Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at $\text{(A)}\ \text{3:40 P.M.} \qquad \text{(B)}\ \text{3:55 P.M.} \qquad \text{(C)}\ \text{4:10 P.M.} \qquad \text{(D)}\ \text{4:25 P.M.} \qquad \text{(E)}\ \text{4:40 P.M.}$

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.

A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear? [list=1] [*] 34 [*] 55 [*] 89 [*] None of these [/list]

On a blackboard a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects the following rules: $-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$; $-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$; $-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner. Find which player has a winning strategy in each of the following cases: a) $n_0=120$; b) $n_0=\dsp \frac {3^{2002}-1}2$; c) $n_0=\dsp \frac{3^{2002}+1}2$.

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Let $ABC$ be an acute angled triangle. The arc between $A$ and $B$ of the circumcircle of $ABC$ is reflected through the line $AB$, and the arc between $A$ and $C$ of the circumcircle of $ABC$ is reflected over the line $AC$. Obviously these two reflected arcs intersect at the point $A$. Prove that they also intersect at another point inside the triangle $ABC$.

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.

[i](5 pts)[/i] Let $n$ be a positive integer, $K = \{1, 2, \dots, n\}$, and $\sigma : K \rightarrow K$ be a function with the property that $\sigma(i) = \sigma(j)$ if and only if $i = j$ (in other words, $\sigma$ is a \textit{bijection}). Show that there is a positive integer $m$ such that \[ \underbrace{\sigma(\sigma( \dots \sigma(i) \dots ))}_\textrm{$m$ times} = i \] for all $i \in K$.

Construct a holomorphic function $f(z) = \sum \limits_{n = 0} ^ \infty a_n z^n$ ( | z | <1 ) in the unit circle that can be analytically continued to all points of the unit circle except one point, and for which the sequence $\{a_n\}$ has two limit points, $\infty$ and a finite value.

Solve in the real numbers the equation $ \cos \left( \pi\log_3 (x+6) \right)\cdot \cos \left( \pi \log_3 (x-2) \right) =1. $

Find the sum of all distinct four-digit numbers that contain only the digits $1, 2, 3, 4,5$, each at most once.

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

In unit square $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let $M$ be the midpoint of $\overline{CD}$, with $\overline{AM}$ intersecting $\overline{BD}$ at $F$ and $\overline{BM}$ intersecting $\overline{AC}$ at $G$. Find the area of quadrilateral $MFEG$.

Let $ABC$ be an acute triangle with $AB<AC$ and let $O$ be its circumcenter. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let $E$ be the intersection of the line $AB$ with the perpendicular line to $AO$ through $D$. Let $F$ be the intersection of the perpendicular line to $OC$ through $C$ with the line parallel to $AC$ and passing through $E$. Finally, let the lines $CE$ and $DF$ intersect in $G$. Show that $AG$ and $BF$ are parallel.

Let $\mathcal{S}$ denote the set of positive integer sequences (with at least two terms) whose terms sum to $2019$. For a sequence of positive integers $a_1, a_2, \dots, a_k$, its \emph{value} is defined to be \[V(a_1, a_2, \dots, a_k) = \frac{a_1^{a_2} a_2^{a_3} \cdots a_{k-1}^{a_k}}{a_1! a_2! \cdots a_k!}.\] Then the sum of the values over all sequences in $\mathcal{S}$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $1000$. [i]Proposed by Sean Li[/i]

If $ \sqrt{x^2+2y+1} +\sqrt[3]{y^3+3x^2+3x+1} $ is rational, then $ x=y. $

Positive integers $m, n$ are given such that $\sqrt{2}<\frac{m}{n}<\sqrt{2}+\frac{1}{2}$ and $m$ is even. Prove that there exist positive integers $k<m$ and $l<n$ such that $$|\frac{k}{l}-\sqrt{2}|<\frac{m}{n}-\sqrt{2}$$

The sides $ [AB]$ and $ [AC]$ of the triangle $ ABC$ are tangent to the incircle with center $ I$ of the $ \triangle ABC$ at the points $ M$ and $ N$, respectively. The internal bisectors of the $ \triangle ABC$ drawn form $ B$ and $ C$ intersect the line $ MN$ at the points $ P$ and $ Q$, respectively. Suppose that $ F$ is the intersection point of the lines $ CP$ and $ BQ$. Prove that $ FI\perp BC$.

Consider $(f_n)_{n\ge 0}$ the Fibonacci sequence, defined by $f_0 = 0$, $f_1 = 1$, $f_{n+1} = f_n + f_{n-1}$ for all positive integers $n$. Solve the following equation in positive integers $$nf_nf_{n+1} = (f_{n+2} - 1)^2.$$ .