Olympiad Problem Sets 1¶

Here’s a set of Olympiad-level problems, suitable for national and international mathematical olympiads, like the USA(J)MO, USAMO, or IMO. These problems involve proof-based solutions and cover advanced topics in algebra, geometry, number theory, and combinatorics.

1. Algebra (Functional Equation)¶

Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \):

\[ f(x + y) + f(x - y) = 2f(x)f(y) \]

2. Geometry¶

Let \( ABC \) be a triangle, and let \( I \) be its incenter. The incircle touches \( BC \), \( CA \), and \( AB \) at \( D \), \( E \), and \( F \), respectively. The lines \( AI \), \( BI \), and \( CI \) intersect the incircle again at \( P \), \( Q \), and \( R \), respectively. Prove that \( P \), \( Q \), and \( R \) are collinear.

3. Number Theory¶

Prove that for any positive integer \( n \), the number \( n^5 - n^3 + n \) is divisible by 5.

4. Combinatorics¶

In a tournament with \( n \) players, every pair of players plays exactly one match, and there are no ties. A player is called a champion if they defeat every other player they play against. Prove that for any tournament, there exists a player who is either a champion or can be defeated by a champion.

5. Algebra (Inequalities)¶

Let \( a, b, c \) be non-negative real numbers such that \( a + b + c = 1 \). Prove that:

\[ a^2 + b^2 + c^2 + 2abc \leq 1 \]

6. Geometry¶

Let \( ABC \) be a triangle with \( AB = AC \). The angle bisector of \( \angle BAC \) intersects \( BC \) at \( D \). The circle with center \( D \) and radius \( DB \) intersects \( AB \) at \( P \) (other than \( B \)). Prove that \( AP = AC \).

7. Number Theory¶

Prove that there are infinitely many pairs of positive integers \( (x, y) \) such that:

\[ x^2 - y^5 = 1 \]

8. Combinatorics¶

Given a set of 100 points on a plane, no three of which are collinear, each pair of points is connected by a line segment. The line segments are colored either red or blue. Prove that there exists a monochromatic triangle.

9. Algebra (Polynomials)¶

Let \( P(x) \) be a polynomial with integer coefficients. Suppose there exists an integer \( a \) such that \( P(a) \) is divisible by \( p \), where \( p \) is a prime.

Prove that for any integer \( b \), if \( b \equiv a \pmod{p} \), then \( P(b) \equiv P(a) \pmod{p} \).

10. Geometry¶

Let \( O \) be the circumcenter of an acute triangle \( ABC \). The perpendicular from \( A \) to \( BC \) meets the circumcircle of triangle \( ABC \) again at \( P \).

Prove that the reflection of \( O \) across \( BC \) lies on the line \( AP \).

11. Number Theory (Diophantine Equations)¶

Solve the equation \( x^2 + y^2 + z^2 + w^2 = 2024 \) in integers.

12. Combinatorics¶

In a group of \( 2n \) people, every person knows exactly \( n+1 \) people. Prove that there exists a pair of people who do not know each other but have a common acquaintance.

These problems are designed for highly skilled mathematics students and require rigorous logical reasoning, creativity, and deep insight. Each solution typically involves several steps and can span multiple areas of mathematics.

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