Let $ABC$ be an isosceles triangle with $\angle{ABC} = \angle{ACB} = 80^\circ$. Let $D$ be a point on $AB$ such that $\angle{DCB} = 60^\circ$ and $E$ be a point on $AC$ such that $\angle{ABE} = 30^\circ$. Find $\angle{CDE}$ in degrees.

Find the sum of all positive integers $n$ for which \[\frac{15\cdot n!^2+1}{2n-3}\] is an integer. [i]Proposed by Andrew Gu.[/i]

Let $ABC$ be a scalene triangle with circumcenter $O$, and let $D$ and $E$ be points inside angle $\measuredangle BAC$ such that $A$ lies on line $DE$, and $\angle ADB=\angle CBA$ and $\angle AEC=\angle BCA$. Let $M$ be the midpoint of $BC$ and $K$ be a point such that $OK$ is perpendicular to $AO$ and $\angle BAK=\angle MAC$. Finally, let $P$ be the intersection of the perpendicular bisectors of $BD$ and $CE$. Show that $KO=KP$. [i]Proposed by Victor Domínguez[/i]

There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that \[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\] for all $n \ge 1$. What is $f(2018^{2019})$?

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]

A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$ is good, for any positive integer $ a$. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...[/hide]

For each $n\in \mathbb{Z}_{>0}$ let $a_n$ be the closest integer to $\sqrt{n}$. Compute the general term of the sequence: $(b_n)_{n\geqslant 1}$ with \[b_n=\sum_{k=1}^{n^2} a_k.\] [i]Pall Dalyay[/i]

Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$. A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.

Prove that $\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor$ for all postive integer $n$.

In triangle $ABC$, given lines $l_{b}$ and $l_{c}$ containing the bisectors of angles $B$ and $C$, and the foot $L_{1}$ of the bisector of angle $A$. Restore triangle $ABC$.

Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.

Let $ABC$ be a triangle with $m(\widehat{ABC}) = 90^{\circ}$. The circle with diameter $AB$ intersects the side $[AC]$ at $D$. The tangent to the circle at $D$ meets $BC$ at $E$. If $|EC| =2$, then what is $|AC|^2 - |AE|^2$ ? $\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 12 \qquad\textbf{(E)}\ 10 \qquad\textbf{(E)}\ \text{None}$

Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. \begin{eqnarray*} \text{X}&\quad\text{X}\quad&\text{X} \\ \text{X}&\quad\text{X}\quad&\text{X} \end{eqnarray*} If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column? $\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{2}{5} \qquad \textbf{(C) } \frac{7}{15} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}$

Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.

[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio \[ \frac{EG}{GF} \] [b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$. Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$

Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint? (A Shapovalov)

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the $12$ entries will be "Possible"? \begin{tabular}{|c|c|c|c|c|} \cline{2-5} \multicolumn{1}{c|}{} & \textbf{zero} & \textbf{exactly one} & \textbf{exactly two} & \textbf{more than two}\\ \hline \textbf{zero slope} & ? & ? & ? & ?\\ \hline \textbf{nonzero rational slope} & ? & ? & ? & ?\\ \hline \textbf{irrational slope} & ? & ? & ? & ?\\ \hline \end{tabular} $ \textbf{(A) }4 \qquad \textbf{(B) }5 \qquad \textbf{(C) }6 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9 \qquad $

Prove that if the positive real numbers $a, b, c$ satisfy the equation \[a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),\] then there is a triangle $ABC$ with internal angles $\alpha, \beta, \gamma$ such that \[\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.\]

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

In a room, there are $100$ light switches, labeled with the positive integers ${1,2, . . . ,100}$. They’re all initially turned off. On the $i$ th day for $1 \le i \le 100$, Bob flips every light switch with label number $k$ divisible by $i$ a total of $\frac{k}{i}$ times. Find the sum of the labels of the light switches that are turned on at the end of the $100$th day.

Assume $a,b,c$ are odd integers. Show that the quadratic equation \[ ax^2 + bx + c = 0 \] has no rational solutions. (A number is said to be [i]rational[/i], if it can be written as a fraction: $\frac{\text{integer}}{\text{integer}}$.)

$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$

Let's call a corner the figure that is obtained by removing one cell from a $2 \times 2$ square. Cut the $6 \times 6$ square into corners so that no two of them form a $2 \times 3$ or $3 \times 2$ rectangle together.