A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

Is it possible to divide a $8 \times 8$ chessboard into $32$ 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 a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.

The function $f(x)$ has the property that $f(x) = -\frac{1}{f(x-1)}.$ Given that $f(0)=-\frac{1}{21},$ find the value of $f(2021).$ [i]Proposed by Ada Tsui[/i]

Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

What is the least real value of the expression $\sqrt{x^2-6x+13} + \sqrt{x^2-14x+58}$ where $x$ is a real number? $ \textbf{(A)}\ \sqrt {39} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ \frac {43}6 \qquad\textbf{(D)}\ 2\sqrt 2 + \sqrt {13} \qquad\textbf{(E)}\ \text{None of the above} $

A carpet dealer,who has a lot of carpets in the market,is available to exchange a carpet of dimensions $a\cdot b$ either with a carpet with dimensions $\frac{1}{a}\cdot \frac{1}{b}$ or with two carpets with dimensions $c\cdot b$ and $\frac{a}{c}\cdot b$ (the customer can select the number $c$).The dealer supports that,at the beginning he had a carpet with dimensions greater than $1$ and,after some exchanges like the ones we described above,he ended up with a set of carpets,each one having one dimension greater than $1$ and one smaller than $1$.Is this possible? [i]Note:The customer can demand from the dealer to consider a carpet of dimensions $a\cdot b$ as one with dimensions $b\cdot a$.[/i]

For how many integral values of $x$ can a triangle of positive area be formed having side lengths $ \log_{2} x, \log_{4} x, 3$? $\textbf{(A) } 57\qquad \textbf{(B) } 59\qquad \textbf{(C) } 61\qquad \textbf{(D) } 62\qquad \textbf{(E) } 63$

Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.