Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$? Recall that Pascal’s triangle is the pattern of numbers that begins as follows [img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img] where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$. Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.

Prove that there exist infinitely many distinct positive integers, $x$ and $y$, such that $$x^3+y^2|x^2+y^3$$

Let \( I \) and \( M \) be the incenter and the centroid of a scalene triangle \( ABC \), respectively. A line passing through point \( I \) parallel to \( BC \) intersects \( AC \) and \( AB \) at points \( E \) and \( F \), respectively. Reconstruct triangle \( ABC \) given only the marked points \( E, F, I, \) and \( M \). [i]Proposed by Hryhorii Filippovskyi[/i]

Let $\sigma_n$ be a permutation of $\{1,\ldots,n\}$; that is, $\sigma_n(i)$ is a bijective function from $\{1,\ldots,n\}$ to itself. Define $f(\sigma)$ to be the number of times we need to apply $\sigma$ to the identity in order to get the identity back. For example, $f$ of the identity is just $1$, and all other permutations have $f(\sigma)>1$. What is the smallest $n$ such that there exists a $\sigma_n$ with $f(\sigma_n)=k$?

Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$. (i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$. (ii) Show that $ BM\equal{}BC$.

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

If we have a number $x$ at a certain step, then at the next step we have $x+1$ or $-\frac 1x$. If we start with the number $1$, which of the following cannot be got after a finite number of steps? $ \textbf{(A)}\ -2 \qquad\textbf{(B)}\ \dfrac 12 \qquad\textbf{(C)}\ \dfrac 53 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of above} $

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks. [i]Proposed by James Lin[/i]

Let $n \ge 2$ be a positive integer. For every number from $1$ to $n$, there is a card with this number and which is either black or white. A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card. For instance, when $n=5$ and the initial configuration is $(1B, 2B, 3W, 4B,5B)$, the magician can choose $1B, 3W$ on the first move to obtain $(3W, 2B, 3W, 4B, 5B)$ and then $3W, 4B$ on the second move to obtain $(4B, 2B, 3W, 4B, 5B)$. Determine in terms of $n$ all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.

Each morning of her five-day workweek, Jane bought either a $ 50$-cent muffin or a $ 75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

For complex numbers $u=a+bi$ and $v=c+di$, define the binary operation $\otimes$ by \[u\otimes v=ac+bdi.\] Suppose $z$ is a complex number such that $z\otimes z=z^{2}+40$. What is $|z|$? $\textbf{(A)}~\sqrt{10}\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~2\sqrt{6}\qquad\textbf{(D)}~6\qquad\textbf{(E)}~5\sqrt{2}$

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?

Given 2015 balls. Astri and Budi will play a game. At first, Astri will choose two different numbers $a$ and $b$ from the set $S = \{ 1, 2, 3, \dots, 30 \}$. Budi will then choose another 2 different numbers $c$ and $d$ from the remaining 28 numbers in set $S$. By taking turns, starting from Astri, they take balls with the following rules: (1) Astri could only take $a$ or $b$ balls. (2) Budi could only take $c$ or $d$ balls. until someone couldn't take any balls satisfying the condition given (and that person will lose). Prove that Budi could choose $c,d$ such that he has a strategy to ensure his victory on this game.

$20$ football teams participate in the championship. What minimal number of the games should be played to provide the property: [i] from the three arbitrary teams we can find at least on pair that have already met in the championship.[/i]

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f(0)=0,f(1)=2013$ and \[(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))\] Note: $f^2(x)=(f(x))^2$

Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.

Let $p,q,r$ be distinct prime numbers and let \[A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} \] Find the least $n\in\mathbb{N}$ such that for any $B\subset A$ where $|B|=n$, has elements $x$ and $y$ such that $x$ divides $y$. [i]Ioan Tomescu[/i]

Emilia and Julieta have a pile of 2024 cards and play the following game: they take turns, and each player removes a number of cards that must be a power of two, i.e., \(1, 2, 4, 8, \dots\). The player who removes the last card wins. Julieta starts the game. Prove that there exists a strategy for Julieta that guarantees her victory, no matter how Emilia plays.

Let $k$ be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white. A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly $k$ times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs. Let $m$ be the minimal $k$ value such that Vera can guarantee that she wins no matter what Marco does. For $k=m$, let $t$ be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most $t$ turns. Compute $100m + t$. [i]Proposed by Ashwin Sah[/i]