Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use? $\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf{(C) }91\qquad\textbf{(D) }96\qquad \textbf{(E) }120$

Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$

Find all positive integers $ x$ and $ y$ such that $ 5^{x}-3^{y}= 16$.

A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$? $ \textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad $

Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total? [img]https://cdn.artofproblemsolving.com/attachments/0/8/e46929e7ec3fff9be4892ef954ae299e0cb8c7.png[/img]

Let $ABC$ be an acute triangle and let $D$ be the foot of the $A$-bisector. Moreover, let $M$ be the midpoint of $AD$. The circle $\omega_1$ with diameter $AC$ meets $BM$ at $E$, while the circle $\omega_2$ with diameter $AB$ meets $CM$ at $F$. Assume that $E$ and $F$ lie inside $ABC$. Prove that $B$, $E$, $F$, $C$ are concyclic.

How many three-digit positive integers have an odd number of even digits? $\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

Let be a natural number $ a\ge 2. $ [b]a)[/b] Show that there is no infinite sequence $ \left( k_n \right)_{n\ge 1} $ of pairwise distinct natural numbers greater than $ 1 $ having the property that the sequence $ \left( a^{1/k_n} \right)_{n\ge 1} $ is a geometric progression. [b]b)[/b] Show that there are finite sequences $ \left( l_i \right)_i, $ of any length, of pairwise distinct natural numbers greater than $ 1 $ with the property that $ \left( a^{1/l_i} \right)_{i} $ is a geometric progression. [i]Bogdan Enescu[/i]

Assume that the number of offspring for every man can be $0,1,\ldots, n$ with with probabilities $p_0,p_1,\ldots,p_n$ independently from each other, where $p_0+p_1+\cdots+p_n=1$ and $p_n\neq 0$. (This is the so-called Galton-Watson process.) Which positive integer $n$ and probabilities $p_0,p_1,\ldots,p_n$ will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation?

Let $S$ be the sum of all distinct real solutions of the equation \[ \sqrt{x + 2015} = x^2 - 2015. \] Compute $\lfloor 1/S \rfloor$. Recall that if $r$ is a real number, then $\lfloor r \rfloor$ (the [i]floor[/i] of $r$) is the greatest integer that is less than or equal to $r$.

An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality $$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$ [i]Proposed by Anastasija Trajanova[/i]

The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30 ^\circ$ larger than the other. What is the degree measure of the largest angle in the triangle? $ \textbf{(A)}\ 69 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 108 $

Let $a, b, c$ be reals which satisfy $a+b+c+ab+bc+ac+abc=>7$, prove that $$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}=>6$$

Let $S = \{1,2,3,4,5\}$. Find the number of functions $f : S \to S$ such that $f ^{50}(x)= x$ for all $x \in S$.

Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.

In a school there are $2021$ doors with the numbers $1,2,\dots,2021$. In a day $2021$ students play the following game: Initially all the doors are closed, and each student receive a card to define the order, there are exactly $2021$ cards. The numbers in the cards are $1,2,\dots,2020,2021$. The order will be student $1$ first, student $2$ will be the second, and going on. The student $k$ will change the state of the doors $k,2k,4k,8k,\dots,2^pk$ with $2^pk\leq 2021\leq 2^{p+1}k$. Change the state is [b]if the door was close, it will be open and vice versa.[/b] a) After the round of the student $16$, determine the configuration of the doors $1,2,\dots,16$ b) After the round of the student $2021$, determine how many doors are closed.

$\omega$ is a circle inside angle $\measuredangle BAC$ and it is tangent to sides of this angle at $B,C$.An arbitrary line $ \ell $ intersects with $AB,AC$ at $K,L$,respectively and intersect with $\omega$ at $P,Q$.Points $S,T$ are on $BC$ such that $KS \parallel AC$ and $TL \parallel AB$.Prove that $P,Q,S,T$ are concyclic.(I.Bogdanov,P.Kozhevnikov)

We say "$s$ grows to $r$" if there exists some integer $n>0$ such that $s^n = r.$ Call a real number $r$ "sparcs" if there are only finitely many real numbers $s$ that grow to $r.$ Find all real numbers that are sparse.

A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then: $\textbf{(A)}\ x=\dfrac{1}{2}h \qquad \textbf{(B)}\ x=\dfrac{bh}{b+h} \qquad \textbf{(C)}\ x=\dfrac{bh}{2h+b} \qquad \textbf{(D)}\ x=\sqrt{\dfrac{hb}{2}} \qquad \textbf{(E)}\ x=\dfrac{1}{2}b$

The sides of a right triangle are $a, a+d,$ and $a+2d$, with $a$ and $d$ both positive. The ratio of $a$ to $d$ is: $ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:4 \qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 3:1\qquad\textbf{(E)}\ 3:4 $

Let $ABC$ be an isosceles triangle with $AB = BC$. Point $E$ lies on the side $AB$, and $ED$ is the perpendicular from $E$ to $BC$. It is known that $AE = DE$. Find $\angle DAC$.

We have $x_i >i$ for all $1 \le i \le n$. Find the minimum value of $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$

The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence: $$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$ Given that $a_{2021} = 2021$, find $a_1$.

For every integer $n \ge 1$, the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$. Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$. Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$. (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.) [i]Proposed by Lewis Chen[/i]