An [i]iterative average[/i] of the numbers $1$, $2$, $3$, $4$, and $5$ is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure? $ \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} $

At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 22 \times 23$ in order to make it a perfect square?

1. Can a $7 \times 7~$ square be tiled with the two types of tiles shown in the figure? (Tiles can be rotated and reflected but cannot overlap or be broken) 2. Find the least number $N$ of tiles of type $A$ that must be used in the tiling of a $1011 \times 1011$ square. Give an example of a tiling that contains exactly $N$ tiles of type $A$. [asy] size(4cm, 0); pair a = (-10,0), b = (0, 0), c = (10, 0), d = (20, 0), e = (20, 10), f = (10, 10), g = (0, 10), h = (0, 20), ii = (-10, 20), j = (-10, 10); draw(a--b--c--f--g--h--ii--cycle); draw(g--b); draw(j--g); draw(f--c); draw((30, 0)--(30, 20)--(50,20)--(50,0)--cycle); draw((40,20)--(40,0)); draw((30,10)--(50,10)); label((0,0), "$(A)$", S); label((40,0), "$(B)$", S); [/asy] [i]Proposed by Muralidharan Somasundaran[/i]

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

Given an integer $n>2$ and an integer $a$, if there exists an integer $d$ such that $n\mid a^d-1$ and $n\nmid a^{d-1}+\cdots+1$, we say [i]$a$ is $n-$separating[/i]. Given any n>2, let the [i]defect of $n$[/i] be defined as the number of integers $a$ such that $0<a<n$, $(a,n)=1$, and $a$ is not [i] $n-$separating[/i]. Determine all integers $n>2$ whose defect is equal to the smallest possible value.

[asy]draw((0,0)--(7,0)--(7,1)--(6,1)--(6,2)--(5,2)--(5,3)--(4,3)--(4,4)--(3,4)--(3,3)--(2,3)--(2,2)--(1,2)--(1,1)--(0,1)--(0,0)--cycle); draw((1,0)--(6,0)--(6,1)--(5,1)--(5,2)--(4,2)--(4,3)--(3,3)--(3,2)--(2,2)--(2,1)--(1,1)--(1,0)--cycle); draw((2,0)--(5,0)--(5,1)--(4,1)--(4,2)--(3,2)--(3,1)--(2,1)--cycle); draw((3,0)--(3,1)--(4,1)--(4,0)--cycle); fill((1,0)--(1,1)--(2,1)--(2,0)--cycle,black); fill((3,0)--(3,1)--(4,1)--(4,0)--cycle,black); fill((5,0)--(5,1)--(6,1)--(6,0)--cycle,black); fill((2,1)--(2,2)--(3,2)--(3,1)--cycle,black); fill((4,1)--(4,2)--(5,2)--(5,1)--cycle,black); fill((3,2)--(3,3)--(4,3)--(4,2)--cycle,black);[/asy] A "stair-step" figure is made up of alternating black and white squares in each row. Rows $ 1$ through $ 4$ are shown. All rows begin and end with a white square. The number of black squares in the $ 37$th row is \[ \textbf{(A)}\ 34 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 38 \]

In a football tournament $2n$ teams play in a round. Every round consists of $n$ pairs of teams that haven’t played with each other yet. Every round’s schedule is determined before the round is held. Find the minimal positive integer $k$ such that the following situation is possible: after $k$ rounds it’s impossible to schedule the next round.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions: $f(x) + f(y) \ge xy$ for all real $x, y$ and for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.

Three balloon vendors each offer two types of balloons - one offers red & blue, one offers blue & yellow, and one offers yellow & red. I like each vendor the same, so I must buy $7$ balloons from each. How many different possible triples $(x,y,z)$ are there such that I could buy $x$ blue, $y$ yellow, and $z$ red balloons?

In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.

What is the sum of the positive solutions to $2x^2 -\lfloor x \rfloor = 5$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$?

For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

$6$ musicians gathered at a chamber music festival. At each scheduled concert, some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that every $2$ musicians each must play for the other in some concert?

A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.

Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

Suppose $ a,c,d \in N$ and $ d|a^2b\plus{}c$ and $ d\geq a\plus{}c$ Prove that $ d\geq a\plus{}\sqrt[2b] {a}$

Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.) \[ x_i^2 + x_i - 1 = x_{i+1} \]

There are numerous sets of $17$ distinct positive integers that sum to $2018$, such that each integer has the same sum of digits in base $10$. Let $M$ be the maximum possible integer that could exist in any such set. Find the sum of $M$ and the number of such sets that contain $M$.

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?

Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.

If the ratio of the legs of a right triangle is $ 1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is: $ \textbf{(A)}\ 1: 4 \qquad \textbf{(B)}\ 1: \sqrt{2} \qquad \textbf{(C)}\ 1: 2 \qquad \textbf{(D)}\ 1: \sqrt{5} \qquad \textbf{(E)}\ 1: 5$

a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots. b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.

Jackie and Phil have two fair coins and a third coin that comes up heads with probability $ \frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $ \frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.