Tim is thinking of a positive integer between $2$ and $15,$ inclusive, and Ted is trying to guess the integer. Tim tells Ted how many factors his integer has, and Ted is then able to be certain of what Tim's integer is. What is Tim's integer?

Show that for all positive reals $a,b,c$ with $a+b+c=3$, $$\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.$$

Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.

For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. If $m \ge 3$, prove that \[\left[\frac{m(m+1)}{2(2m-1)}\right]=\left[\frac{m+1}{4}\right]\]

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $ 40 \%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $ 30 \%$ of the group are girls. How many girls were initially in the group? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

Find $3$ numbers, each of which is equal to the square of the difference of the other two.

How many distinct ordered pairs of integers $(b, m, t)$ satisfy the equation $b^8+m^4+t^2+1 = 2019$?

Let $ABC$ be an acute-angled triangle in which $D,E,F$ are points on $BC,CA,AB$ respectively such that $AD \perp BC$;$AE = BC$; and $CF$ bisects $\angle C$ internally, Suppose $CF$ meets $AD$ and $DE$ in $M$ and $N$ respectively. If $FM$$= 2$, $MN =1$, $NC=3$, find the perimeter of $\Delta ABC$.

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

A circular disk is divided by $2n$ equally spaced radii($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is $\textbf{(A) }2n+1\qquad\textbf{(B) }2n+2\qquad\textbf{(C) }3n-1\qquad\textbf{(D) }3n\qquad \textbf{(E) }3n+1$

Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$. [i] (Călin Pop & Vlad Robu) [/i]

Suppose in triangle $ABC$ incircle touches the side $BC$ at $P$ and $\angle APB=\alpha$. Prove that : \[\frac1{p-b}+\frac1{p-c}=\frac2{rtg\alpha}\]

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Determine all positive integers $n$ for which there exist pairwise distinct positive real numbers $a_1, a_2, \cdots, a_n$ satisfying $\displaystyle \left\{a_i+\frac{(-1)^i}{a_i}\mid 1\leq i \leq n\right\}=\{a_i\mid 1\leq i \leq n\}$

Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?

Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the next step: he replaces the number $n$ on the board by $\frac12n$ if $n$ is even and by $n^2 + 3$ if $n$ is odd. For how many choices of starting numbers below $2023$ will Felix never write a number of more than four digits on the board?

For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$.

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

In the triangle $ABC$ with sides $AC = b$ and $AB = c$, the extension of the bisector of angle $A$ intersects it's circumcircle at point with $W$. Circle $\omega$ with center at $W$ and radius $WA$ intersects lines $AC$ and $AB$ at points $D$ and $F$, respectively. Calculate the lengths of segments $CD$ and $BF$. (Evgeny Svistunov) [img]https://cdn.artofproblemsolving.com/attachments/7/e/3b340afc4b94649992eb2dccda50ca8f3f7d1d.png[/img]

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

If $ \sum_{m=-\infty}^{+\infty} |a_m| < \infty$, then what can be said about the following expression? \[ \lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.\] [i]P. Turan[/i]

Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]

We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between $1 km/h$, $2km/h$,..., and $n km/h$, each one with a distinct velocity, and each guinea pig has a counter starting from $0$. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each $1\leq i\leq n$ we have that the $i-$th guinea pig has $i-1$ in its counter.

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$. Prove that points $P,Q,P_1$, and $Q_1$ are concyclic. [i]Proposed by Anton Trygub, Ukraine[/i]

Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$. a) Show that $super sextalternados$ numbers don't exist. b) Find the smallest $sextalternado$ number.