Let $A$ be a finite set of positive integers, and for each positive integer $n$ we define \[S_n = \{x_1 + x_2 + \cdots + x_n \;\vert\; x_i \in A \text{ for } i = 1, 2, \dots, n\}\] That is, $S_n$ is the set of all positive integers which can be expressed as sum of exactly $n$ elements of $A$, not necessarily different. Prove that there exist positive integers $N$ and $k$ such that $$\left\vert S_{n + 1} \right\vert = \left\vert S_n \right\vert + k \text{ for all } n\geq N.$$ [i]Proposed by Ariel García[/i]

Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$. Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.

Let $ABC$ be a triangle with $AB \neq AC$, let $I$ be its incenter, $\gamma$ its inscribed circle and $D$ the midpoint of $BC$. The tangent to $\gamma$ from $D$ different to $BC$ touches $\gamma$ in $E$. Prove that $AE$ and $DI$ are parallel.

Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?

In a regular square room of side length $2\sqrt{2}$ ft, two cats that can see $2$ feet ahead of them are randomly placed into the four corners such that they do not share the same corner. If the probability that they don't see the mouse, also placed randomly into the room can be expressed as $\frac{a-b\pi}{c},$ where $a,b,c$ are positive integers with a greatest common factor of $1,$ then find $a+b+c.$ [i]Proposed by Ada Tsui[/i]

Let $a,b,c$ be positive integers satisfying $\gcd(a,b,c)=1$ and $$\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}$$ is an integer. Prove that $abc$ is a perfect square.

Let points $P_1, P_2, P_3$, and $P_4$ be arranged around a circle in that order. (One possible example is drawn in Diagram 1.) Next draw a line through $P_4$ parallel to $P_1P_2$, intersecting the circle again at $P_5$. (If the line happens to be tangent to the circle, we simply take $P_5 =P_4$, as in Diagram 2. In other words, we consider the second intersection to be the point of tangency again.) Repeat this process twice more, drawing a line through $P_5$ parallel to $P_2P_3$, intersecting the circle again at $P_6$, and finally drawing a line through $P_6$ parallel to $P_3P_4$, intersecting the circle again at $P_7$. Prove that $P_7$ is the same point as $P_1$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/fa8c1b88f78c09c3afad2c33b50c2be4635a73.png[/img]

The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that: a) $g_a,g_b,g_c \ge \frac{2}{3}r$ b) $g_a+g_b+g_c \ge 3r$

Let $ABCD$ be a parallelogram (non-rectangle) and $\Gamma$ is the circumcircle of $\triangle ABD$. The points $E$ and $F$ are the intersections of the lines $BC$ and $DC$ with $\Gamma$ respectively. Define $P=ED\cap BA$, $Q=FB\cap DA$ and $R=PQ\cap CA$. Prove that $$\frac{PR}{RQ}=(\frac{BC}{CD})^2$$

Find the value of \[(2+0+2+4)+\left(2^0+2^4\right)+\left(2^{0^{2^4}}\right).\]

Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations: \[ \begin{cases} \sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\ \sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2 \end{cases} \] Find the maximum value of \(x + y\).

In how many distinguishable rearrangements of the letters ABCCDEEF does the A precede both C's, the F appears between the 2 C's, and the D appears after the F?

Given $\triangle ABC$ with incenter $I$. Line $AI$ meets $BC$ at $D$. The incenter of $\triangle ABD, \triangle ADC$ are $E,F$, respectively. Line $DE$ meets the circumcircle of $\triangle BCE$ at$ P(\neq E)$ and line $DF$ meets the circumcircle of $\triangle BCF$ at$ Q(\neq F)$. Show that the midpoint of $BC$ lies on the circumcircle of $\triangle DPQ$.

$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.

Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$: \[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Let $x_0 = 1$ and $x_1 = 2003$ and define the sequence $(x_n)_{n \ge 0}$ by: $x_{n+1} =\frac{x^2_n + 1}{x_{n-1}}$ , $\forall n \ge 1$ Prove that for every $n \ge 2$ the denominator of the fraction $x_n$, when $x_n$ is expressed in lowest terms is a power of $2003$.

Let $ m,n$ be two integers such that $ \varphi(m) \equal{}\varphi(n) \equal{} c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.

SEEMOUS 2016 COMPETITION PROBLEMS

We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.

In a convex quadrilateral $ABCD$, the lines $AB$ and $DC$ intersect at point $P{}$ and the lines $AD$ and $BC$ intersect at point $Q{}$. The points $E{}$ and $F{}$ are inside the quadrilateral $ABCD$ such that the circles $(ABE), (CDE), (BCF),(ADF)$ intersect at one point $K{}$. Prove that the circles $(PKF)$ and $(QKE)$ intersect a second time on the line $PQ$.

Determine all functions $f$ from the reals to the reals for which (1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)

Compute $99(99^2+3) + 3\cdot99^2$. [i]Proposed by Evan Chen[/i]

In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$. [asy] pair A, B, C, D; A = origin; real Bcoord = 3*sqrt(2) + sqrt(6); B = Bcoord/2*dir(180); C = sqrt(6)*dir(120); draw(A--B--C--cycle); D = (C-B)/2.4 + B; draw(A--D); label("$A$", A, dir(0)); label("$B$", B, dir(180)); label("$C$", C, dir(110)); label("$D$", D, dir(130)); [/asy] $ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$

$m$ and $n$ are positive integers. If the number \[ k=\dfrac{(m+n)^2}{4m(m-n)^2+4}\] is an integer, prove that $k$ is a perfect square.