Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent. [i]Pablo Jaén, Panama[/i]

Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.

A club provides 30 snacks to 18 members, and each member orders 3 different snacks. It is known that every snack is ordered by at least one member, and that any two members order at most one same snack. Is it possible to find 12 snacks, such that the snacks ordered by any member is not completely in these 12 snacks?

Some cells of an infinite chessboard (infinite in all directions) are coloured blue so that at least one of the $100$ cells in any $10 \times 10$ rectangular grid is blue. Prove that, for any positive integer $n$, it is possible to select $n$ rows and $n$ columns so that all of the $n^2$ cells in their intersections are blue.

A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either: $\bullet$ grades another problem of the same student, or $\bullet$ grades the same problem of the student $s_{i-1}$ or $s_{i+1}$ (if $i > 1$ and $i < 40$, respectively). He grades each problem exactly once, starting with the first problem of $s_1$ and ending with the third problem of $s_{40}$. Let $N$ be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when $N$ is divided by $100$.

[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img] [b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$. (a) Give an example of a lucky set with five elements. (b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote] [b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that (a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$, (b) $p(x) > 0$ for all real $x$, (c) $q(x) > 0$ for all real $x$. [b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions. [b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define the sequence $a_1, a_2, a_3, ...$ by $a_1 = 1$, $a_n = a_{n-1} + a_{[n/2]}$. Does the sequence contain infinitely many multiples of $7$?

Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is [asy] pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); transform t = rotate(-45,(3,.5)); pair e = t*a,f=t*b,g=t*c,h=t*d; pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); draw(a--b^^c--d^^e--f^^g--h); filldraw(i--j--l--k--cycle,blue); label("$\alpha$",i+(-.5,.2)); //commented out labeling because it doesn't look right. //path lbl1 = (a+(.5,.2))--(c+(.5,-.2)); //draw(lbl1); //label("$1$",lbl1);[/asy] $\text{(A)} \ \sin \alpha \qquad \text{(B)} \ \frac{1}{\sin \alpha} \qquad \text{(C)} \ \frac{1}{1 - \cos \alpha} \qquad \text{(D)} \ \frac{1}{\sin^2 \alpha} \qquad \text{(E)} \ \frac{1}{(1 - \cos \alpha)^2}$

An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of $1$ unit either up or to the right. A lattice point $(x, y)$ with $0 \le x, y \le 5$ is chosen uniformly at random. Compute the expected number of up-right paths from $(0, 0)$ to$ (5,5)$ not passing through $(x, y)$.

A natural number is called $nice$ if it doesn't contain 0 and if we add the product of its digit to the number, we obtain number with the same product of its digits. Prove that there is a nice 2015-digit number.

$\triangle KWU$ is an equilateral triangle with side length $12$. Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$. If $\overline{KP} = 13$, find the length of the altitude from $P$ onto $\overline{WU}$. [i]Proposed by Bradley Guo[/i]

A solid can be separated from a liquid by all the following means EXCEPT $ \textbf{(A) }\text{decantation} \qquad\textbf{(B) }\text{distillation}\qquad$ $\textbf{(C) }\text{filtration}\qquad\textbf{(D) }\text{hydration}\qquad$

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

A projectile is launched across flat ground at an angle $\theta$ to the horizontal and travels in the absence of air resistance. It rises to a maximum height $H$ and lands a horizontal distance $R$ away. What is the ratio $H/R$? (A) $\tan \theta$ (B) $2 \tan \theta$ (C) $\frac{2}{\tan \theta}$ (D) $\frac{1}{2}\tan \theta$ (E) $\frac{1}{4}\tan \theta$

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.

Let $ABCD$ be a rhombus. Let $K$ be a point on the line $CD$, other than $C$ or $D$, such that $AD = BK$. Let $P$ be the point of intersection of $BD$ with the perpendicular bisector of $BC$. Prove that $A, K$ and $P$ are collinear.

Given a point $P$ in the circumcircle $\omega$ of an equilateral triangle $ABC$, prove that the segments $PA$, $PB$, and $PC$ form a triangle $T$. Let $R$ be the radius of the circumcircle $\omega$ and let $d$ be the distance between $P$ and the circumcenter. Find the area of $T$.

What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4\sqrt{3}-6 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3}+6$

An $n \times n$ array of numbers is given. $n$ is odd and each number in the array is $1$ or $-1$. Prove that the number of rows and columns containing an odd number of $-1$s cannot total $n$.

Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by \[F(M) = BM +CM-\sqrt{3}AM\]

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

Find the volume of the four-dimensional hypersphere $x^2 +y^2 +z^2 +t^2 =r^2$ and the hypervolume of its interior $x^2 +y^2 +z^2 +t^2 <r^2$

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

Compute the number of positive integer divisors of $2121$ with a units digit of $1$. [i]2021 CCA Math Bonanza Individual Round #1[/i]