Issue #1 - September 17, 2020

Problems A (Not open to Grade 11 and 12 students)

Note: Problems are not in order of difficulty.

Problem 1A. Proposed by Vedaant Srivastava
Let $x$ and $y$ be positive reals such that $x+y=6$. Find the minimum value of $$\left(x+\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$$ and determine the values of $x, y$ which yield such a minimum.

Problem 2A. Proposed by Max Jiang
Find the largest $x\in\mathbb{N}$ such that $2020^x$ divides $202020!$.

Problem 3A. Proposed by Nicolas Sullivan
In December 2020, an alien civilization is discovered. In trying to communicate, we discover that they use a mathematical operation $\star$, and they only tell us that $a\star a = 1$ and $a\star(bc) = a\star b + a\star c$, for any positive real $a, b, c,\, a \neq 1$. Find $$\begin{equation*} \sum_{n=1}^{2019} \left[2020\star{\left(1+\frac{1}{n}\right)}\right]. \end{equation*}$$ Problem 4A. Proposed by Ken Jiang
Alice and Bob are playing a game. They take turns drawing cards from separate 52-card decks. After her turn, Alice places her card back into her deck and reshuffles it, while Bob does not. The player who draws the ace of spades first is the winner. If Alice goes first, what is the probability that she will win?

Problem 5A. Proposed by Alexander Monteith-Pistor
Let $\bigtriangleup ABC$ be an isosceles right triangle with right angle at $A$ and incenter $I$. The point $P$ is randomly chosen inside $\bigtriangleup ABC$. Find the probability that $\bigtriangleup PAI$ is an acute triangle (all of its angles are acute).

Problem 6A. Proposed by Andy Kim
Prove that all six-digit palindromes are divisible by 11.

Problem 7A. Proposed by Kevin Li
If $x^2+\frac{y^2}{4}= 8x+y+8$, find the largest value of $5x+ 6y$.

Problem 8A. Proposed by DC
The area of the triangle ABC is $S$ and the altitude to $a$ is $h_a$. Prove that the following relationship is true: $$\frac{a}{h_a^2bc} + \frac{b}{h_a^2ac}+ \frac{c}{h_a^2ab}= \frac{a(a^2+b^2+c^2)}{4S^2bc}.$$

Problem 9A. Proposed by DC
In any triangle ABC, prove that $$b^2sin^2C+c^2sin^2B=2bc(cosA +cosB\cdot cosC).$$

Problem 10A. Proposed by DC
Find all three digit numbers $\overline{abc}$ such that the last three digits of $(\overline{abc})^3$ are $\overline{abc}$.
 

Problems B (Open to all Grades)

Note: Problems are not in order of difficulty.

Problem 1B. Proposed by Vedaant Srivastava
Johnny the sketchy vendor has infinitely many stuffed toys lined up in a row. From left to right, he has 1 teddy bear costing 1 dollar, 2 teddy bears costing 2 dollars, 3 teddy bears costing 3 dollars, and so on. Show that the price of the $n$-th teddy bear in the row from the left is $$\left \lfloor \sqrt{2n}+\frac{1}{2}\right \rfloor$$ dollars.

Problem 2B. Proposed by Max Jiang
Compute $$\sum_{k=1}^{\infty}\dfrac{k^2}{2^k}.$$

Problem 3B. Proposed by Nicolas Sullivan
Let $F_n$ be the $n$th number in the Fibonacci sequence, defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for any natural $n$. If $\varphi = \lim_{n\to\infty} \frac{F_{n+1}}{F_n}$, show that for any $n\in\mathbb{Z}^{+}$, $$\begin{equation*} 27F_{n-1}F_nF_{n+1} < (\varphi^{-1} F_{n-1} + \varphi F_{n+1})^3. \end{equation*}$$ Problem 4B. 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. 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 6B. Proposed by Nikola Milijevic
Prove that there exist infinitely many positive integers $n$ such that $3^n + 2$ and $5^n + 2$ are both composite.

Problem 7B. Proposed by Nikola Milijevic
Prove that equation $x^5 - x = 3 - y^4$ has no solutions in integers x and y.

Problem 8B. Proposed by Andy Kim
Let a, b, and c be positive real numbers. Given that $abc = 1$, prove that $$\frac{a^3 - 1}{b^2 c^2} + \frac{b^3 - 1}{c^2 a^2} + \frac{c^3 - 1}{a^2 b^2} \geq 0$$

Problem 9B. Proposed by Kevin Li
There are $n$ students at Main High School. Each student is friends with at least $\lceil\frac{n}{k}\rceil$ other students (friendships are mutual),for some positive integral $k$. Define a friend chain to be a series of students $a_1$,$a_2$,$a_3$,$\ldots$,$a_{i-1}$,$a_i$ where $i$ is a positive integer, such that $a_1$ is friends with $a_2$, $a_2$ is friends with $a_3$, and so on until $a_{i-1}$ is friends with $a_i$. We call a friend group a set of students such that there exists a friend chain between every two students in the set. No student outside of the friend group is friends with any members of the friend group, and any student who has a friend chain to a member of a friend group is also part of that friend group. Show that there cannot exist $k$ or more friend groups.

Problem 10B. Proposed by DC
Find the values for $m$ such that the equation $$\begin{equation*} x^3-2mx^2+3x+9=0 \end{equation*}$$ has one double root.