Difference between revisions of "2025 IMO Problems"

Revision as of 19:56, 19 July 2025

Problems of the 2025 IMO.

Day I

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 $n\ge3$ 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 $a+b\le n+1$ , the point $(a,b)$ is on at least one of the lines; and
exactly $k$ of the $n$ lines are sunny.

Solution

Problem 2

Let $\Omega$ and $\Gamma$ be circles with centers $M$ and $N$ , respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$ . Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$ . Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$ , such that points $C$ , $M$ , $N$ , and $D$ lie on the line in that order. Let $P$ be the circumcenter of triangle $ACD$ . Line $AP$ intersects $\Omega$ again at $E\neq A$ . Line $AP$ intersects $\Gamma$ again at $F\neq A$ . Let $H$ be the orthocenter of triangle $PMN$ .

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

(The orthocenter of a triangle is the point of intersection of its altitudes.)

Solution

Problem 3

Let $\mathbb{N}$ denote the set of positive integers. A function $f: \mathbb{N} \rightarrow \mathbb{N}$ is said to be bonza if

$f(a)$ 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 $n$ .

Solution

Day II

Problem 4

A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1,a_2,\dots$ consists of positive integers, each of which has at least three proper divisors. For each $n\ge1$ , the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$ .

Determine all possible values of $a_1$ .

Solution

Problem 5

Alice and Bazza are playing the inekoalaty game, a two‑player 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 $x_1 + x_2 + \cdots + x_n \le \lambda n$ .

If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that $x_1^2 + x_2^2 + \cdots + x_n^2 \le n$

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.

Solution

Problem 6

Consider a 2025 x 2025 grid of unit squares. Matlida 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 Matlida 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.

Solution

@@ Line 3: / Line 3: @@
 ==Day I==
 ===Problem 1===
-A line in the plane is called <i>sunny</i> if it is not parallel to any of the <math>x</math>–axis, the <math>y</math>–axis, or the line <math>x+y=0</math>.
+A line in the plane is called <i>sunny</i> if it is <b>not</b> parallel to any of the <math>x</math>–axis, the <math>y</math>–axis, and the line <math>x+y=0</math>.
 Let <math>n\ge3</math> be a given integer. Determine all nonnegative integers <math>k</math> such that there exist <math>n</math> distinct lines in the plane satisfying both of the following:
-* For all positive integers <math>a</math> and <math>b</math> with <math>a+b\le n+1</math>, the point <math>(a,b)</math> lies on at least one of the lines; and
+* for all positive integers <math>a</math> and <math>b</math> with <math>a+b\le n+1</math>, the point <math>(a,b)</math> is on at least one of the lines; and
-* Exactly <math>k</math> of the <math>n</math> lines are sunny.
+* exactly <math>k</math> of the <math>n</math> lines are sunny.
 [[2025 IMO Problems/Problem 1|Solution]]
 ===Problem 2===
-Let <math>\Omega</math> and <math>\Gamma</math> be circles with centres <math>M</math> and <math>N</math>, respectively, such that the radius of <math>\Omega</math> is less than the radius of <math>\Gamma</math>. Suppose <math>\Omega</math> and <math>\Gamma</math> intersect at two distinct points <math>A</math> and <math>B</math>. Line <math>MN</math> intersects <math>\Omega</math> at <math>C</math> and <math>\Gamma</math> at <math>D</math>, so that <math>C, M, N, D</math> lie on <math>MN</math> in that order. Let <math>P</math> be the circumcentre of triangle <math>ACD</math>. Line <math>AP</math> meets <math>\Omega</math> again at <math>E\neq A</math> and meets <math>\Gamma</math> again at <math>F\neq A</math>. Let <math>H</math> be the orthocentre of triangle <math>PMN</math>.
+Let <math>\Omega</math> and <math>\Gamma</math> be circles with centers <math>M</math> and <math>N</math>, respectively, such that the radius of <math>\Omega</math> is less than the radius of <math>\Gamma</math>. Suppose circles <math>\Omega</math> and <math>\Gamma</math> intersect at two distinct points <math>A</math> and <math>B</math>. Line <math>MN</math> intersects <math>\Omega</math> at <math>C</math> and <math>\Gamma</math> at <math>D</math>, such that points <math>C</math>, <math>M</math>, <math>N</math>, and <math>D</math> lie on the line in that order. Let <math>P</math> be the circumcenter of triangle <math>ACD</math>. Line <math>AP</math> intersects <math>\Omega</math> again at <math>E\neq A</math>. Line <math>AP</math> intersects <math>\Gamma</math> again at <math>F\neq A</math>. Let <math>H</math> be the orthocenter of triangle <math>PMN</math>.
 Prove that the line through <math>H</math> parallel to <math>AP</math> is tangent to the circumcircle of triangle <math>BEF</math>.
-(The <i>orthocentre</i> of a triangle is the point of intersection of its altitudes.)
+(The <i>orthocenter</i> of a triangle is the point of intersection of its altitudes.)
 [[2025 IMO Problems/Problem 2|Solution]]
 ===Problem 3===
-Let <math>\mathbb{N}</math> denote the set of positive integers. A function <math>f: \mathbb{N} \rightarrow \mathbb{N}</math> is said to be bonza if <math>f(a)</math> divides <math>b^{a} - f(b)^{f(a)}</math> for all positive integers <math>a</math> and <math>b</math>. Determine the smallest real constant <math>c</math> such that <math>f(n) \leq cn</math> for all bonza functions <math>f</math> and all positive integers <math>n</math>.
+Let <math>\mathbb{N}</math> denote the set of positive integers. A function <math>f: \mathbb{N} \rightarrow \mathbb{N}</math> is said to be <i>bonza</i> if <center><math>f(a)</math> divides <math>b^{a} - f(b)^{f(a)}</math></center> for all positive integers <math>a</math> and <math>b</math>. Determine the smallest real constant <math>c</math> such that <math>f(n) \leq cn</math> for all bonza functions <math>f</math> and all positive integers <math>n</math>.
 [[2025 IMO Problems/Problem 3|Solution]]
@@ Line 28: / Line 28: @@
 ==Day II==
 ===Problem 4===
-A proper divisor of a positive integer <math>N</math> is a positive divisor of <math>N</math> other than <math>N</math> itself.
+A <i>proper divisor</i> of a positive integer <math>N</math> is a positive divisor of <math>N</math> other than <math>N</math> itself.
 The infinite sequence <math>a_1,a_2,\dots</math> consists of positive integers, each of which has at least three proper divisors. For each <math>n\ge1</math>, the integer <math>a_{n+1}</math> is the sum of the three largest proper divisors of <math>a_n</math>.
@@ Line 37: / Line 37: @@
 ===Problem 5===
-Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number <math>\lambda</math> which is known to both players. On the <math>n</math>th turn of the game (starting with <math>n=1</math>) the following happens:
+Alice and Bazza are playing the <i>inekoalaty</i> game, a two‑player game whose rules depend on a positive real number <math>\lambda</math> which is known to both players. On the <math>n</math>th turn of the game (starting with <math>n=1</math>) the following happens:
-- If <math>n</math> is odd, Alice chooses a nonnegative real number <math>x_n</math> such that <math>x_1 + x_2 + \cdots + x_n \le \lambda n</math>.
+*If <math>n</math> is odd, Alice chooses a nonnegative real number <math>x_n</math> such that <math>x_1 + x_2 + \cdots + x_n \le \lambda n</math>.
-- If <math>n</math> is even, Bazza chooses a nonnegative real number <math>x_n</math> such that <math>x_1^2 + x_2^2 + \cdots + x_n^2 \le n</math>
+*If <math>n</math> is even, Bazza chooses a nonnegative real number <math>x_n</math> such that <math>x_1^2 + x_2^2 + \cdots + x_n^2 \le n</math>
 If a player cannot choose a suitable <math>x_n</math>, 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.
@@ Line 59: / Line 59: @@
 * [[2025 IMO]]
 * [[IMO Problems and Solutions, with authors]]
-* [[Mathematics competition resources]]
+* [[Mathematics competitions resources]]
 {{IMO box|year=2025|before=[[2024 IMO]]|after=[[2026 IMO]]}}

2025 IMO (Problems) • Resources
Preceded by 2024 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 2026 IMO
All IMO Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "2025 IMO Problems"

Revision as of 19:56, 19 July 2025

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also