Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle? $\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$

Let $a$ and $b$ be real numbers such that $\sin a+\sin b=\dfrac{\sqrt2}2$ and $\cos a+\cos b=\dfrac{\sqrt6}2$. Find $\sin(a+b)$. $\textbf{(A) }\dfrac12\qquad\textbf{(B) }\dfrac{\sqrt2}2\qquad\textbf{(C) }\dfrac{\sqrt3}2\qquad\textbf{(D) }\dfrac{\sqrt6}2\qquad\textbf{(E) }1$

Let $s(n)$ denote the sum of digits of a positive integer $n$. Prove that $s(9^n) > 9$ for all $n\geq 3$.

The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level (Figure). We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: (a) The top level is painted with only one color. (b) The $3$ doorways at the front on the second level are painted with the same color. (c) The $3$ doorways at the front on the first level are painted with the same color. (d) Each of the remaining $14$ doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors. How many ways are there to paint the first level? How many ways are there to paint the entire tower model? [img]https://cdn.artofproblemsolving.com/attachments/f/9/2249f8595a8efe711680f3dfb8ff959c140a21.png[/img]

Let $S$ be the set of all integers $k$, $1\leq k\leq n$, such that $\gcd(k,n)=1$. What is the arithmetic mean of the integers in $S$?

Provided that $\{a_i\}^{28}_{i=1}$ are the $28$ distinct roots of $29x^{28} + 28x^{27} + ... + 2x + 1 = 0$, then the absolute value of $\sum^{28}_{i=1}\frac{1}{(1-a_i)^2}$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively. a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel. b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.

Ben writes the string $$\underbrace{111\ldots 11}_{2020 \text{ digits}}$$on a blank piece of paper. Next, in between every two consecutive digits, he inserts either a plus sign $(+)$ or a multiplication sign $(\times)$. He then computes the expression using standard order of operations. Find the number of possible distinct values that Ben could have as a result. [i]Proposed by Taiki Aiba[/i]

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

The game of backgammon has a "doubling" cube, which is like a standard 6-faced die except that its faces are inscribed with the numbers 2, 4, 8, 16, 32, and 64, respectively. After rolling the doubling cube four times at random, we let $a$ be the value of the first roll, $b$ be the value of the second roll, $c$ be the value of the third roll, and $d$ be the value of the fourth roll. What is the probability that $\frac{a + b}{c + d}$ is the average of $\frac{a}{c}$ and $\frac{b}{d}$ ?

In trapezium $ABCD$, $BC \perp AB$, $BC\perp CD$, and $AC\perp BD$. Given $AB=\sqrt{11}$ and $AD=\sqrt{1001}$. Find $BC$

Given is an $n\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$, such that for any initial choice of integers, I can make a finite number of moves so that there are at least $N(n)$ even numbers on the board.

The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits

find all positive integer $n$, satisfying $$\frac{n(n+2016)(n+2\cdot 2016)(n+3\cdot 2016) . . . (n+2015\cdot 2016)}{1\cdot 2 \cdot 3 \cdot . . . . . \cdot 2016}$$ is positive integer.

Let $p,q$ denote distinct prime numbers and $k,l$ positive integers. Find all positive divisors of the numbers: (a) $A = p^k$ (b) $B=p^kq^l$ (c) $1944$

What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?

Suppose that $f : Z\times Z \to R$, satisfies the equation $f(x, y) = f(3x+y, 2x+ 2y)$ for all $x, y \in Z$. Determine the maximal number of distinct values of $f(x, y)$ for $1 \le x, y \le 100$.

Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

What is the smallest number of disks radius $\frac12$ that can cover a disk radius $1$?

Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

If the price of an article is increased by percent $ p$, then the decrease in percent of sales must not exceed $ d$ in order to yield the same income. The value of $ d$ is: $ \textbf{(A)}\ \frac{1}{1\plus{}p} \qquad\textbf{(B)}\ \frac{1}{1\minus{}p} \qquad\textbf{(C)}\ \frac{p}{1\plus{}p} \qquad\textbf{(D)}\ \frac{p}{p\minus{}1} \qquad\textbf{(E)}\ \frac{1\minus{}p}{1\plus{}p}$

Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$.