Let $ABC$ be a triangle such that $AB = AC$ and let $M$ be a point interior to the triangle. If $BM$ meets $AC$ at $D$. show that $\frac{DM}{DA}=\frac{AM}{AB}$ if and only if $\angle AMB = 2\angle ABC$. Michel Bataille

Let $ABC$ be an arbitrary triangle, and let $M, N, P$ be any three points on the sides $BC, CA, AB$ such that the lines $AM, BN, CP$ concur. Let the parallel to the line $AB$ through the point $N$ meet the line $MP$ at a point $E$, and let the parallel to the line $AB$ through the point $M$ meet the line $NP$ at a point $F$. Then, the lines $CP, MN$ and $EF$ are concurrent. [hide=MOP 97 problem]Let $ABC$ be a triangle, and $M$, $N$, $P$ the points where its incircle touches the sides $BC$, $CA$, $AB$, respectively. The parallel to $AB$ through $N$ meets $MP$ at $E$, and the parallel to $AB$ through $M$ meets $NP$ at $F$. Prove that the lines $CP$, $MN$, $EF$ are concurrent. [url=https://artofproblemsolving.com/community/c6h22324p143462]also[/url][/hide]

Define the unit $N$-hypercube to be the set of points $[0, 1]^N \subset R^N$ . For example, the unit $0$-hypercube is a point, and the unit $3$-hypercube is the unit cube. Define a $k$-face of the unit $N$-hypercube to be a copy of the $k$-hypercube in the exterior of the $N$-hypercube. More formally, a $k$-face of the unit $N$-hypercube is a set of the form $$\prod_{i=1}^{N} S_i$$ where $S_i$ is either $\{0\}$, $\{1\}$, or $[0, 1]$ for each $1 \le i \le N$, and there are exactly $k$ indices $i$ such that $S_i = [0, 1]$. The expected value of the dimension of a random face of the unit $ 8$-hypercube (where the dimension of a face can be any value between $0$ and $N$) can be written in the form $p/q$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

Find all real solutions to the equation $(x^2 + 3x + 1)^{x^2-x-6} = 1$.

The number of digits in the number $N=2^{12}\times 5^8$ is $\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad \textbf{(E) }20$

Given a circle $\Omega$ tangent to side $AB$ of angle $\angle BAC$ and lying outside this angle. We consider circles $w$ inscribed in angle $BAC$. The internal tangent of $\Omega$ and $w$, different from $AB$, touches $w$ at a point $K$. Let $L$ be the point of tangency of $w$ and $AC$. Prove that all such lines $KL$ pass through a fixed point without depending on the choice of circle $w$.

Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.·

Find all integer $n$ such that $n-5$ divide $n^2+n-27$.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.

Martu wants to build a set of cards with the following properties: • Each card has a positive integer on it. • The number on each card is equal to one of $5$ possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same. Determine the fewest number of cards Martu's set can have and give an example for that number.

Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive? (4 points) Boris Frenkin

On this monthly calendar, the date behind one of the letters is added to the date behind $C$. If this sum equals the sum of the dates behind $A$ and $B$, then the letter is [asy] unitsize(12); draw((1,1)--(23,1)); draw((0,5)--(23,5)); draw((0,9)--(23,9)); draw((0,13)--(23,13)); for(int a=0; a<6; ++a) { draw((4a+2,0)--(4a+2,14)); } label("Tues.",(4,14),N); label("Wed.",(8,14),N); label("Thurs.",(12,14),N); label("Fri.",(16,14),N); label("Sat.",(20,14),N); label("C",(12,10.3),N); label("$\textbf{A}$",(16,10.3),N); label("Q",(12,6.3),N); label("S",(4,2.3),N); label("$\textbf{B}$",(8,2.3),N); label("P",(12,2.3),N); label("T",(16,2.3),N); label("R",(20,2.3),N);[/asy] $ \text{(A)}\ \text{P}\qquad\text{(B)}\ \text{Q}\qquad\text{(C)}\ \text{R}\qquad\text{(D)}\ \text{S}\qquad\text{(E)}\ \text{T} $

Let $a_k$ be the number of nonnegative integers $ n $ with the properties: a) $n\in[0, 10^k)$ has exactly $ k $ digits, such that he zeroes on the first positions of $ n $ are included in the decimal writting. b) the digits of $ n $ can be permutated such that the new number is divisible by $11.$ Show that $a_{2m}=10a_{2m-1}$ for every $m\in\mathbb{N}.$

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ Let $A'$ be the midpoint of $BC,$ $M$ be the midpoint of the height $AD$ and $P$ be the intersection of $BM$ and $AA'.$ Prove that $\tan\angle PCB=\sin C\cdot\cos C.$ [i]Daniel Văcărețu[/i]

Let $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ be nine strictly positive real numbers. We set \[S_1 = a_1b_2c_3, \quad S_2 = a_2b_3c_1, \quad S_3 = a_3b_1c_2;\]\[T_1 = a_1b_3c_2, \quad T_2 = a_2b_1c_3, \quad T_3 = a_3b_2c_1.\] Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements. Prove that \[S_1 + S_2 + S_3 = T_1 + T_2 + T_3.\]

Let circles $\Gamma_1$ and $\Gamma_2$ have radii $r_1$ and $r_2$, respectively. Assume that $r_1 < r_2$. Let $T$ be an intersection point of $\Gamma_1$ and $\Gamma_2$, and let $S$ be the intersection of the common external tangents of $\Gamma_1$ and $\Gamma_2$. If it is given that the tangents to $\Gamma_1$ and $ \Gamma_2$ at $T$ are perpendicular, determine the length of $ST$ in terms of $r_1$ and $r_2$.

Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\] [i]Poland (Tomasz Kobos)[/i]

Let $ABCD$ be the isosceles trapezium with bases $AB$ and $CD$, such that $AC = BC$. The point $M$ is the midpoint of side $AD$. Prove that $\sphericalangle ACM = \sphericalangle CBD$.

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

Let $n$ be a positive integer less than $1000$. The remainders obtained when dividing $n$ by $2, 2^2, 2^3, ... , 2^8$, and $2^9$ , are calculated. If the sum of all these remainders is $137$, what are all the possible values ​​of $n$?

The number $\overline{13\ldots 3}$, with $k>1$ digits $3$, is a prime. Prove that $6\mid k^{2}-2k+3$.

Which of the following cannot be the number of positive integer divisors of the number $n^2+1$, where $n$ is an integer? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of above} $