Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Thursday, December 31st, to submit solutions.
Problem 15A. (not solved yet) Proposed by Alexander Monteith-Pistor
Let $x_0 = 0$. For $n \ge 1$: if $x_{n-1}$ is even, $x_n$ is randomly chosen from the multiples of $3$ between $x_{n-1} - 10$ and $x_{n-1} + 10$ (inclusive); if $x_{n-1}$ is odd, $x_n$ is randomly chosen from the multiples of $4$ between $x_{n-1} - 10$ and $x_{n-1} + 10$ (inclusive). Find the probability that $x_{5}$ is even.
Problem 20A. (not solved yet) Proposed by DC
In a cyclic quadrilateral ABCD, P is the intersection of the diagonals and E, M, G and L are the midpoints on sides AB, BC, CD and DA. By projecting the medians PE, PM, PG and PL on sides AB, BC, CD and DA, four segments are obtained: EF, MN, GH and LI. Prove that $$ AB\times FE + CD \times HG= AD \times IL + CB \times MN.$$
Problem 21A. Proposed by Max Jiang
Alice, Bob, Carl, and Daniel like flipping coins. They are playing a game where they repeatedly flip coins simultaneously until each player has gotten at least 1 heads. What is the expected value of the number of times each player will flip his or her coin?
Problem 22A. Proposed by Proposed by Nikola Milijevic
For a fixed natural number n, which k would maximize the following expression: \( \binom{3n+k}{2n} \binom{3n-k}{2n} \)
Problem 23A. Proposed by Nikola Milijevic
For how many natural numbers $n$ is the expression \( n! +3 \) a perfect cube?
Problem 24A. Proposed by Vedaant Srivastava
Let $\{a_1, a_2, a_3, \dots \}$ be sequence of rational numbers such that $a_1 = 2$ and
$$a_n = \frac{3}{2}\left(\frac{a_{n-1}}{3}+\frac{1}{a_{n-1}}\right)$$
for $n \ge 2$. Determine an explicit formula (in terms of $n$) for $a_n$.
Problem 25A. Proposed by Nicholas Sullivan
Consider isosceles triangle $ABC$ ($AB = AC$, with point $D$ on $AB$ and $E$ on $AC$ such that $AD = DE = EB = BC$. Find $\mathrm{m}(\angle ABC)$.
Problem 26A. Proposed by Frederick Pu
Remark: The original submission was modified by AE.
Suppose you lived on an island where every islander $x$ can be described by an ordered
list of 169 real number attributes, $(x_1, x_2, \dots, x_{169})$. One of the islanders, Bob, has attributes $B = (1, 2, \dots, 169)$. We define an intelligence function, $I : \mathbb{R}^{169} \mapsto \mathbb{R}$, which takes an islander's 169 attributes as an input and outputs their intelligence. Prove that there exists an intelligence function such that no islander has a higher intelligence than Bob.
Problem 27A. Proposed by DC
In any triangle ABC, find the ratio $$\frac{sinA+sinB+sinC}{cotA+cotB+cotC}$$ function of the altitudes in the triangle.
Note 1: Problems are not in order of difficulty.
Note 2: You will have until midnight on Thursday, December 31st, to submit solutions.
Problem 4B. (not solved yet) Proposed by Ken Jiang
Alice and Bob are playing a new game. Starting from $N$, they take turns counting down $F_i$ numbers, where $F_i$ must be a member of the Fibonacci sequence. Alice goes first, and the player who counts to $1$ is the winner. Show that there are infinite values of $N$ such that, no matter how Alice plays, Bob can win.
Problem 5B. (not solved yet) Proposed by Alexander Monteith-Pistor
Let $S = \{0, 1, 2, ..., 2020\}$ and $f : S \to S$ satisfy
$$f(x)f(y)f(xy) = f(f(x+y))$$
for all $x, y \in S$ with $xy \le 2020$ and $x + y \le 2020$. Find the maximum possible value of
$$\sum_{i=0}^{2020} f(i)$$
Problem 13B. (not solved yet) Proposed by Alexander Monteith-Pistor
Let $p(x)$ be a polynomial with integer coefficients satisfying
$$p(x+2020)^{2020} = p(x^{2020}) + 2020$$
Find all possible values of $p(0)$.
Problem 21B. Proposed by Alexander Monteith-Pistor
Let $A_n$ denote the number of tuples $(a_1, ..., a_n)$ of positive integers which satisfy $a_1 = 1$ and $a_{k+1} \mid 2a_{k}$ for all $1 \le k \le n-1$ (note $A_0 = 1$). Prove that
$$A_{n+1} = \sum_{i=0}^{n} A_iA_{n-i}$$
Problem 22B. Proposed by Alexander Monteith-Pistor
Find the number of different black and white colourings of an $n \times n$ grid such that every square has exactly one black neighbour. Two squares are said to be neighbours if they share an edge.
Problem 23B. Proposed by Max Jiang
Mabel has a bag of $n$ marbles whose weights are $1,2,3,\ldots,n$. She draws the marbles one by one without replacement out of the bag. At any point, the probably of drawing any given marble is proportional to its weight. For example, if the bag had 3 marbles of weights 1, 2, and 4, the probability of drawing the marble of weight 1, 2, and 4 are $\frac{1}{7},\frac{2}{7}$, and $\frac{4}{7}$, respectively. Find a closed form formula for the probability that Mabel draws the marbles in order of increasing weight.
Problem 24B. Proposed by Proposed by Vedaant Srivastava
Donnie holds 5 exclusive parties in order to increase his popularity among his 538 friends. At each party, some of his 538 friends were present and some were not. The dollar ticket prices at each event were 2, 3, 3, 4, and 5 respectively. Each event made over 1000 dollars in ticket sales. Prove that Donnie can choose two people out of his 538 friends such that at least one of them was present at each party.
Problem 25B. Proposed by Proposed by Andy Kim
Let
$$p(x) = x^5 + x^4 + ax^3 + bx^2 + cx + d$$
be a real polynomial with $1 + i$ and $2 + i$ as roots, where $i^2 = -1$. Find $a + b + c + d$.
Problem 26B. Proposed by Proposed by Andy Kim
Alice and Bob each have a bucket with 1 liter of water in it to start. Every minute, Alice moves half of the water in her bucket to Bob's, and Bob moves a fourth of the water in his bucket to Alice's. Assuming Alice and Bob are immortal, find the limiting value of the amount of water in each bucket.
Problem 27B. Proposed by Nicholas Sullivan
Consider a continuous function $f(x)$ that satisfies $f(x)f(y)=f(x)+f(y)-f(xy)$, for any real numbers $x,y\neq 0$. If $f(2)=\frac{1}{2}$, then find all possible functions $f(x)$.
Problem 28B. Proposed by DC
Find the relationship between $a$ and $b$ such that the maximum and the minimum of the function defined on real numbers:
$$\frac{x^2+2ax+1}{x^2+2bx+1}$$ are satisfying $\max=-2\min$.