Given an acute triangle $ABC$ and points $D$, $E$, $F$ on sides $BC$, $CA$ and $AB$, respectively. If the lines $DA$, $EB$ and $FC$ are the angle bisectors of triangle $DEF$, prove that the three lines are the altitudes of triangle $ABC$.

Let $a_1$, $a_2$, $\ldots$, $a_n$ be a geometric progression with $a_1 = \sqrt{2}$ and $a_2 = \sqrt[3]{3}$. What is \[\displaystyle{\frac{a_1+a_{2013}}{a_7+a_{2019}}}?\]

Let $\eta \in [0, 1]$ be a relative measure of material absorbence. $\eta$ values for materials combined together are additive. $\eta$ for a napkin is $10$ times that of a sheet of paper, and a cardboard roll has $\eta = 0.75$. Justin can create a makeshift cup with $\eta = 1$ using $50$ napkins and nothing else. How many sheets of paper would he need to add to a cardboard roll to create a makeshift cup with $\eta = 1$?

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

Determine all positive integers $ n$ such that it is possible to tile a $ 15 \times n$ board with pieces shaped like this: [asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]

A taxi ride costs $\$1.50$ plus $\$0.25$ per mile traveled. How much does a $5$-mile taxi ride cost? $ \textbf{(A)}\ \$2.25 \qquad\textbf{(B)}\ \$2.50 \qquad\textbf{(C)}\ \$2.75 \qquad\textbf{(D)}\ \$3.00 \qquad\textbf{(E)}\ \$ 3.25$

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.

Consider a half-plane with the boundary line $p$ and two points $M,N$ in it such that the distances $Mp$ and $Np$ are different. Construct a trapezoid $MNPQ$ with area $MN^2$ such that $P,Q\in p.$ Discuss conditions of solvability.

Albatross and Frankinfueter both own a circle. Frankinfueter also has stolen from Prof. Trugweg a ruler. Before that, Trugweg had two points with a distance of $1$ drawn his (infinitely large) board. For a natural number $n$, let A $(n)$ be the number of the construction steps that Albatross needs at least to create two points with a distance of $n$ to construct. Similarly, Frankinfueter needs at least $F(n)$ steps for this. How big can $\frac{A (n)}{F (n)}$ become? There are only the following three construction steps: a) Mark an intersection of two straight lines, two circles or a straight line with one circle. b) Pierce at a marked point $P$ and draw a circle around $P$ through one marked point . c) Draw a straight line through two marked points (this implies possession of a ruler ahead!).

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? $\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

Find the number of ordered triples of positive integers $(a, b, c)$, where $a, b,c$ is a strictly increasing arithmetic progression, $a + b + c = 2019$, and there is a triangle with side lengths $a, b$, and $c$.

Let $w,x,y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold: [list] [*] $\frac{1}{w}+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3,$ [*] $wx+wy+wz+xy+xz+yz=14,$ [*] $(w+x)^3+(w+y)^3+(w+z)^3+(x+y)^3+(x+z)^3+(y+z)^3=2160, \text{ and}$ [*] $w+x+y+z+i\sqrt{n} \in \mathbb{R}.$ [/list] Compute $n$. [i]Proposed by Luke Robitaille[/i]

In the $25$ squares of a $5 \times 5$ square, zeros are initially written. in the every minute Ionel chooses two squares with a common side. If they are written in them the numbers $a$ and $b$, then he writes instead the numbers $a + 1$ and $b + 1$ or $a - 1$ and $b - 1$. Over time he noticed that the sums of the numbers in each line were equal, as well as the sums of the numbers in each column are equal. Prove that this observation was made after an even number of minutes.

Let $n$ be an positive integer and $\sigma = (a_1, \dots, a_n)$ a permutation of $\{1, \dots, n\}$. The [i]cadence number[/i] of $\sigma$ is the number of maximal decrescent blocks. For example, if $n = 6$ and $\sigma = (4, 2, 1, 5, 6, 3)$, then the cadence number of $\sigma$ is $3$, because $\sigma$ has $3$ maximal decrescent blocks: $(4, 2, 1)$, $(5)$ and $(6, 3)$. Note that $(4, 2)$ and $(2, 1)$ are decrescent, but not maximal, because they are already contained in $(4, 2, 1)$. Compute the sum of the cadence number of every permutation of $\{1, \dots, n\}$.

Let $ABC$ be a triangle. Construct point $X$ such that $BX=BA$ and $X$ and $C$ lies on the same side of line $AB$. Construct point $Y$ such that $CY=CA$ and $Y$ and $B$ lies on different sides of line $AC$. Suppose that triangle $BAX$ and triangle $CAY$ are similar, prove that the circumcenter of triangle $AXY$ lies on the circumcircle of triangle $ABC$.

The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$ \[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]

The figure $F$ is the intersection of $N$ circles (they may have different radii). Find the maximal number of curvilinear “sides” which $F$ can have. Curvilinear sides of $F$ are the arcs (of the given circumferences) that constitute the boundary of $F$. (Their ends are the “vertices” of $F$ - the points of intersection of given circumferences that lie on the boundary of $F$.) (N Brodsky)

Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.

$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.

Determine the smallest positive integer $n$ with the following property: on a board $n \times n$, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers $1,2,3 , ... , n^2$, a number in each square, so if $B$ is the sum of the numbers written in the white squares and $P$ is the sum of the numbers written in the black squares, then $\frac {B}{P} = \frac{2021}{4321}$.

Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.

A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$. [i](Ahaan Rungta, 2 points)[/i]

Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.