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.

$2000$ consecutive integers (not necessarily positive) are written on the board. A student takes several turns. On each turn, he partitions the $2000$ integers into $1000$ pairs, and substitutes each pair by the difference arid the sum of that pair (note that the difference does not need to be positive as the student may choose to subtract the greater number from the smaller one; in addition, all the operations are carried simultaneously). Prove that the student will never again write $2000$ consecutive integers on the board.

Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.

Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition: For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and$ AZ$ have length $1$ (it means that $AX = AY = AZ = 1$).

If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, then $m$ equals: $\textbf{(A)}\ -\frac{1}{2}\qquad \textbf{(B)}\ -\frac{2}{3}\qquad \textbf{(C)}\ -\frac{3}{2} \qquad \textbf{(D)}\ -2 \qquad \textbf{(E)}\ \text{none of these}$

A ball is placed on a plane and a point on the ball is marked. Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling. Prove that it is possible. Time allowed for this problem was 90 minutes.

Let $n$ be a positive integer. Suppose that the diophantine equation \[z^n = 8 x^{2009} + 23 y^{2009} \] uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.

Someone draws at least three lines on paper. Each cuts the other lines two by two. No three lines pass through one point. He chooses a line and counts the intersection points on either side of the line. The numbers of intersections turn out to be the same. He chooses another line. Now the intersections number on one side appears to be six times as large as that on the other side. What is the minimum number of lines where this is possible? [hide=original wording of second sentence]De lijnen snijden elkaar twee aan twee.[/hide]