Olympiad problems

ICMC 6, 1

The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building. How long will it be until Atlantis becomes completely submerged? [i]Proposed by Ethan Tan[/i]

ICMC 7, 1

Tags: number theory , Fibonacci , ICMC

Let $F_n{}$ denote the $n{}$-th Fibonacci number. Prove that $3^{2023}$ divides \[3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.\][i]Proposed by Dylan Toh[/i]

ICMC 4, 1

Tags: geometry , Countable , uncountable , ICMC , college contests

A set of points in the plane is called [i]sane[/i] if no three points are collinear and the angle between any three distinct points is a rational number of degrees. (a) Does there exist a countably infinite sane set $\mathcal{P}$? (b) Does there exist an uncountably infinite sane set $\mathcal{Q}$? [i]Proposed by Tony Wang[/i]

ICMC 6, 2

Tags: geometry , 3D geometry , tetrahedron , ICMC , college contests

Show that if the distance between opposite edges of a tetrahedron is at least $1$, then its volume is at least $\frac{1}{3}$. [i]Proposed by Simeon Kiflie[/i]

ICMC 7, 2

Tags: combinatorics , lattice points , ICMC

Fredy starts at the origin of the Euclidean plane. Each minute, Fredy may jump a positive integer distance to another lattice point, provided the jump is not parallel to either axis. Can Fredy reach any given lattice point in 2023 jumps or less? [i]Proposed by Tony Wang[/i]

ICMC 6, 4

Tags: college contests , combinatorics , graph theory , ICMC

Let $\mathcal{G}$ be a simple graph with $n$ vertices and $m$ edges such that no two cycles share an edge. Prove that $2m < 3n$. [i]Note[/i]: A [i]simple graph[/i] is a graph with at most one edge between any two vertices and no edges from any vertex to itself. A [i]cycle[/i] is a sequence of distinct vertices $v_1, \dots, v_n$ such that there is an edge between any two consecutive vertices, and between $v_n$ and $v_1$. [i]Proposed by Ethan Tan[/i]

ICMC 5, 2

Tags: ICMC , algebra , analysis , calculus , college contests

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

ICMC 7, 4

Tags: geometry , 3D geometry , ICMC

Points $A, B, C,$ and $D{}$ lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron $ABCD.$ [i]Proposed by Fredy Yip[/i]

ICMC 7, 3

Tags: combinatorics , ICMC

There are 105 users on the social media platform Mathsenger, every pair of which has a direct messaging channel. Prove that each messaging channel may be assigned one of 100 encryption keys, such that no 4 users have the 6 pairwise channels between them all being assigned the same encryption key. [i]Proposed by Fredy Yip[/i]

ICMC 4, 1

Tags: ICMC , college contests , Sets , combinatorics , counting

Let $S$ be a set with 10 distinct elements. A set $T$ of subsets of $S$ (possibly containing the empty set) is called [i]union-closed[/i] if, for all $A, B \in T$, it is true that $A \cup B \in T$. Show that the number of union-closed sets $T$ is less than $2^{1023}$. [i]Proposed by Tony Wang[/i]

ICMC 6, 5

Tags: geometry , combinatorics , ICMC , college contests , expected value , area of a triangle

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

ICMC 4, 5

Tags: number theory , Powers of 2 , elementary-number-theory , ICMC , college contests

Find all composite positive integers $m$ such that, whenever the product of two positive integers $a$ and $b$ is $m$, their sum is a power of $2$. [i]Proposed by Harun Khan[/i]

ICMC 5, 1

Tags: combinatorics , college contests , counting , ICMC

Let $T_n$ be the number of non-congruent triangles with positive area and integer side lengths summing to $n$. Prove that $T_{2022}=T_{2019}$. [i]Proposed by Constantinos Papachristoforou[/i]

ICMC 4, 2

Tags: linear algebra , Matrices , Nilpotent , ICMC , college contests , matrix

Let $A$ be a square matrix with entries in the field $\mathbb Z / p \mathbb Z$ such that $A^n - I$ is invertible for every positive integer $n$. Prove that there exists a positive integer $m$ such that $A^m = 0$. [i](A matrix having entries in the field $\mathbb Z / p \mathbb Z$ means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of $p$.)[/i] [i]Proposed by Tony Wang[/i]

ICMC 5, 5

Tags: probability , expected value , ICMC , college contests

A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down $2/2=1$. Find the expected sum of all numbers it writes down, given that it is finite. [i]Proposed by Ethan Tan[/i]

ICMC 6, 3

Tags: ICMC , combinatorics , Fibonacci , college contests

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

ICMC 4, 6

Tags: combinatorics , ICMC , college contests

There are $n+1$ squares in a row, labelled from 0 to $n$. Tony starts with $k$ stones on square 0. On each move, he may choose a stone and advance the stone up to $m$ squares where $m$ is the number of stones on the same square (including itself) or behind it. Tony's goal is to get all stones to square $n$. Show that Tony cannot achieve his goal in fewer than $\frac{n}{1} + \frac{n}{2} + \cdots + \frac{n}{k}$ moves. [i]Proposed by Tony Wang[/i]

ICMC 5, 3

Tags: linear algebra , college contests , Determinants , ICMC

Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$. [i]Proposed by Ethan Tan[/i]

ICMC 5, 6

Tags: geometry , combinatorics , covering , college contests , ICMC

Is it possible to cover a circle of area $1$ with finitely many equilateral triangles whose areas sum to $1.01$, all pointing in the same direction? [i]Proposed by Ethan Tan[/i]

ICMC 6, 6

Tags: ICMC , college contests , real analysis , Convergence , Sequences , floor function

Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

ICMC 5, 4

Tags: reflection , Groups , number theory , mod p , college contests , ICMC

Let $p$ be a prime number. Find all subsets $S\subseteq\mathbb Z/p\mathbb Z$ such that 1. if $a,b\in S$, then $ab\in S$, and 2. there exists an $r\in S$ such that for all $a\in S$, we have $r-a\in S\cup\{0\}$. [i]Proposed by Harun Khan[/i]

ICMC 7, 6

Tags: number theory , limits , bijection , ICMC

Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection of the positive integers. Prove that at least one of the following limits is true: \[\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.\][i]Proposed by Dylan Toh[/i]

ICMC 6, 5

Tags: college contests , real analysis , continuous function , function , ICMC

Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times? [i]Proposed by Ethan Tan[/i]

2023 SG Originals, Q1

Tags: college contests , combinatorial geometry , ICMC

Two straight lines divide a square of side length $1$ into four regions. Show that at least one of the regions has a perimeter greater than or equal to $2$. [i]Proposed by Dylan Toh[/i]

ICMC 5, 1

Tags: geometry , combinatorial geometry , college contests , ICMC

Let $S$ be a set of $2022$ lines in the plane, no two parallel, no three concurrent. $S$ divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area? [i]Proposed by Tony Wang[/i]