[b]p1.[/b] Let $f(x) = \frac{3x^3+7x^2-12x+2}{x^2+2x-3}$ . Find all integers $n$ such that $f(n)$ is an integer. [b]p2.[/b] How many ways are there to arrange $10$ trees in a line where every tree is either a yew or an oak and no two oak trees are adjacent? [b]p3.[/b] $20$ students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? [b]p4.[/b] Let $f_0(x) = x + |x - 10| - |x + 10|$, and for $n \ge 1$, let $f_n(x) = |f_{n-1}(x)| - 1$. For how many values of $x$ is $f_{10}(x) = 0$? [b]p5.[/b] $2$ red balls, $2$ blue balls, and $6$ yellow balls are in a jar. Zion picks $4$ balls from the jar at random. What is the probability that Zion picks at least $1$ red ball and$ 1$ blue ball? [b]p6.[/b] Let $\vartriangle ABC$ be a right-angled triangle with $\angle ABC = 90^o$ and $AB = 4$. Let $D$ on $AB$ such that $AD = 3DB$ and $\sin \angle ACD = \frac35$ . What is the length of $BC$? [b]p7.[/b] Find the value of of $$\dfrac{1}{1 +\dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ ...}}}}}$$ [b]p8.[/b] Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area? [b]p9.[/b] How many perfect cubes exist in the set $\{1^{2018},2^{2017}, 3^{2016},.., 2017^2, 2018^1\}$? [b]p10.[/b] Let $n$ be the number of ways you can fill a $2018\times 2018$ array with the digits $1$ through $9$ such that for every $11\times 3$ rectangle (not necessarily for every $3 \times 11$ rectangle), the sum of the $33$ integers in the rectangle is divisible by $9$. Compute $\log_3 n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine, with proof, the number of ordered triples $\left(A_{1}, A_{2}, A_{3}\right)$ of sets which have the property that (i) $A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},$ and (ii) $A_{1} \cap A_{2} \cap A_{3}=\emptyset.$ Express your answer in the form $2^{a} 3^{b} 5^{c} 7^{d},$ where $a, b, c, d$ are nonnegative integers.

In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.

Determine all triples $(x,y, p)$ with $x$, $y$ positive integers and $p$ a prime number verifying the equation $p^x -y^p = 1$.

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$. Prove that, $PQ$ bisects $BC$.

Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.

The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as: $\textbf{(A) }2^{100}-1\qquad \textbf{(B) }2^{100}(x-1)-(x-2)\qquad \textbf{(C) }2^{100}(x-3)\qquad$ $\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad \textbf{(E) }2^{100}(x+1)-(x+2)$

Alice has three daughters, each of whom has two daughters, each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)

In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$, the circle through $A$ touching the line $BC$ at $B$ has centre $Q$. Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$, respectively. Show that $R^2 = OP \cdot OQ$.

Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.

[b]2/1/32.[/b] Is it possible to fill in a $2020$ x $2020$ grid with the integers from $1$ to $4,080,400$ so that the sum of each row is $1$ greater than the previous row?

There are $n$ points in the plane such that at least $99\%$ of quadrilaterals with vertices from these points are convex. Can we find a convex polygon in the plane having at least $90\%$ of the points as vertices? [i]Proposed by Morteza Saghafian - Iran[/i]

Consider the acute-angled triangle $ABC$. Let $X$ be a point on the side $BC$, and $Y$ be a point on the side $CA$. The circle $k_1$ with diameter $AX$ cuts $AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$. (i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$. (ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$. Prove that the orthocentre of $ABC$ lies on the line $PQ$.

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

How many square inches of paint are needed to fully paint a regular $6$-sided die with side length $2$ inches, except for the $\frac13$-inch diameter circular dots marking $1$ through $6$ (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.

Let $m,n\in\mathbb Z_{\ge 0},$ $a_0,a_1,\ldots ,a_m,b_0,b_1,\ldots ,b_n\in\mathbb R_{\ge 0}$ For any integer $0\le k\le m+n,$ define $c_k:=\max_{i+j=k}a_ib_j.$ Proof $$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$ [i]Created by Yinghua Ai[/i]

Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

Problem 19 The diagram below shows a 24×24 square ABCD. Points E and F lie on sides AD and CD, respectively, so that DE = DF = 8. Set X consists of the shaded triangle ABC with its interior, while set Y consists of the shaded triangle DEF with its interior. Set Z consists of all the points that are midpoints of segments connecting a point in set X with a point in set Y . That is, Z = {z | z is the midpoint of xy for x ∈ X and y ∈ Y}. Find the area of the set Z. For diagram to http://www.purplecomet.org/welcome/practice

$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.