Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters, that her garden will take up?

Let $ABCD$ be a convex trapezoid such that $\angle BAD = \angle ADC = 90^o$, $AB = 20$, $AD = 21$, and $CD = 28$. Point $P \ne A$ is chosen on segment $AC$ such that $\angle BPD = 90^o$. Compute $AP$.

Let $m$ be a positive integer not divisible by 3. Prove that there are infinitely many positive integers $n$ such that $s(n)$ and $s(n+1)$ are divisible by $m$, where $s(x)$ is the sum of digits of $x$. [i]Dorel Miheț[/i]

A sequence of $X$s and $O$s is given, such that no three consecutive characters in the sequence are all the same, and let $N$ be the number of characters in this sequence. Maia may swap two consecutive characters in the sequence. After each swap, any consecutive block of three or more of the same character will be erased (if there are multiple consecutive blocks of three or more characters after a swap, then they will be erased at the same time), until there are no more consecutive blocks of three or more of the same character. For example, if the original sequence were $XXOOXOXO$ and Maia swaps the fifth and sixth character, the end result will be $$XXOOOXXO \to XXXXO \to O.$$ Find the maximum value $N$ for which Maia can’t necessarily erase all the characters after a series of swaps. Partial credit will be awarded for correct proofs of lower and upper bounds on $N$.

A floor has $2 \times 11$ dimension, and this floor covering with $1 \times 2$ rectangles. (No two rectangles overlap). How many cases we done this job?

Find all pair of positive integers $(x, y)$ satisfying the equation \[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\]

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

A special kind of chess knight is in the origin of an infinite grid. It can make one of twelve different moves: it can move directly up, down, left, or right one unit square, or it can move $1$ units in one direction and $3$ units in an orthogonal direction. How many different squares can it be on after $2$ moves?

Consider $P$ a regular $9$-sided convex polygon with each side of length $1$. A diagonal at $P$ is any line joining two non-adjacent vertices of $P$. Calculate the difference between the lengths of the largest and smallest diagonal of $P$.

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

If $x, y$ and $z$ are real numbers such that $x^2 + y^2 + z^2 = 2$, prove that $x + y + z \le xyz + 2$.

Find the last digit of \[7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.\]

There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ? [i](i)[/i] Every route includes exactly three ports; [i](ii)[/i] No two routes contain the same three ports; [i](iii)[/i] The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.

An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle in point $M$. Find the locus of circumcenters of triangles $AMK$.

Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.

Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.

Let $ z_1,z_2,z_3 $ be nonzero complex numbers and pairwise distinct, having the property that $\left( z_1+z_2\right)^3 =\left( z_2+z_3\right)^3 =\left( z_3+z_1\right)^3. $ Show that $ \left| z_1-z_2\right| =\left| z_2-z_3\right| =\left| z_3-z_1\right| . $

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Given some $m\times n$ table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.

We define $\mathbb F_{101}[x]$ as the set of all polynomials in $x$ with coefficients in $\mathbb F_{101}$ (the integers modulo $101$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb F_{101}$ for each nonnegative integer $k$. For example, $(x+3)(100x+5)=100x^2+2x+15$ in $\mathbb F_{101}[x]$ because the corresponding coefficients are equal modulo $101$. We say that $f(x)\in\mathbb F_{101}[x]$ is \emph{lucky} if it has degree at most $1000$ and there exist $g(x),h(x)\in\mathbb F_{101}[x]$ such that \[f(x)=g(x)(x^{1001}-1)+h(x)^{101}-h(x)\] in $\mathbb F_{101}[x]$. Find the number of lucky polynomials. [i]Proposed by Michael Ren.[/i]

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power: $$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$ [i]Proposed by Navid Safaei[/i]

In $\Delta ABC$, $AB=6$, $BC=7$ and $CA=8$. Let $D$ be the mid-point of minor arc $AB$ on the circumcircle of $\Delta ABC$. Find $AD^2$

Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides? [asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("$10$",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("$252^{\circ}$",(0.05,0.05),NE);[/asy] [asy] import three; picture mainframe; defaultpen(fontsize(11pt)); picture conePic(picture pic, real r, real h, real sh) { size(pic, 3cm); triple eye = (11, 0, 5); currentprojection = perspective(eye); real R = 1, y = 2; triple center = (0, 0, 0); triple radPt = (0, R, 0); triple negRadPt = (0, -R, 0); triple heightPt = (0, 0, y); draw(pic, arc(center, radPt, negRadPt, heightPt, CW)); draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8")); draw(pic, center--radPt, linetype("8 8")); draw(pic, center--heightPt, linetype("8 8")); draw(pic, negRadPt--heightPt--radPt); label(pic, (string) r, center--radPt, dir(270)); if (h != 0) { label(pic, (string) h, heightPt--center, dir(0)); } if (sh != 0) { label(pic, (string) sh, heightPt--radPt, dir(0)); } return pic; } picture pic1; pic1 = conePic(pic1, 6, 0, 10); picture pic2; pic2 = conePic(pic2, 6, 10, 0); picture pic3; pic3 = conePic(pic3, 7, 0, 10); picture pic4; pic4 = conePic(pic4, 7, 10, 0); picture pic5; pic5 = conePic(pic5, 8, 0, 10); picture aux1; picture aux2; picture aux3; add(aux1, pic1.fit(), (0,0), W); label(aux1, "$\textbf{(A)}$", (0,0), 22W, linewidth(4)); label(aux1, "$\textbf{(B)}$", (0,0), 3E); add(aux1, pic2.fit(), (0,0), 35E); add(aux2, aux1.fit(), (0,0), W); label(aux2, "$\textbf{(C)}$", (0,0), 3E); add(aux2, pic3.fit(), (0,0), 35E); add(aux3, aux2.fit(), (0,0), W); label(aux3, "$\textbf{(D)}$", (0,0), 3E); add(aux3, pic4.fit(), (0,0), 35E); add(mainframe, aux3.fit(), (0,0), W); label(mainframe, "$\textbf{(E)}$", (0,0), 3E); add(mainframe, pic5.fit(), (0,0), 35E); add(mainframe.fit(), (0,0), N); [/asy]

Let $p(x)$ be the polynomial with leading coefficent 1 and rational coefficents, such that \[p\left(\sqrt{3 + \sqrt{3 + \sqrt{3 + \ldots}}}\right) = 0,\] and with the least degree among all such polynomials. Find $p(5)$.