Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$. Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality. (Proposed by Karl Czakler)

Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

$ABC$ 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 $n$