to the right of \( S \) so that the remaining balls form a contiguous row. Determine the smallest positive integer \( N \) satisfying the following property: for every configuration of balls and for every integer \( k \) satisfying \( 0 \leq k \leq 2025 \), there exists a sequence of at most \( N \) balanced removals after which there are exactly \( 2k \) remaining balls.

C4. Bluey is placing lollies on a \( 1000 \times 1000 \) grid. Initially, every cell of the grid is empty. At each step, Bluey chooses a row or column in which the total number of lollies is a multiple of 3, chooses an empty cell in this row or column, and places a lolly in it. If there is no such row or column, Bluey stops.

Determine the maximum number of lollies that Bluey can place.

C5. Let \( a_1, a_2, \ldots, a_n \) be a permutation of \( 1, 2, \ldots, n \). A pair \( (i, j) \) with \( i < j \) is called an inversion if \( a_i > a_j \). An inversion is called odd if \( i - j \) is odd.

(a) Prove that the total number of inversions is at most 5 times the number of odd inversions.

(b) There is a permutation such that the total number of inversions is more than 4.99 times the number of odd inversions.

C6. On a \( 45 \times 45 \) square grid there is an echidna. From any cell, the echidna can move to any other cell that shares a side. The echidna makes a sequence of 2024 moves, after which it has visited each cell exactly once. Prove that it is possible to write the numbers from 1 to 2025, one in each cell, in such a way that: For any pair of adjacent cells in the same row, the cell containing the larger number was visited earlier. For any pair of adjacent cells in the same column, the cell containing the larger number was visited later.

C7. Let \( P \) be a regular 100–gon. An Aussie triangulation is formed by dividing \( P \) into a set \( T \) of non-overlapping triangles such that: there are exactly 2025 different points, including the 100 vertices of \( P \) that are a vertex of at least one triangle in \( T \), and no three of these 2025 points are collinear. A quokkalateral is a convex quadrilateral that consists of two triangles in \( T \) sharing a common side. Two quokkalaterals may share a triangle in \( T \). Determine the minimum number of quokkalateral among all Aussie triangulation.

C8. Consider a \( 2025 \times 2025 \) 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.

3 Geometry

G1. Let \(ABC\) be a triangle such that \(\angle A\) is obtuse. Let \(D\) and \(E\) be points on sides \(AB\) and \(AC\), respectively, such that \(BDEC\) is cyclic. Suppose the circumcircle of triangle \(ADE\) intersects side \(BC\) at two points \(X\) and \(Y\), where \(B\), \(X\), \(Y\) and \(C\) lie in that order. Let the circumcircles of triangles \(BDX\) and \(CEY\) be \(\Omega\) and \(\Gamma\), respectively. Let \(P\) be a point such that \(PD\) is tangent to \(\Omega\) and \(PE\) is tangent to \(\Gamma\). Let \(Q\) be a point such that \(QB\) is tangent to \(\Omega\) and \(QC\) is tangent to \(\Gamma\). Prove that points \(A\), \(P\) and \(Q\) are collinear.

G2. Let \(ABC\) be an acute-angled triangle. Let \(I_B\) and \(I_C\) be the excenters of triangle \(ABC\) opposite \(B\) and \(C\), respectively. Let \(E\) be a point on line \(BA\) such that \(\angle EI_B B = 90^\circ\). Let \(F\) be a point on line \(CA\) such that \(\angle C I_C F = 90^\circ\). Let the circle with center \(I_B\) and radius \(I_B E\) intersect the segment \(I_B B\) at \(X\). Let the circle with center \(I_C\) and radius \(I_C F\) intersect the segment \(I_C C\) at \(Y\). Prove that \(BY\) is perpendicular to \(CX\).

The excenter of a triangle \(ABC\) opposite vertex \(B\) is the center of the circle that is tangent to line segment \(AC\), to ray \(BA\) beyond \(A\), and to ray \(BC\) beyond \(C\). The excenter opposite \(C\) is similarly defined.

G3. Let \(ABC\) be an acute triangle with \(AB < AC\). Let \(\Gamma\) be the circumcircle of \(ABC\). The perpendicular from \(B\) to \(AC\) intersects \(\Gamma\) again at \(D \neq B\). Let \(Q\) be the point on \(\Gamma\), different from \(B\), such that \(AB = AQ\). Lines \(BQ\) and \(AC\) intersect at \(R\). Line \(DR\) intersects \(\Gamma\) again at \(S \neq D\). Segment \(BC\) intersects the circumcircle of triangle \(QRS\) at \(T\). Lines \(TQ\) and \(RS\) intersect at \(X\). Prove that \(CX\) is perpendicular to \(AB\).

G4. Let \(\Omega\) and \(\Gamma\) be circles with centres \(M\) and \(N\), respectively, such that the radius of \(\Omega\) is less than the radius of \(\Gamma\). Suppose \(\Omega\) and \(\Gamma\) intersect at two distinct points \(A\) and \(B\). Line \(MN\) intersects \(\Omega\) at \(C\) and \(\Gamma\) at \(D\), so that \(C, M, N, D\) lie on \(MN\) in that order. Let \(P\) be the circumcentre of triangle \(ACD\). Line \(AP\) meets \(\Omega\) again at \(E \neq A\) and meets \(\Gamma\) again at \(F \neq A\). Let \(H\) be the orthocentre of triangle \(PMN\).

Prove that the line through \(H\) parallel to \(AP\) is tangent to the circumcircle of triangle \(BEF\).

G5. Let \(ABCD\) be a convex quadrilateral with \(\angle ADC > 90^\circ\). Suppose that points \(X\) and \(Y\) lie on diagonal \(AC\) such that

\[
\angle ADX = \angle YDC = 90^\circ, \quad \angle CAD = \angle CBX, \quad \text{and} \quad \angle DCA = \angle YBA.
\]

Let \(O\) be the circumcentre of triangle \(BXY\). Suppose point \(R \neq O\) lies on the perpendicular bisector of \(AC\) such that lines \(OR\) and \(AC\) are parallel.

Prove that line \(RO\) bisects angle \(\angle DRB\).

G6. Let \(ABC\) be a triangle with \(\angle B > \angle A > \angle C\). Let \(\Omega\) be the circumcircle of triangle \(ABC\). The lines tangent to \(\Omega\) at points \(B\) and \(C\) intersect at \(P\). Lines \(BP\) and \(AC\) intersect at \(E\). Lines \(CP\) and \(AB\) intersect at \(F\). Let \(D\) be the reflection of point \(A\) in line \(EF\). The circumcircles of triangles \(DEB\) and \(DFC\) intersect again at \(Q \neq D\). Lines \(DP\) and \(BC\) intersect at \(Z\).

Prove that points \(A\), \(Z\) and \(Q\) are collinear.

G7. Let \(ABCD\) be a cyclic quadrilateral with circumcircle \(\Omega\) such that \(CD > BD > BC\). The tangents to \(\Omega\) at \(B\) and \(C\) meet at a point \(S\). A line \(\ell\) is called coastal if:

+ \(\ell\) does not pass through \(A\),
+ \(\ell\) intersects ray \(SD\) at a point \(X_\ell\) beyond \(D\), and
+ there is a circle tangent to line \(\ell\), line \(X_\ell A\), segment \(SB\) and segment \(SC\).

Prove that there is a circle that is tangent to \(\Omega\) and all coastal lines.

4 Number Theory

N1. Let n and k be positive integers such that \( n < k < 2n \). Suppose \( a_1, a_2, \ldots, a_k \) are positive integers such that \( 2^{a_1} + 2^{a_2} + \cdots + 2^{a_k} \) is divisible by \( 2^n - 1 \).

Show that at least three of \( a_1, a_2, \ldots, a_k \) have the same remainder when divided by \( n \).

N2. In an \( n \times n \) board, for each \( 1 \leq i \leq n \) and \( 1 \leq j \leq n \), the cell in the \( i \)th row from the bottom and \( j \)th column from the left contains \( \gcd(i, j) \) leaves. A koala travels from the bottom left corner to the top right corner. At each step, the koala moves up or to the right by a single cell. The koala eats the leaves in each cell it visits, including the bottom left and top right cells.

Determine the maximum number of leaves that the koala can eat.

N3. A proper divisor of a positive integer \( N \) is a positive divisor of \( N \) other than \( N \) itself.

The infinite sequence \( a_1, a_2, \cdots \) consists of positive integers, each of which has at least three proper divisors. For each \( n \geq 1 \), the integer \( a_{n+1} \) is the sum of the three largest proper divisors of \( a_n \).

Determine all possible values of \( a_1 \).

N4. Find all functions \( f : \mathbb{N} \to \mathbb{Z} \) such that for every positive integers \( n \) and \( d \) such that \( d \mid n \), there exits a positive integer \( e \mid n \) such that \( n \mid d + f(e) \).

N5. The sequence of triples of positive integers \( (a_0, b_0, c_0), (a_1, b_1, c_1), \ldots, (a_{2025}, b_{2025}, c_{2025}) \) satisfies
\[
(a_{i+1}, b_{i+1}, c_{i+1}) = (a_i + \gcd(b_i, c_i), b_i + \gcd(c_i, a_i), c_i + \gcd(a_i, b_i))
\]
for \( 0 \leq i \leq 2024 \). Determine the smallest possible value of \( a_{2025} \).

Here \( \gcd(x, y) \) denotes the greatest common divisor of integers \( x \) and \( y \).

N6. For positive integers \( n \) and \( k \), let \( f_k(n) \) denote the remainder when \( n \) is divided by \( 2^k - 1 \). A positive integer \( n \) is grouse if
\[
f_1(n) \leq f_2(n) \leq f_3(n) \leq f_4(n) \leq \ldots
\]
(a) Prove that there are infinitely many grouse numbers.

(b) Prove that there exists a positive integer \( M \) such that there are at most \( M^{1/2025} \) grouse numbers less than \( M \).

N7. Let \( \mathbb{N} \) denote the set of positive integers. A function \( f : \mathbb{N} \to \mathbb{N} \) is said to be bonza if
\[
f(a) \text{ divides } b^a - f(b)^{f(a)}
\]
for all positive integers \( a \) and \( b \).

Determine the smallest real constant \( c \) such that \( f(n) \leq cn \) for all bonza functions \( f \) and all positive integers \( n \).

N8. Prove that there are finitely many prime numbers \( p \) such that
• \( p - 8 \) is a perfect square, and
• for all odd positive integers \( k < \sqrt{p} \), the number \( p - k^2 \) has at most two distinct prime factors.