Let $a,b,c$ be positive reals. Prove that $$a^ab^bc^c \geq a^bb^cc^a$$

Prove that the inequality $$\left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2$$ holds for any positive numbers $a_1,\ldots,a_n$ and determine when equality occurs.

There are $m$ boys and $n$ girls participate in a duet singing contest ($m,n\geq 2$). At the contest, each section there will be one show. And a show including some boy-girl duets where each boy-girl couple will sing with each other no more than one song and each participant will sing at least one song. Two show $A$ and $B$ are considered different if there is a boy-girl couple sing at show $A$ but not show $B$. The contest will end if and only if every possible shows are performed, and each show is performed exactly one time. a) A show is called [i]depend[/i] on a participant $X$ if we cancel all duets that $X$ perform, then there will be at least one another participant will not sing any song in that show. Prove that among every shows that depend on $X$, the number of shows with odd number of songs equal to the number of shows with even number of songs. b) Prove that the organizers can arrange the shows in order to make sure that the numbers of songs in two consecutive shows have different parity.

Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.

An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?

In an acute triangle $ABC$, the bisector $AL$, the altitude $BH$, and the perpendicular bisector of the side $AB$ intersect at one point. Find the value of the angle $BAC$.

Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part? $\textbf{(A)}\ -2 \qquad\textbf{(B)}\ -\sqrt{3}+i \qquad\textbf{(C)}\ -\sqrt{2}+\sqrt{2}i \qquad\textbf{(D)}\ -1+\sqrt{3}i \qquad\textbf{(E)}\ 2i$

The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$

A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.

[asy]size(150); draw((0,0)--(1,0)--(1,1.5)--(0,1.5)--cycle); draw((2,0)--(3,0)--(3,1.5)--(2,1.5)--cycle); draw((4,0)--(5,0)--(5,1.5)--(4,1.5)--cycle); draw((2,2.5)--(3,2.5)--(3,4)--(2,4)--cycle); draw((4,2.5)--(5,2.5)--(5,4)--(4,4)--cycle); label("3",(0.5,0.5),N); label("4",(2.5,0.5),N); label("6",(4.5,0.5),N); label("P",(2.5,3),N); label("Q",(4.5,3),N);[/asy] Five cards are lying on a table as shown. Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over? \[ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \text{P} \qquad \textbf{(E)}\ \text{Q} \qquad \]

Let $p$ be some prime number. a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$. b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).

11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$. ([i]A. Golovanov[/i])

Determine all finite increasing arithmetic progressions in which each term is the reciprocal of a positive integer and the sum of all the terms is $1$.

We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

Konstantin has a pile of $100$ pebbles. In each move, he chooses a pile and splits it into two smaller ones until he gets $100 $ piles each with a single pebble. (a) Prove that at some point, there are $30$ piles containing a total of exactly $60$ pebbles. (b) Prove that at some point, there are $20$ piles containing a total of exactly $60$ pebbles. (c) Prove that Konstantin may proceed in such a way that at no point, there are $19$ piles containing a total of exactly $60$ pebbles.

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment AB. Analogously, the second line intersects the first circle at C and C′ and the second circle at D and D′. Prove that the points A, B, C, D are concyclic if and only if the points A′, B′, C′, D′ are concyclic.

Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points? [i]Sergey Dvoryaninov[/i]

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that \[\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992\] What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$

Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$