$ABC$ is a triangle where $AB = 10$, $BC = 14$, and $AC = 16$. Let $DEF$ be the triangle with smallest area so that $DE$ is parallel to $AB$, $EF$ is parallel to $BC$, $DF$ is parallel to $AC$, and the circumcircle of $ABC$ is $DEF$’s inscribed circle. Line $DA$ meets the circumcircle of $ABC$ again at a point $X$. Find $AX^2$ .

Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$. Determine the minimum value of the red number at the end of this procedure.

A non-empty subset of $\{1,2, ..., n\}$ is called [i]arabic [/i] if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of $\{1,2, ..., n\}$ has the same parity as $n$.

Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations? i)$k$ is a prime number greater than $2$; ii) $k$ is odd; iii) $k$ is even.

Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

Consider $v,w$ two distinct non-zero complex numbers. Prove that \[|zw+\bar{w}|\le |zv+\bar{v}|,\] for any $z\in\mathbb{C},|z|=1$, if and only if there exists $k\in [-1,1]$ such that $w=kv$. [i]Dan Marinescu[/i]

Olya and Tolya are playing a game on $[0,1]$ segment. In the beginning it is white. In the first round Tolya chooses a number $0 \leq l \leq 1$, and then Olya chooses a subsegment of $[0,1]$ of length $l$ and recolors every its point to the opposite color(white to black, black to white). In the next round players change roles, etc. The game lasts $2024$ rounds. Let $L$ be the sum of length of white segments after the end of the game. If $L > \frac{1}{2}$ Olya wins, otherwise Tolya wins. Which player has a strategy to guarantee his win? [i]A. Naradzetski[/i]

Let $ ABC$ be a triangle with $ \angle BAC\equal{}60^{\circ}$. The incircle of $ ABC$ is tangent to $ AB$ at $ D$. Construct a circle with radius $ DA$ and cut the incircle of $ ABC$ at $ E$. If $ AF$ is an altitude, prove that $ AE\ge AF$.

In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? $ \textbf{(A)}\ 5462 \qquad\textbf{(B)}\ 5586 \qquad\textbf{(C)}\ 5664 \qquad\textbf{(D)}\ 5720 \qquad\textbf{(E)}\ 5848 $

If $ V \equal{} gt \plus{} V_0$ and $ S \equal{} \frac {1}{2}gt^2 \plus{} V_0t$, then $ t$ equals: $ \textbf{(A)}\ \frac {2S}{V \plus{} V_0} \qquad\textbf{(B)}\ \frac {2S}{V \minus{} V_0} \qquad\textbf{(C)}\ \frac {2S}{V_0 \minus{} V} \qquad\textbf{(D)}\ \frac {2S}{V} \qquad\textbf{(E)}\ 2S \minus{} V$

Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$ . [i]Sam Bealing, United Kingdom[/i]

If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$

For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$ Find the value of the expression $x^2 + y^2+ z^2$.

[u]Bases[/u] Many of you may be familiar with the decimal (or base $10$) system. For example, when we say $2013_{10}$, we really mean $2\cdot 10^3+0\cdot 10^2+1\cdot 10^1+3\cdot 10^0$. Similarly, there is the binary (base $2$) system. For example, $11111011101_2 = 1 \cdot 2^{10}+1 \cdot 2^9+1 \cdot 2^8+1 \cdot 2^7+1 \cdot 2^6+0 \cdot 2^5+1 \cdot 2^4+1 \cdot 2^3+1 \cdot 2^2+0 \cdot 2^1+1 \cdot 2^0 = 2013_{10}.$ In general, if we are given a string $(a_na_{n-1} ... a_0)_b$ in base $b$ (the subscript $b$ means that we are in base $b$), then it is equal to $\sum^n_{i=0} a_ib^i$. It turns out that for every positive integer $b > 1$, every positive integer $k$ has a unique base $b$ representation. That is, for every positive integer $k$, there exists a unique $n$ and digits $0 \le a_0,..., a_n < b$ such that $(a_na_{n-1} ... a_0)_b = k$. We can adapt this to bases $b < -1$. It actually turns out that if $b < -1$, every nonzero integer has a unique base b representation. That is, for every nonzero integer $k$, there exists a unique $n$ and digits $0 \le a_0,..., a_n < |b|$ such that $(a_na_{n-1} ... a_0)_b = k$. The next five problems involve base $-4$. Note: Unless otherwise stated, express your answers in base $10$. [b]p6.[/b] Evaluate $1201201_{-4}$. [b]p7.[/b] Express $-2013$ in base $-4$. [b]p8.[/b] Let $b(n)$ be the number of digits in the base $-4$ representation of $n$. Evaluate $\sum^{2013}_{i=1} b(i)$. [b]p9.[/b] Let $N$ be the largest positive integer that can be expressed as a $2013$-digit base $-4$ number. What is the remainder when $N$ is divided by $210$? [b]p10.[/b] Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \ne 4$ such that the base $-4$ representation of $n$ is the same as the base $b$ representation of $n$.

An alphabet $A$ has $16$ letters. A message is written using the alphabet and, to encrypt the message, a permutation $f : A \to A$ is applied to each letter. Let $n(f)$ be the smallest positive integer $k$ such that every message $m$, encrypted by applying $f$ to the message $k$ times, produces $m$. Compute the largest possible value of $n(f)$.

(a) Let $n$ is a positive integer. Calculate $\displaystyle \int_0^1 x^{n-1}\ln x\,dx$.\\ (b) Calculate $\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).$

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$. Show that $a^2+b^2-ab$ is not a square.

The quadrilateral $ABCD$ is inscribed in the circle ω. The diagonals $AC$ and $BD$ intersect at the point $O$. On the segments $AO$ and $DO$, the points $E$ and $F$ are chosen, respectively. The straight line $EF$ intersects ω at the points $E_1$ and $F_1$. The circumscribed circles of the triangles $ADE$ and $BCF$ intersect the segment $EF$ at the points $E_2$ and $F_2$ respectively (assume that all the points $E, F, E_1, F_1, E_2$ and $F_2$ are different). Prove that $E_1E_2 = F_1F_2$. $(N. Sedrakyan)$

for a positive integer $n$, there are positive integers $a_1, a_2, ... a_n$ that satisfy these two. (1) $a_1=1, a_n=2020$ (2) for all integer $i$, $i$satisfies $2\leq i\leq n, a_i-a_{i-1}=-2$ or $3$. find the greatest $n$

Given any five nonnegative real numbers with the sum $1$, show that they can be arranged around a circle in such a way that the five products of two consecutive numbers sum up to at most $1/5$.

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

The circle $C_1$ and $C_2$ are externally tangent to each other at $T$. A line passing through $T$ meets $C_1$ at $A$ and meets $C_2$ at $B$. The line which is tangent to $C_1$ at $A$ meets $C_2$ at $D$ and $E$. If $D \in [AE]$, $|TA|=a$, $|TB|=b$, what is $|BE|$? $ \textbf{(A)}\ \sqrt{a(a+b)} \qquad\textbf{(B)}\ \sqrt{a^2+b^2+ab} \qquad\textbf{(C)}\ \sqrt{a^2+b^2-ab} \qquad\textbf{(D)}\ \sqrt{a^2+b^2} \qquad\textbf{(E)}\ \sqrt{(a+b)b} $

Prove that there does not exist a ring with exactly 5 regular elements. ($ a$ is called a regular element if $ ax \equal{} 0$ or $ xa \equal{} 0$ implies $ x \equal{} 0$.) A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.