Let $n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$ distinct points in the plane. Let $\mathcal S$ denote the union of all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if it is always possible to find points $A$ and $B$ in $\mathcal S$ such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line $P_1P_n$) and $P_1P_n = kAB$, where (1) $k = 2.5$; (2) $k = 3$.

Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$, inclusive, such that, if $$q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}},$$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$. [i]Proposed by Ankit Bisain[/i]

Find all triples of integers $(x, y, z)$ such that $$x^2 + y^2 + z^2 = 16(x + y + z).$$

Select a random real number $m$ from the interval $(\frac{1}{6}, 1)$. A track is in the shape of an equilateral triangle of side length $50$ feet. Ch, Hm, and Mc are all initially standing at one of the vertices of the track. At the time $t = 0$, the three people simultaneously begin walking around the track in clockwise direction. Ch, Hm, and Mc walk at constant rates of $2, 3$, and $4$ feet per second, respectively. Let $T$ be the set of all positive real numbers $t_0$ satisfying the following criterion: [i]If we choose a random number $t_1$ from the interval $[0, t_0]$, the probability that the three people are on the same side of the track at the time $t = t_1$ is precisely $m$.[/i] The probability that $|T| = 17$ (i.e., $T$ has precisely $17$ elements) equals $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $a,\ b$ be real numbers. Suppose that a function $f(x)$ satisfies $f(x)=a\sin x+b\cos x+\int_{-\pi}^{\pi} f(t)\cos t\ dt$ and has the maximum value $2\pi$ for $-\pi \leq x\leq \pi$. Find the minimum value of $\int_{-\pi}^{\pi} \{f(x)\}^2dx.$ [i]2010 Chiba University entrance exam[/i]

[b]2/1/27.[/b] Suppose $a, b,$ and $c$ are distinct positive real numbers such that \begin{align*}abc=1000, \\ bc(1-a)+a(b+c)=110.\end{align*} If $a<1$, show that $10<c<100$.

Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs. a) Determine the maximum possible area of a trio. b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon. c) Determine the total number of triangles associated with the hexagon.

In a triangle $ABC$, $\angle C = 60^o$, $D, E, F$ are points on the sides $BC, AB, AC$ respectively, and $M$ is the intersection point of $AD$ and $BF$. Suppose that $CDEF$ is a rhombus. Prove that $DF^2 = DM \cdot DA$

When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$? $\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$

Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

A circle with radius $1$ and center $(0, 1)$ lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of $35$ degrees relative to the positive $x$-axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with $n$ points on the circle. What is $n$?

For every $k\leq n$ define $r_k$ the residue of $2^n$ modulo $k$. Prove that $\sum r_i> \frac{n*log_2(\frac{n}{3})}{2}-n$, for any $n\geq 2$

Define the recursive sequence $1, 4, 13, \ldots$ by $s_1 = 1$ and $s_{n+1} = 3s_n + 1$ for all positive integers $n$. The element $s_{18} = 193710244$ ends in two identical digits. Prove that all the elements in the sequence that end in two or more identical digits come in groups of three consecutive elements that have the same number of identical digits at the end.

Prove that there is no function $f:\mathbb{R}_{\ge0}\rightarrow\mathbb{R}$ satisfying: $f(x+y^2)\ge f(x)+y$ for all two nonnegative real numbers $x,y$.

Hamilton Avenue has eight houses. On one side of the street are the houses numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An eccentric postman starts deliveries at house 1 and delivers letters to each of the houses, finally returning to house 1 for a cup of tea. Throughout the entire journey he must observe the following rules. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout, each house except house 1 is visited exactly once (house 1 is visited twice) and the postman at no time is allowed to cross the road to the house directly opposite. How many different delivery sequences are possible?

Let $n> 1$ be a positive integer. Danielle chooses a number $N$ of $n$ digits but does not tell her students and they must find the sum of the digits of $N$. To achieve this, each student chooses and says once a number of $n$ digits to Danielle and she tells how many digits are in the correct location compared with $N$. Find the minimum number of students that must be in the class to ensure that students have a strategy to correctly find the sum of the digits of $N$ in any case and show a strategy in that case.

A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________.

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

There are $200$ boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box $X$ which contains a number $c$ such that $c = a + b$ for some numbers $a$ and $b$ which are contained in some other boxes. Then she picks a positive integer $k > 1$. Finally, she removes $c$ from $X$ and replaces it with $kc$. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.

Let $ a_i,b_i$ $ (i\equal{}1,2,...,n)$ be real numbers such that the $ a_i$ are distinct, and suppose that there is a real number $ \alpha$ such that the product $ (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n)$ is equal to $ \alpha$ for each $ i$. Prove that there is a real number $ \beta$ such that $ (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j)$ is equal to $ \beta$ for each $ j$.

Convex quadrilateral $ABCD$ is described on a circle $\omega$, and is not a trapezius inscribed in a circle. Let the tangency points of $\omega$ and $AB, BC, CD, DA$ be $K, L, M, N$ respectively. A circle with a center $I_K$, different from $\omega$ is tangent to the segement $AB$ and lines $AD, BC$. A circle with center $I_L$, different from $\omega$ is tangent to segment $BC$ and lines $AB, CD$. A circle with center $I_M$, different from $\omega$ is tangent to segment $CD$ and lines $AD, BC$. A circle with center $I_N$, different from $\omega$ is tangent to segment $AD$ and lines $AB, CD$. Prove that the lines $I_KK, I_LL, I_MM, I_NN$ are concurrent.

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.