Let a triangle $ABC$ be given with $AB <AC$. Let the inscribed center of the triangle be $I$. The perpendicular bisector of side $BC$ intersects the angle bisector of $BAC$ at point $S$ and the angle bisector of $CBA$ at point $T$. Prove that the points $C, I, S$ and $T$ lie on a circle. (Karl Czakler)

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.

Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is $|BC|$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6\sqrt 2 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 5\sqrt 3 $

If the coefficient of the third term in the binomial expansion of $(1 - 3x)^{1/4}$ is $-a/b$, where $ a$ and $b$ are relatively prime integers, find $a+b$.

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

The numbers $1,2,...,n$ ($n \ge 5$) are written on the circle in the clockwise order. Per move it is allowed to exchange any couple of consecutive numbers $a, b$ to the couple $\frac{a+b}{2}, \frac{a+b}{2}$. Is it possible to make all numbers equal using these operations?

Johan is swimming laps in the pool. At $\text{12:17}$ he realized that he had just finished one-third of his workout. By $\text{12:22}$ he had completed eight more laps, and he realized that he had just finished five-elevenths of his workout. After $\text{12:22}$ how many more laps must Johan swim to complete his workout?

A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$. The wheel is perfectly circular and has radius $5$. After the three laps, how many revolutions around its axis has the wheel been turned through?

There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code). In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click. Find the smallest number of attempts that is sufficient to open the safe in all cases.

How many two-digit positive integers $ N$ have the property that the sum of $ N$ and the number obtained by reversing the order of the digits of $ N$ is a perfect square? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

Find the number of triples $(a, b, c)$ of positive integers such that (a) $a b$ is a prime; (b) $b c$ is a product of two primes; (c) $a b c$ is not divisible by square of any prime and (d) $a b c \leq 30$.

Prove that there do not exist pairwise distinct complex numbers $a, b, c,$ and $d$ such that $$a^3-bcd=b^3-acd=c^3-abd=d^3-abc.$$

On a $49\times 69$ rectangle formed by a grid of lattice squares, all $50\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector $\overrightarrow{AB}$, or by the vector $\overrightarrow{BA}$. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is $\overrightarrow{0}$, then he wins; else, the second player wins. Which player has a winning strategy?

Consider the collection of different squares which may be formed by sets of four points chosen from the $12$ labelled points in the diagram on the right. For each possible area such a square may have, determine the number of squares which have this area. Make sure to explain why your list is complete. [img]https://cdn.artofproblemsolving.com/attachments/b/a/faf00c2faa7b949ab2894942f8bd99505543e8.png[/img]

Let $a,b,c,x,y$ be positive real numbers such that $abc=1$. Prove that $$\frac{a^5}{xc+yb}+\frac{b^5}{xa+yc}+\frac{c^5}{xb+ya}\geq \frac{9}{(x+y)(a^2+b^2+c^2)}.$$

Find the smallest natural number that is the sum of $9$ consecutive natural numbers, is the sum of $10$ consecutive natural numbers and is also the sum of $11$ consecutive natural numbers.

Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths $6$, $8$, and $10$ and that no part of him is below the $x$-axis, find the minimum possible value of $k$.

Suppose \[ n(n\plus{}1)a_{n\plus{}1}\equal{}n(n\minus{}1)a_n\minus{}(n\minus{}2)a_{n\minus{}1} \] for every positive integer $ n\ge1$. Given that $ a_0\equal{}1,a_1\equal{}2$, find \[ \frac{a_0}{a_1}\plus{}\frac{a_1}{a_2}\plus{}\frac{a_2}{a_3}\plus{}\dots\plus{}\frac{a_{50}}{a_{51}}. \]

The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers $x$, $y$, and $z$ on the board, and does the following. First, $x$ is erased and replaced with $y$, after which $y$ is erased and replaced with $z$, and finally $z$ is erased and replaced with $x$. The higher power repeats this process some finite number of times. For example, if $(x,y,z)=(2,4,5)$ is chosen, followed by $(x,y,z)=(1,4,3)$, the board would change in the following manner: \[12345678 \rightarrow 14352678 \rightarrow 43152678\] Compute the number of possible final orderings of the eight numbers.

Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}$.

Consider parallelogram $ABCD$ with $AB > BC$. Point $E$ on $\overline{AB}$ and point $F$ on $\overline{CD}$ are marked such that there exists a circle $\omega_1$ passing through $A$, $D$, $E$, $F$ and a circle $\omega_2$ passing through $B$, $C$, $E$, $F$. If $\omega_1$, $\omega_2$ partition $\overline{BD}$ into segments $\overline{BX}$, $\overline{XY}$ , $\overline{Y D}$ in that order, with lengths $200$, $9$, $80$, respectively, compute $BC$.

There are $N$ acute triangles on the plane. Their vertices are all integer points, their areas are all equal to $2^{2020}$, but no two of them are congruent. Find the maximum possible value of $N$. Note: $(x,y)$ is an integer point if and only if $x$ and $y$ are both integers. [i]Proposed by CSJL[/i]

Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them.

Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.