If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$. [i]Proposed by Deyuan Li and Andrew Milas[/i]

You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is [i]good [/i] if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice. Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a [i]good [/i] coloring.

Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}$.

The difference between two prime numbers is $11$. Find their sum.

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds: \[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

Let $ABCD$ be a parallelogram. Let $W, X, Y,$ and $Z$ be points on sides $AB, BC, CD,$ and $DA$, respectively, such that the incenters of triangles $AWZ, BXW, CYX,$ and $DZY$ form a parallelogram. Prove that $WXYZ$ is a parallelogram.

How many permutations $\pi$ of $\left\{1,2,\ldots,7\right\}$ are there such that $\pi\left(k\right)\leq2k$ for $k=1,\ldots,7$? A permutation $\pi$ of a set $S$ is a function from $S$ to itself such that if $a\neq b$, then $\pi\left(a\right)\neq\pi\left(b\right)$. For example, $\pi\left(x\right)=x$ and $\pi\left(x\right)=8-x$ are permutations of $\left\{1,2,\ldots,7\right\}$ but $\pi\left(x\right)=1$ is not. [i]2019 CCA Math Bonanza Individual Round #7[/i]

The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$, where $n$ and $k$ are integers and $0\leq k<2013$. What is $k$? Recall $2013=3\cdot 11\cdot 61$.

$4(299)+3(299)+2(299)+298=$ $\text{(A)}\ 2889 \qquad \text{(B)}\ 2989 \qquad \text{(C)}\ 2991 \qquad \text{(D)}\ 2999 \qquad \text{(E)}\ 3009$

In four years Kay will be twice as old as Gordon. Four years after that Shaun will be twice as old as Kay. Four years after that Shaun will be three times as old as Gordon. How many years old is Shaun now?

On the Island of Camelot live $13$ grey, $15$ brown and $17$ crimson chameleons . If two chameleons of different colours meet , they both simultaneously change colour to the third colour (e .g . if a grey and brown chameleon meet each other they both change to crimson) . Is it possible that they will eventually all be the same colour? (V . G . Ilichev)

PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Given $100$ points on the plane. Prove that you can cover them with a family of circles with the sum of their diameters less than $100$ and the distance between any two of the circles more than one.

In a table $4\times 4$ we put $k$ blocks such that i) Each block covers exactly 2 cells ii) Each cell is covered by, at least, one block iii) If we delete a block; there is, at least, one cell that is not covered. Find the maximum value of $k$. Note: The blocks can overlap.

Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $ 50\%$ discount as children. The two members of the oldest generation receive a $ 25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $ \$6.00$, is paying for everyone. How many dollars must he pay? $ \textbf{(A)}\ 34 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 48$

Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.) $ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$ \[f(x+y^2+z)=f(f(x))+yf(y)+f(z). \]

A trapezoid $ABCD$ with bases $AD$ and $BC$ is such that $AB = BD$. Let $M$ be the midpoint of $DC$. Prove that $\angle MBC$ = $\angle BCA$.

The total area of all the faces of a rectangular solid is $22 \text{cm}^2$, and the total length of all its edges is $24 \text{cm}$. Then the length in $\text{cm}$ of any one of its internal diagonal is A. $\sqrt{11}$ B. $\sqrt{12}$ C. $\sqrt{13}$ D. $\sqrt{14}$ E. Not uniquely determined

The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway? $\text{(A)} \ 100 \qquad \text{(B)} \ 120 \qquad \text{(C)} \ 200 \qquad \text{(D)} \ 240 \qquad \text{(E)} \ 400$

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.