During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as \[\dfrac{a-b\sqrt c}d,\] where all four variables are positive integers, $c$ is a multple of no perfect square greater than $1$, $a$ is coprime with $d$, and $b$ is coprime with $d$. Find the value of $a+b+c+d$.

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

In a new edition of QoTD duels, $n \ge 2$ ranked contestants (numbered 1 to $n$) play a round robin tournament (i.e. each pair of contestants compete against each other exactly once); no draws are possible. Define an upset to be a pair $(i, j)$ where$ i > j$ and contestant $i$ wins against contestant $j$. At the end of the tournament, contestant $i$ has $s_i$ wins for each $1 \le i \le n$. The result of the tournament is defined as the $n$-tuple $(s_1, s_2, \cdots , s_n)$. An $n$-tuple $S$ is called interesting if, among the distinct tournaments that produce $S$ as a result, the number of tournaments with an odd number of upsets is not equal to the number of tournaments with an even number of upsets. Find the number of interesting $n$-tuples in terms of $n$. [i](Two tournaments are considered distinct if the outcome of some match differs.)[/i]

Let \( A \) and \( B \) be two square matrices with complex entries such that \( A + B = AB \), \( A = A^* \), and \( A \) has all distinct eigenvalues. Prove that there exists a polynomial \( P \) with complex coefficients such that \( P(A) = B \).

Find six points $A_1, A_2, \ldots , A_6$ in the plane, such that for each point $A_i, i = 1, 2, \ldots , 6$ there are exactly three of the remaining five points exactly $1$ cm from $A_i$.

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

Find the number of ordered triples of integers $(a,b,c)$ such that $$a^2 + b^2 + c^2 - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2$$ and $|a|, |b|, |c| \le 2021.$

Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are arranged in a circle. It turned out that for any $i = 1, 2, \ldots, 2024$, the following condition holds: $a_ia_{i+1} < a_{i+2}$. (Here we assume that $a_{2025} = a_1$ and $a_{2026} = a_2$). What largest number of positive integers could there be among these numbers $a_1, a_2, \ldots, a_{2024}$? [i]Proposed by Mykhailo Shtandenko[/i]

Consider $2$ positive integers $a,b$ such that $a+2b=2020$. (a) Determine the largest possible value of the greatest common divisor of $a$ and $b$. (b) Determine the smallest possible value of the least common multiple of $a$ and $b$.

Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms. [i]Derek Liu[/i]

Let the inscribed circle $ \omega $ of triangle $ ABC $ touch the side $ BC $ at the point $ A '$. Let $ AA '$ intersect $ \omega $ at $ P \neq A $. Let $ CP $ and $ BP $ intersect $ \omega $, respectively, at points $ N $ and $ M $ other than $ P $. Prove that $ AA ', BN $ and $ CM $ intersect at one point.

Suppose P is a point in the interior of a triangle ABC, that x; y; z are the distances from P to A; B; C, respectively, and that p; q; r are the per- pendicular distances from P to the sides BC; CA; AB, respectively. Prove that $xyz \geq 8pqr$; with equality implying that the triangle ABC is equilateral.

[b]p1.[/b] Is it possible to write $5$ different fractions that add up to $1$, such that their numerators are equal to one and their denominators are natural numbers? [b]p2.[/b] The following is known about two numbers $x$ and $y$: if $x\ge 0$, then $y = 1 -x$; if $y\le 1$, then $x = 1 + y$; if $x\le 1$, then $x = |1 + y|$. Find $x$ and $y$. [b]p3.[/b] Five people living in different cities received a salary, some more, others less ($143$, $233$, $313$, $410$ and $413$ rubles). Each of them can send money to the other by mail. In this case, the post office takes $10\%$ of the amount of money sent for the transfer (in order to receive $100$ rubles, you need to send $10\%$ more, that is, $110$ rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method? [b]p4.[/b] a) List three different natural numbers $m$, $n$ and $k$ for which $m! = n! \cdot k!$ . b) Is it possible to come up with $1999$ such triplets? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Positive real numbers $a,b,c,d$ satisfy the equalities \[a^3+b^3+c^3=3d^3\\ b^4+c^4+d^4=3a^4\\ c^5+d^5+a^5=3b^5. \] Prove that $a=b=c=d$.

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

Trapezoid $ABCD$ with bases $AD$ and $BC$ is inscribed in a circle, $M$ is the intersection of of its diagonals. A straight line passing through $M$ perpendicular to the bases intersects $BC$ at point$ K$, and the circle at point $L$, where $L$ is the one of the two intersection points for which $M$ lies on the segment $KL$. It is known that $MK = a$, $LM = b$. Find the radius of the circle tangent to the segments $AM$, $BM$ and the circle circumscribed around $ABCD$.

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

Let $ABC$ be a triangle and $D$ a point on the side $BC$. The tangent line to the circumcircle of the triangle $ABD$ at the point $D$ intersects the side $AC$ at $E$. The tangent line to the circumcircle of the triangle $ACD$ at the the point $D$ intersects the side $AB$ at $F$. Prove that the point $A$ and the circumcenters of the triangles $ABC$ and $DEF$ are collinear. (Malik Talbi)

The given points are $ A $ and $ B $ and the circle $ k $. Draw a circle passing through the points $ A $ and $ B $ and defining, at the intersection with the circle $ k $, a common chord of a given length $ d $.

Let $C$ denote the set of points $(x, y) \in R^2$ such that $x^2 + y^2 \le1$. A sequence $A_i = (x_i, y_i), |i \ge¸ 0$ of points in $R^2$ is ‘centric’ if it satisfies the following properties: $\bullet$ $A_0 = (x_0, y_0) = (0, 0)$, $A_1 = (x_1, y_1) = (1, 0)$. $\bullet$ For all $n\ge 0$, the circumcenter of triangle $A_nA_{n+1}A_{n+2}$ lies in $C$. Let $K$ be the maximum value of $x^2_{2012} + y^2_{2012}$ over all centric sequences. Find all points $(x, y)$ such that $x^2 + y^2 = K$ and there exists a centric sequence such that $A_{2012} = (x, y)$.

Let $\omega$ be the circumcircle of $\Delta ABC$, where $|AB|=|AC|$. Let $D$ be any point on the minor arc $AC$. Let $E$ be the reflection of point $B$ in line $AD$. Let $F$ be the intersection of $\omega$ and line $BE$ and Let $K$ be the intersection of line $AC$ and the tangent at $F$. If line $AB$ intersects line $FD$ at $L$, Show that $K,L,E$ are collinear points

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]