Issue #2 - October 20, 2020

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

Note: Problems are not in order of difficulty.

Problem 5A. (not solved yet) 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 11A. Proposed by Luca Tu
Call a number C unlucky if there exists a multiset $S \subseteq \mathbb{N}^*$ such that the sum of all the elements in $S$ is $169$, and the product of all the elements in $S$ is $C$. Find the largest unlucky number. (A multiset is a set that can have duplicate elements.)

Problem 12A. Proposed by Luca Tu
In the following calculation, each letter stands for a different digit (e.g. if $X = 0$ then $Y \neq 0$)

	C	L	O	V	E	R
+	C	R	O	C	U	S
=	V	I	O	L	E	T

Decode the equation (find the numbers CLOVER, CROCUS and VIOLET). (Note O in the equation is the letter O and not the digit 0).

Problem 13A. Proposed by Vedaant Srivastava
Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 13$, $B_1C_1 = 14$, $C_1A_1 = 15$. Let $\Gamma_1 : (O_1, r_1)$ be the incircle of $A_1B_1C_1$. Construct $B_2, C_2$ on sides $A_1B_1$ and $A_1C_1$ respectively such that $B_2C_2 \parallel B_1C_1$ and $B_2C_2$ is tangent to $\Gamma_1$.
Now let $\Gamma_2 : (O_2, r_2)$ be the incircle of $\triangle AB_2C_2$. Construct $B_3, C_3$ on sides $AB_2, AC_2$ respectively such that $B_3C_3 \parallel B_2C_2$ and $B_3C_3$ is tangent to $\Gamma_2$.
Continue this process, shading in the circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4, \dots$. What is the total area of the shaded region?

Problem 14A. Proposed by Vedaant Srivastava
Consider an acute triangle $ABC$ with orthocenter $H$. Let $H_A$ be the reflection of $H$ over $BC$. Let $E$ and $F$ be the projections of $H_A$ onto $AB$ and $AC$ respectively. Prove that $EF$ bisects $HH_A$.

Problem 15A. 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 16A. Proposed by Nicholas Sullivan
The soccer ball, otherwise known as the truncated icosahedron, is composed of 12 pentagonal faces and 20 hexagonal faces, such that one pentagon and two hexagons meet at each vertex. If each vertex is labelled with a distinct integer between 1 and 70, inclusive, show that there must be at least one edge whose vertices are not relatively prime..

Problem 17A. Proposed by Nikola Milijevic
Express the sum of A and B in simplest form if \[A=\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}\] and \[B=1+\cfrac{1}{A+\cfrac{1}{A+\cfrac{1}{A+\dots}}}\]

Problem 18A. Proposed by Andy Kim
Find the probability that a randomly chosen 6-digit palindrome with no digits equal to 0 is divisible by 13.

Problem 19A. Proposed by Luca Tu
Solve for $x \in \mathbb{R}$: $x^3+10x^2+25x+6=0$ without using Cardano's formula or its equivalent. Show your work.

Problem 20A. 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 $$ AB\times FE + CD \times HG= AD \times IL + CB \times MN.$$
 

Problems B (Open to all Grades)

 

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) 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 8B. (not solved yet) 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 11B. Proposed by Max Jiang
Find all complex roots of the polynomial \[x^8-4x^7+10x^6-16x^5+19x^4-16x^3+10x^2-4x+1=0.\]

Problem 12B. Proposed by Max Jiang
Instead of an election, Donald and Joe play a grid-coloring game to decide who gets to be the next president. Given a $n\times m$ grid of unit squares, the candidates will take turns coloring $1\times k$ or $k\times 1$ sub-arrays of the grid, where $k$ is a positive integer, such that no cell in the sub-array is already colored. The last player to move is declared the winner (at which point the grid will be completely colored). Find all pairs $(n,m)$ such that the first player to move has a winning strategy.

Problem 13B. 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 14B. Proposed by Ken Jiang
For integer $n>1$, determine which expression is greater: $n^{n!}$ or $(n^n)!$.

Problem 15B. Proposed by Ken Jiang
Let $f_n(x)$ be a function such that: if $x\leq n$ then $f_n(x)=1$ and if $x>n$ then $f_n(x)=\sum\limits_{i=x-n}^{x-1} f_n(i)$. Prove that \[\sum\limits_{i=1}^{\infty} \dfrac{f_n(i)}{2^i}=n\]

Problem 16B. Proposed by Nicolas Sullivan
Let $\Delta ABC$ be an equilateral triangle with side length $1$. Choose points $M$, $N$ and $P$ on $BC$, $CA$ and $AB$ respectively. Find the smallest possible ratio between the area of $\Delta MNP$ and $\Delta ABC$ if \begin{equation*} \begin{split} 64\left[\frac{1}{BM^3} + \frac{1}{CN^3} + \frac{1}{AP^3}\right] + \left[\frac{1}{CM^3} + \frac{1}{AN^3} + \frac{1}{BP^3}\right] = 729. \end{split} \end{equation*}

Problem 17B. Proposed by Nikola Milijevic
Recall that the Fibonacci sequence is defined by $f_{1}=1,f_{2}=1$ and $f_{k}=f_{k-1}+f_{k-2}$ for $k\ge3$. Prove that $f_{n+4} \equiv 2f_{n}$ (mod 3) for all $n \in \mathbb{N}$

Problem 18B. Proposed by Andy Kim
There is an event in Ottawa, which has attendees sitting in a row of $n$ chairs. However, to comply to the social distancing guidelines, there must be a space of at least one empty chair between attendees. Given that there is at least 1 attendee, show that the number of possible seating arrangements is $F_{n+2}-1$. (note: $F_n$ is the $n$'th fibonacci number, which is defined by $F_1 = 1$, $F_2 = 1$, $F_n = F_{n-1} + F_{n-2}$ for natural $n \geq 3$)

Problem 19B. Proposed by Luca Tu
Find the maximum of $\log a \cdot \log c$, given: $$\log a + \log_b c = 3 \text{ and } \log b + \log_a c = 4$$ (where $\log x$ denotes $\log_{10} x$)

Problem 20B. Proposed by DC
Given six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ on a circle such that $\overset{\frown}{A_1A_2}=\overset{\frown}{A_2A_3}$, $\overset{\frown}{A_3A_4}=\overset{\frown}{A_4A_5}$, and $\overset{\frown}{A_5A_6}=\overset{\frown}{A_6A_1}$. Prove that the three points obtained by intersecting $A_4A_6$ with $A_1A_3$, $A_2A_4$ with $A_1A_5$, and $A_3A_5$ with $A_2A_6$ are collinear.