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$.

Determine if there exists a subset $E$ of $Z \times Z$ with the properties: (i) $E$ is closed under addition, (ii) $E$ contains $(0,0),$ (iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$. Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.

Consider the two square matrices \[A=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &-1&-1 &+1& +1 \\ +1 & -1 & -1 & -1 & +1 \\ +1 &+1&-1 &+1&-1 \end{bmatrix} \quad \text{ and } \quad B=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &+1&-1& +1&-1 \\ +1 &-1& -1& +1& +1 \\ +1 & -1& +1&-1 &+1 \end{bmatrix}\] with entries $+1$ and $-1$. The following operations will be called elementary: (1) Changing signs of all numbers in one row; (2) Changing signs of all numbers in one column; (3) Interchanging two rows (two rows exchange their positions); (4) Interchanging two columns. Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.

Let $a, b, c, d, e$ be real numbers such that $a<b<c<d<e.$ The least possible value of the function $f: \mathbb R \to \mathbb R$ with $f(x) = |x-a| + |x - b|+ |x - c| + |x - d|+ |x - e|$ is $$ \mathrm a. ~ e+d+c+b+a\qquad \mathrm b.~e+d+c-b-a\qquad \mathrm c. ~e+d+|c|-b-a \qquad \mathrm d. ~e+d+b-a \qquad \mathrm e. ~e+d-b-a $$

Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be [i]unattainable[/i]. [b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. [b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.

Consider the sequence $(x_n)$ given by $$x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.$$Prove that the sequence $y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1$ is convergent and find its limit.

There are several candles of the same size on the Chapel of Bones. On the first day a candle is lit for a hour. On the second day two candles are lit for a hour, on the third day three candles are lit for a hour, and successively, until the last day, when all the candles are lit for a hour. On the end of that day, all the candles were completely consumed. Find all the possibilities for the number of candles.

Consider a convex polygon having $n$ vertices, $n\geq 4$. We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon, such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon. Prove that there are two more black triangles that white ones.