Olympiad problems

2025 NCJMO, 5

Tags: combinatorics , NC(J)MO

Each element of set $\mathcal{S}$ is colored with multiple colors. A $\textit{rainbow}$ is a subset of $\mathcal{S}$ which has amongst its elements at least $1$ color from each element of $\mathcal{S}$. A $\textit{minimal rainbow}$ is a rainbow where removing any single element gives a non-rainbow. Prove that the union of all minimal rainbows is $\mathcal{S}$. [i]Grisham Paimagam[/i]

2025 NCJMO, 3

Tags: NC(J)MO

Alan has three pins that form a right triangle with legs $1$ and $4$ at first. Every move, he can pick any one of the pins, pick any new point $\mathcal{P}$ on the opposite side, and move the pin to its $\textit{reflection}$ across $\mathcal{P}$. After a series of moves, can the pins eventually form a right triangle with legs $2$ and $3$? [center][img width=75]https://cdn.artofproblemsolving.com/attachments/e/0/f50c28102c8cefd1fd1f4c327fd3f24f12748d.png[/img][/center] [i]Jason Lee[/i]

2025 NCMO, 5

Tags: trigonometry , algebra , NC(J)MO

Let $x$ be a real number. Suppose that there exist integers $a_0,a_1,\dots,a_n$, not all zero, such that \[\sum_{k=0}^n a_k\cos(kx)=\sum_{k=0}^na_k\sin(kx)=0.\] Characterize all possible values of $\cos x$. [i]Grisham Paimagam[/i]

2025 NCMO, 4

Tags: NC(J)MO

Let $P$ be a polynomial. Suppose that there exists a rational $q$ such that $P(m)=q^n$ for infinitely many integers $(m,n)$. Prove that $P(x)=c\cdot Q(x)^k$ for some integer constants $c$ and $k$ and irreducible polynomial $Q$ with rational coefficients. (Here, a polynomial is $\textit{irreducible}$ if it can't be factored into the product of non-constant polynomials with rational coefficients.) [i]Jason Lee[/i]

2025 NCJMO, 1

Tags: NC(J)MO

Cerena, Faith, Edna, and Veronica each have a cube. Aarnő knows that the side lengths of each of their cubes are distinct integers greater than $1$, and he is trying to guess their exact values. Each girl fully paints the surface of her cube in Carolina blue before splitting the entire cube into $1\times1\times1$ cubes. Then, [list=disc] [*] Cerena reveals how many of her $1\times1\times1$ cubes have exactly $0$ blue faces. [*] Faith reveals how many of her $1\times1\times1$ cubes have exactly $1$ blue faces. [*] Edna reveals how many of her $1\times1\times1$ cubes have exactly $2$ blue faces. [*] Veronica reveals how many of her $1\times1\times1$ cubes have exactly $3$ blue faces. [/list] Whose side lengths can Aarnő deduce from these statements? [i]Jason Lee[/i]

2025 NCMO, 2

Tags: geometry , NC(J)MO

In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent. [i]Alan Cheng[/i]

2025 NCJMO, 4

Tags: geometry , NC(J)MO

In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent. [i]Alan Cheng[/i]

2025 NCMO, 1

Tags: combinatorics , NC(J)MO

A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$? [i]Aaron Wang[/i]

2025 NCJMO, 2

Tags: combinatorics , NC(J)MO

A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$? [i]Aaron Wang[/i]

2025 NCMO, 3

Tags: combinatorial geometry , NC(J)MO

Let $\mathcal{S}$ be a set of points in the plane such that for each subset $\mathcal{T}$ of $\mathcal{S}$, there exists a convex $2025$-gon which contains all of the points in $\mathcal{T}$ and none of the rest of the points in $\mathcal{S}$ but not $\mathcal{T}$. Determine the greatest possible number of points in $\mathcal{S}$. [i]Jason Lee[/i]