Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children in $42$. What is the sum of the ages of the oldest three? $ \mathrm{(A) \ } 51 \qquad \mathrm{(B) \ } 54 \qquad \mathrm {(C) \ } 57 \qquad \mathrm{(D) \ } 60 \qquad \mathrm{(E) \ } 63$

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

What is the size of the largest subset $S'$ of $S = \{2^x3^y5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$?

Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds: \[\sum_{P\in\mathcal{C}\cap M} f(P)=0\] [i]Dorel Mihet[/i]

The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?

Positive integers $d_1,d_2,...,d_n$ are divisors of $1995$. Prove that there exist $d_i$ and $d_j$ among them, such the denominator of the reduced fraction $d_i/d_j$ is at least $n$

Let $O$ be the circumcenter of triangle $\triangle ABC$, where $\angle BAC=120^{\circ}$. The tangent at $A$ to $(ABC)$ meets the tangents at $B,C$ at $(ABC)$ at points $P,Q$ respectively. Let $H,I$ be the orthocenter and incenter of $\triangle OPQ$ respectively. Define $M,N$ as the midpoints of arc $\overarc{BAC}$ and $OI$ respectively, and let $MN$ meet $(ABC)$ again at $D$. Prove that $AD$ is perpendicular to $HI$.

Describe all groups $ G $ which have the property that: $$ (\forall H\le G)(\forall x,y\in G)(xy\in H\implies (x,y\in H\vee xy=1)) $$

There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time. 1. Determine whether or not there is the case $ n \equal{} 1$. 2. Determine whether or not there is the case $ n > 1$.

Triangle $ABC$ in which $AB <BC$, is inscribed in a circle $\omega$ and circumscribed about a circle $\gamma$ with center $I$. The line $\ell$ parallel to $AC$, touches the circle $\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. It is known that $PQ = 2BI$. Prove that $AP + 2PB = CP$.

Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\tfrac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\tfrac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)? Let $C$ be the actual answer and $A$ be the answer you will submit. Your score will be given by $\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}$.

Find all non-empty sets $S$ of nonzero real numbers such that a) $S$ has at most $5$ elements b) If $x$ is in $S$, then so are $1- x$ and $\frac{1}{x}$.

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.

The sequence of functions $f_n:[0,1]\to\mathbb R$ $(n\ge2)$ is given by $f_n=1+x^{n^2-1}+x^{n^2+2n}$. Let $S_n$ denote the area of the figure bounded by the graph of the function $f_n$ and the lines $x=0$, $x=1$, and $y=0$. Compute $$\lim_{n\to\infty}\left(\frac{\sqrt{S_1}+\sqrt{S_2}+\ldots+\sqrt{S_n}}n\right)^n.$$

In a planar coordinate system, the points have non-negative integer coordinates numbered according to the figure. E.g. the point $(3,1)$ has the number $12$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/28725d75f281ac4618129067037d751c8d8f83.png[/img] What is the number of the point$(x,y)$?

Prove that the square of any integer cannot end with two fives.

For $i = 1,2, \ldots, 1991$, we choose $n_i$ points and write number $i$ on them (each point has only written one number on it). A set of chords are drawn such that: (i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers. If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers $n_1, n_2, \ldots, n_{1991}$ must satisfy for this to be possible.

Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]

If $ a$ and $ b$ are two unequal positive numbers, then: $ \textbf{(A)}\ \frac {2ab}{a \plus{} b} > \sqrt {ab} > \frac {a \plus{} b}{2} \qquad\textbf{(B)}\ \sqrt {ab} > \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2}$ $ \textbf{(C)}\ \frac {2ab}{a \plus{} b} > \frac {a \plus{} b}{2} > \sqrt {ab} \qquad\textbf{(D)}\ \frac {a \plus{} b}{2} > \frac {2ab}{a \plus{} b} > \sqrt {ab}$ $ \textbf{(E)}\ \frac {a \plus{} b}{2} > \sqrt {ab} > \frac {2ab}{a \plus{} b}$

Four lighthouses are located at points $ A$, $ B$, $ C$, and $ D$. The lighthouse at $ A$ is $ 5$ kilometers from the lighthouse at $ B$, the lighthouse at $ B$ is $ 12$ kilometers from the lighthouse at $ C$, and the lighthouse at $ A$ is $ 13$ kilometers from the lighthouse at $ C$. To an observer at $ A$, the angle determined by the lights at $ B$ and $ D$ and the angle determined by the lights at $ C$ and $ D$ are equal. To an observer at $ C$, the angle determined by the lights at $ A$ and $ B$ and the angle determined by the lights at $ D$ and $ B$ are equal. The number of kilometers from $ A$ to $ D$ is given by $ \displaystyle\frac{p\sqrt{r}}{q}$, where $ p$, $ q$, and $ r$ are relatively prime positive integers, and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$,

An $m \times n$ rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a $2 \times 2$ square. For what $m, n$ can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?

Find all reals $z$ such that $z^4 - z^3 - 2z^2 - 3z - 1= 0$.

Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.