Show that any positive integer has at least as many positive divisors of the form $3k+1$ as of the form $3k-1$. [b](N. 7)[/b]

How many pairs of $2007$-digit numbers $\underline{a_1a_2}\cdots\underline{a_{2007}}$ and $\underline{b_1b_2}\cdots\underline{b_{2007}}$ are there such that $a_1b_1+a_2b_2+\cdots+a_{2007}b_{2007}$ is even? Express your answer as $a \** b^c + d \** e^f$ for integers $a$, $b$, $c$, $d$, $e$, and $f$ with $a \nmid b$ and $d \nmid e$.

Given are on a line three points $A, B, C$ such that $AB = 1$ and $BC = x$. Consider the circles $\Omega_a, \Omega_b$ and $\Omega_c$ which are tangent to the given line at the points $A, B, C$ respectively, and such that $\Omega_b$ is tangent externally with both $\Omega_a$ and $\Omega_c$ in points $M, N$ respectively. Find all values of the radius of the circle $\Omega_b$ for which the triangle $BMN$ is isosceles.

Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$.) [i]Proposed by Yonah Borns-Weil[/i]

If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$, find the maximum of the expression \[(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.\]

On the same set of axes are drawn the graph of $ y\equal{}ax^2\plus{}bx\plus{}c$ and the graph of the equation obtained by replacing $ x$ by $ \minus{}x$ in the given equation. If $ b \neq 0$ and $ c \neq 0$ these two graphs intersect: $ \textbf{(A)}\ \text{in two points, one on the x\minus{}axis and one on the y\minus{}axis}\\ \textbf{(B)}\ \text{in one point located on neither axis} \\ \textbf{(C)}\ \text{only at the origin} \\ \textbf{(D)}\ \text{in one point on the x\minus{}axis} \\ \textbf{(E)}\ \text{in one point on the y\minus{}axis}$

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

$S$ is a family of balls in $\mathbb{R}^{n}$ ($n > 1$) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of $S$ is countable.

On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.

If the natural numbers $a, b, c, d$ verify the relationships: $$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$ $$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$ and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

Let $C$ be a semicircle with diameter $AB$. Circles $S$, $S_1$, $S_2$ with radii $r$, $r_1$, $r_2$, respectively, are tangent to $C$ and the segment $AB$, and moreover $S_1$ and $S_2$ are externally tangent to $S$. Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$

The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $

Source: 2017 Canadian Open Math Challenge, Problem A2 ----- An equilateral triangle has sides of length $4$cm. At each vertex, a circle with radius $2$cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$. Determine $a$. [center][asy] size(2.5cm); draw(circle((0,2sqrt(3)/3),1)); draw(circle((1,-sqrt(3)/3),1)); draw(circle((-1,-sqrt(3)/3),1)); draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle); fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray); draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle); fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray); draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle); fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray); [/asy][/center]

Let $\{x\} = x- \lfloor x \rfloor$ . Consider a function f from the set $\{1, 2, . . . , 2020\}$ to the half-open interval $[0, 1)$. Suppose that for all $x, y,$ there exists a $z$ so that $\{f(x) + f(y)\} = f(z)$. We say that a pair of integers $m, n$ is valid if $1 \le m, n \le 2020$ and there exists a function $f$ satisfying the above so $f(1) = \frac{m}{n}$. Determine the sum over all valid pairs $m, n$ of ${m}{n}$.

A restricted rook (RR) is a fictional chess piece that can move horizontally or vertically (like a rook), except that each move is restricted to a neighboring square (cell). If RR can only (with at most one exception) move up and to the right, how many possible distinct paths are there to move RR from the bottom left square to the top right square of a standard 8-by-8 chess board? Note that RR may visit some squares more than once. A path is the sequence of squares visited by RR on its way.

Given an infinite positive integer sequence $\{x_i\}$ such that $$x_{n+2}=x_nx_{n+1}+1$$ Prove that for any positive integer $i$ there exists a positive integer $j$ such that $x_j^j$ is divisible by $x_i^i$. [i]Remark: Unfortunately, there was a mistake in the problem statement during the contest itself. In the last sentence, it should say "for any positive integer $i>1$ ..."[/i]

A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result? [asy] unitsize(20); for(int a = 0; a < 10; ++a) { draw((0,a)--(8,a)); } for (int b = 0; b < 9; ++b) { draw((b,0)--(b,9)); } draw((0,0)--(0,-.5)); draw((1,0)--(1,-1.5)); draw((.5,-1)--(1.5,-1)); draw((2,0)--(2,-.5)); draw((4,0)--(4,-.5)); draw((5,0)--(5,-1.5)); draw((4.5,-1)--(5.5,-1)); draw((6,0)--(6,-.5)); draw((8,0)--(8,-.5)); fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black); fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black); fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black); fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black); fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black); label("$2$",(1.5,8.2),N); label("$4$",(3.5,8.2),N); label("$5$",(4.5,8.2),N); label("$7$",(6.5,8.2),N); label("$8$",(7.5,8.2),N); label("$9$",(0.5,7.2),N); label("$11$",(2.5,7.2),N); label("$12$",(3.5,7.2),N); label("$13$",(4.5,7.2),N); label("$14$",(5.5,7.2),N); label("$16$",(7.5,7.2),N); [/asy] $\text{(A)}\ 36 \qquad \text{(B)}\ 64 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 120$

Compute the sum $\sum^{200}_{n=1}\frac{1}{n(n+1)(n+2)}$ .

The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$. \[\begin{tabular}{ccccccc}& 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0\\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}\] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \text{More than } 10$

We have a triangle with $ \angle BAC=\angle BCA. $ The point $ E $ is on the interior bisector of $ \angle ABC $ so that $ \angle EAB =\angle ACB. $ Let $ D $ be a point on $ BC $ such that $ B $ is on the segment $ CD $ (endpoints excluded) and $ BD=AB. $ Show that the midpoint of $ AC $ is on the line $ DE. $

Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions: (a) the two triangles have the same area; (b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers. Determine if there are infinitely many pairs of triangles that are pals of each other.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.

Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a>b$ and no square dividing $a$ or $b$. Find $100a+10b+c$. [i]Proposed by Michael Kural[/i]