On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$. [i]Nicolae Bourbacut[/i]

Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$ [i]Proposed by Andrew Wu[/i]

Let a,b,c Postive real numbers such that $a+b+c\geq 6$. Find the minimum value $A=\sum_{cyc}{a^2}$+$\sum_{cyc}{\frac{a}{b^2+c+1}}$

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

What is the number of structural isomers with the molecular formula $\ce{C6H14}$? $ \textbf{(A) }\text{Three} \qquad\textbf{(B) }\text{Four} \qquad\textbf{(C) }\text{Five} \qquad\textbf{(D) }\text{Six}\qquad$

Suppose that $ \mathcal F$ is a family of subsets of $ X$. $ A,B$ are two subsets of $ X$ s.t. each element of $ \mathcal{F}$ has non-empty intersection with $ A, B$. We know that no subset of $ X$ with $ n \minus{} 1$ elements has this property. Prove that there is a representation $ A,B$ in the form $ A \equal{} \{a_1,\dots,a_n\}$ and $ B \equal{} \{b_1,\dots,b_n\}$, such that for each $ 1\leq i\leq n$, there is an element of $ \mathcal F$ containing both $ a_i, b_i$.

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Given triangle $ PQR$ with $ \overline{RS}$ bisecting $ \angle R$, $ PQ$ extended to $ D$ and $ \angle n$ a right angle, then: [asy]path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false) { pair M,N; path mark; M=t*0.03*unit(A-B)+B; N=t*0.03*unit(C-B)+B; if(flip) mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B)); else mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B)); return mark; } unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair P=(0,0), R=(3,2), Q=(4,0); pair S0=bisectorpoint(P,R,Q); pair Sp=extension(P,Q,S0,R); pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp); pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R); draw(P--R--Q); draw(R--Sp); draw(P--D--M); draw(anglemark2(Sp,P,R,17)); label("$p$",P+(0.35,0.1)); draw(anglemark2(R,Q,P,11)); label("$q$",Q+(-0.17,0.1)); draw(anglemark2(R,Np,D,8,true)); label("$n$",Np+(+0.12,0.07)); draw(anglemark2(R,M,D,13,true)); label("$m$",M+(+0.25,0.03)); draw(anglemark2(M,D,P,29)); label("$d$",D+(-0.75,0.095)); pen f=fontsize(10pt); label("$R$",R,N,f); label("$P$",P,S,f); label("$S$",Sp,S,f); label("$Q$",Q,S,f); label("$D$",D,S,f);[/asy]$ \textbf{(A)}\ \angle m \equal{} \frac {1}{2}(\angle p \minus{} \angle q) \qquad \textbf{(B)}\ \angle m \equal{} \frac {1}{2}(\angle p \plus{} \angle q)$ $ \textbf{(C)}\ \angle d \equal{} \frac {1}{2} (\angle q \plus{} \angle p) \qquad \textbf{(D)}\ \angle d \equal{} \frac {1}{2}\angle m \qquad \textbf{(E)}\ \text{none of these is correct}$

A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season?

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Determine $ \gcd (n^2\plus{}2,n^3\plus{}1)$ for $ n\equal{}9^{753}$.

Assume that a set $M$ of real numbers is the union of finitely many disjoint intervals with the total length greater than $1$. Prove that $M$ contains a pair of distinct numbers whose difference is an integer.

A frictionless roller coaster ride is given a certain velocity at the start of the ride. At which point in the diagram is the velocity of the cart the greatest? Assume a frictionless surface. [asy]pair A = (1.7,3.9); pair B = (3.2,2.7); pair C = (5,1.2); pair D = (8,2.7); size(8cm); path boundary = (0,0.5)--(8,0.5)--(8,5)--(0,5)--cycle; path track = (0,3.2)..A..(3,3)..B..(4,1.8)..C..(6,1.5)..(7,2.3)..D; path sky = (0,5)--track--(8,5)--cycle; for (int a=0; a<=8; ++a) { draw((a,0)--(a,5), black+1); } for (int a=0; a<=5; ++a) { draw((0,a)--(8,a), black+1); } for (int a=-100; a<=100; ++a) { draw((0,a)--(8,a+8)); } for (int a=-100; a<=100; ++a) { draw((8,a)--(0,a+8)); } fill(sky,white); draw(track, black+3); clip(boundary); label("$A$", A, dir(120)); label("$B$", B, dir(60)); label("$C$", C, dir(90)); label("$D$", D, dir(135));[/asy] $ \textbf {(A) } \text {A} \qquad \textbf {(B) } \text {B} \qquad \textbf {(C) } \text {C} \qquad \textbf {(D) } \text {D} \\ \textbf {(E) } \text {There is insufficient information to decide} $ [i]Problem proposed by Kimberly Geddes[/i]

A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps: $ (A)$ $ 1$ to right and $ 3$ up, and $ (B)$ $ 2$ to the left and $ 4$ down. $ (a)$ The start position of the kangaroo is $ (0,0)$. Show that it can jump to the point $ (19,95)$ and determine the number of jumps needed. $ (b)$ Show that if the start position is $ (1,0)$, then it cannot reach $ (19,95)$. $ (c)$ If the start position is $ (0,0)$, find all points $ (m,n)$ with $ m,n \ge 0$ which the kangaroo can reach.

Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

Find the sum of the x-coordinates of the distinct points of intersection of the plane curves given by $x^2 = x + y + 4$ and $y^2 = y - 15x + 36$.

Let $ f$ be a function satisfying $ f(xy) \equal{} f(x)/y$ for all positive real numbers $ x$ and $ y$. If $ f(500) \equal{} 3$, what is the value of $ f(600)$? $ \textbf{(A)} \ 1 \qquad \textbf{(B)} \ 2 \qquad \textbf{(C)} \ \displaystyle \frac {5}{2} \qquad \textbf{(D)} \ 3 \qquad \textbf{(E)} \ \displaystyle \frac {18}{5}$

Let be a natural number $ n\ge 2. $ Prove that there exists an unique bipartition $ \left( A,B \right) $ of the set $ \{ 1,2\ldots ,n \} $ such that $ \lfloor \sqrt x \rfloor\neq y , $ for any $ x,y\in A , $ and $ \lfloor \sqrt z \rfloor\neq t , $ for any $ z,t\in B. $ [i]Costin Bădică[/i]

Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular

The sum $$\sum_{n=0}^{2016\cdot2017^2}2018^n$$ can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$.

The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form \[f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),\] where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.

For each integer $n\geq 3$, find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)