In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.

Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB|=|AC|$ such that $[CD]$ is an angle bisector and $m(\widehat{ABC})=40^\circ$. Let $F$ be a point on the extension of $[AB]$ after $B$ such that $|BC|=|AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is $m(\widehat{FBG})$? $ \textbf{(A)}\ 150^\circ \qquad\textbf{(B)}\ 135^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 105^\circ \qquad\textbf{(E)}\ \text{None of above} $

Consider the sequence $u_0, u_1, u_2, ...$ defined by $u_0 = 0, u_1 = 1,$ and $u_n = 6u_{n - 1} + 7u_{n - 2}$ for $n \ge 2$. Show that there are no non-negative integers $a, b, c, n$ such that $$ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.$$

Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$

Two medians drawn from vertices A and B of triangle ABC are perpendicular. Prove that side AB is the shortest side of ABC.

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

Brandon wants to split his orchestra of $20$ violins, $15$ violas, $10$ cellos, and $5$ basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?

Let $k\ge 2$ be an integer and $a_1,a_2,\cdots,a_k$ be $k$ non-zero reals. Prove that there are finitely many pairs of pairwise distinct positive integers $(n_1,n_2,\cdots,n_k)$ such that $$a_1\cdot n_1!+a_2\cdot n_2!+\cdots+a_k\cdot n_k!=0.$$

For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,...$. For how many values of $k$ does $S_k$ contain the term $2005$?

Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$

Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$

I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.

Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear. [i]Author: Waldemar Pompe, Poland[/i]

We call a natural number [i]almost a square[/i] if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.

We define [i]similar numbers[/i] as positive integers that have exactly the same digits (but possibly in another order). For example, $1241$, $2114$ and $4211$ are similar numbers, but $1424$ is not similar to the other three. Determine whether there exist three similar numbers, each with $300$ digits (all digits being non-zero), such that the sum of two of them equals the third. If the answer is yes, provide an example; if not, justify why it is impossible.

Sixteen dots are arranged in a four by four grid as shown. The distance between any two dots in the grid is the minimum number of horizontal and vertical steps along the grid lines it takes to get from one dot to the other. For example, two adjacent dots are a distance 1 apart, and two dots at opposite corners of the grid are a distance 6 apart. The mean distance between two distinct dots in the grid is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$. [center][img]https://i.snag.gy/c1tB7z.jpg[/img][/center]

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

Let $a$, $b$, and $c$ be such that the coefficient of the $x^ay^bz^c$ term in the expansion of $(x+2y+3z)^{100}$ is maximal (no other term has a strictly larger coefficient). Find the sum of all possible values of $1,000,000a+1,000b+c$.

Consider the $2\times3$ rectangle below. We fill in the small squares with the numbers $1,2,3,4,5,6$ (one per square). Define a [i]tasty[/i] filling to be one such that each row is [b]not[/b] in numerical order from left to right and each column is [b]not[/b] in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? \begin{tabular}{|c|c|c|c|} \hline & & \\ \hline & & \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Lightning #4.2[/i]

On side, $BC, AB$ of a parallelogram $ABCD$ lie points $M,N$ respectively such that $|AM| =|CN|$. Let $P$ be the intersection of $AM$ and $CN$. Prove that the angle bisector of $\angle APC$ passes through $D$.

Evaluate the infinite sum \[\sum_{n \equal{} 0}^\infty \binom{2n}{n}\frac {1}{5^n}.\]

On the table there are several cards. Each card has an integer number written on it. Beto performs the following operation several times: he chooses two cards from the table, calculates the difference between the numbers written on them, writes the result on his notebook and removes those two cards from the table. He can perform this operation as many times as he wants, as long as there are at least two cards on the table. After this, Beto multiplies all the numbers that he wrote on his notebook. Beto's goal is that the result of this multiplication is a multiple of $7^{100}$. a) Prove that if there are $207$ cards initially on the table then Beto can always achieve his goal, no matter what the numbers on the cards are. b) If there are $128$ cards initially on the table, is it true that Beto can always achieve his goal?

The rectangular table has four rows. The first one contains arbitrary natural numbers (some of them may be equal). The consecutive lines are filled according to the rule: we look through the previous row from left to the certain number $n$ and write the number $k$ if $n$ was met $k$ times. Prove that the second row coincides with the fourth one.

A deck of $52$ cards is stacked in a pile facing down. Tom takes the small pile consisting of the seven cards on the top of the deck, turns it around, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down, since the seven cards at the bottom now face up. Tom repeats this move until all cards face down again. In total, how many moves did Tom make?