Source: 2018 Canadian Open Math Challenge Part C Problem 1 ----- At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center] $\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$, [color=transparent](A.)[/color]layer of this pyramid? $\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. [color=transparent](B.)[/color]How many cans are on the bottom layer of the prism? $\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle. [color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...) [color=transparent](C.)[/color]For example, a prism could be composed of the following layers: [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center] Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.

Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

Lukas is playing pool on a table shaped like an equilateral triangle. The pockets are at the corners of the triangle and are labeled $C$, $H$, and $T$. Each side of the table is $16$ feet long. Lukas shoots a ball from corner $C$ of the table in such a way that on the second bounce, the ball hits $2$ feet away from him along side $CH$. a. How many times will the ball bounce before hitting a pocket? b. Which pocket will the ball hit? c. How far will the ball travel before hitting the pocket?

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

Problem: A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). In how many ways can this be done so that all friends are invited at least once?

In any set of $181$ square integers, prove that one can always find a subset of $19$ numbers, sum of whose elements is divisible by $19$.

The diagonals of a convex quadrilateral $ABCD$ are equal to each other and intersect at point $M$. Points $K$ and $L$ are taken on $AB$ and $CD$, respectively, so that $\frac{AK}{KB}=\frac{DL}{LC}$. Lines $AB$ and $KD$ intersect at point $P$. Prove that $MP$ is the bisector of angle $AMD$.

Find all integers $a,b,c $ such that $a+b+c=abc$

Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$. $ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$. $ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that $f(km)+f(kn)-f(k)f(mn)\ge 1$ for all $k,m,n\in\mathbb{N}$.

Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .

Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.

prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure. [b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common. we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.

The general term of a sequence of numbers is defined as $a_n =\frac{1}{n^2 - n}$, for every integer $n \ge 3$. That is, $a_3 =\frac16$, $a_4 =\frac{1}{12}$, $a_5 =\frac{1}{20}$, and so on. Find a general expression for the sum $S_n$, which is the sum of all terms from $a_3$ until $a_n$.

Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials) such that [list] [*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$; [*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does. [/list] [i]Carl Schildkraut[/i]

A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will close the third locker, open the sixth, close the ninth. . . . Student 5 then goes through and "flips"every 5th locker. This process continues with all students with odd numbers $n<100$ going through and "flipping" every $n$th locker. How many lockers are open after this process?

Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.

Determine the polynomial P(X) satisfying simoultaneously the conditions: a) The remainder obtained when dividing P(X) to the polynomial X^3 −2 is equal to the fourth power of quotient. b) P(−2) + P(2) = −34.

A grasshopper can jump along a checkered strip for $8, 9$ or $10$ cells in any direction. A natural number $n$ is called jumpable if the grasshopper can start from some cell of a strip of length $n$ and visit every cell exactly once. Find at least one non-jumpable number $n > 50$. [i](Egor Bakaev)[/i]

Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.

Two externally tangent circles have radius $2$ and radius $3$. Two lines are drawn, each tangent to both circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines?

Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.