Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that \[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]

Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, \[f(xf(y))+f(y)=f(x+y)+f(xy).\] [i]Milan Haiman[/i]

Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$. Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.

Let be three real positive numbers $ \alpha ,\beta ,\gamma $ and let $ M,N $ be points on the sides $ AB,BC, $ respectively, of a triangle $ ABC, $ such that $ \frac{MA}{MB} =\frac{\alpha }{\beta } $ and $ \frac{NB}{NC} =\frac{\beta }{\gamma } . $ Also, let $ P $ be the intersection of $ CM $ with $ AN. $ Show that: $$ \frac{1}{\alpha }\overrightarrow{PA} +\frac{1}{\beta }\overrightarrow{PB} +\frac{1}{\gamma }\overrightarrow{PC} =0 $$

Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.

We call Golomb ruler a ruler of length $l$, bearing $k+1\geq 2$ marks $0<a_1<\ldots <a_{k-1}<l$, such that the lengths that can be measured using marks on the ruler are consecutive integers starting with $1$, and each such length be measurable between just two of the gradations of the ruler. Find all Golomb rulers.

There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!) (a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$. (b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$. [i]Proposed by Ray Li[/i]

Let $A_{n,k}$ denote the set of lattice paths in the coordinate plane of upsteps $u=[1,1]$, downsteps $d=[1,-1]$, and flatsteps $f=[1,0]$ that contain $n$ steps, $k$ of which are slanted ($u$ or $d$). A sharp turn is a consecutive pair of $ud$ or $du$. Let $B_{n,k}$ denote the set of paths in $A_{n,k}$ with no upsteps among the first $k-1$ steps, and let $C_{n,k}$ denote the set of paths in $A_{n,k}$ with no sharps anywhere. For example, $fdu$ is in $B_{3,2}$ but not in $C_{3,2}$, while $ufd$ is in $C_{3,2}$ but not $B_{3,2}$. For $1 \le k \le n$, prove that the sets $B_{n,k}$ and $C_{n,k}$ contains the same number of elements.

Triangle $WSE$ has side lengths $WS=13$, $SE=15$, and $WE=14$. Points $J$ and $H$ lie on $\overline{SE}$ such that $SJ=JH=HE=5$. Let the angle bisector of $\angle{WES}$ intersect $\overline{WH}$ and $\overline{WJ}$ at points $M$ and $T$, respectively. Find the area of quadrilateral $JHMT$.

Real numbers $a,b,c,d$ such that $|a|>1$ , $|b|>1$ , $|c|>1$ , $|d|>1$ and $ab(c+d)+dc(a+b)+a+b+c+d=0$ then prove that $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}+\frac{1}{d-1} >0$

For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.

In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.

Consider the graph of the function $y = (1 -x^2)^3$. Find the set of points $M(x,y)$ through which you can draw at least $6$ lines touching this graph.

Moist air is less dense than dry air at the same temperature and barometric pressure. Which is the best explanation for this observation? $ \textbf{(A)}\hspace{.05in}\ce{H2O} \text{ is a polar molecule but } \ce{N2} \text{ and } \ce{O2} \text{ are not} \qquad$ $\textbf{(B)}\hspace{.05in} \ce{H2O} \text{has a higher boiling point than } \ce{N2} \text{or} \ce{O2}\qquad$ $\textbf{(C)}\hspace{.05in}\ce{H2O} \text{has a lower molar mass than} \ce{N2} \text{or} \ce{O2}\qquad$ $\textbf{(D)}\hspace{.05in}\ce{H2O} \text{has a higher heat capacity than} \ce{N2} \text{or} \ce{O2}\qquad$

3. Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints. Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .

Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.

A triangle $ABC$ has side lengths $AB=8$ and $BC=10.$ Given that the altitude to side $BC$ has length $4,$ what is the length of the altitude to side $AB?$

Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.

Let triangle $ABC$ with incenter $I$ satisfy $AB = 3$, $AC = 4$, and $BC = 5$. Suppose that $D$ and $E$ lie on $AB$ and $AC$, respectively, such that $D$, $I$, and $E$ are collinear and $DE \perp AI$. Points $P$ and $Q$ lie on side $BC$ such that $IP = BP$ and $IQ = CQ$, and lines $DP$ and $EQ$ meet at $S$. Compute $SI^2$.

Let $(x_n)_{n\in \mathbb{N}}$ be a sequence defined recursively such that $x_1=\sqrt{2}$ and $$x_{n+1}=x_n+\frac{1}{x_n}\text{ for }n\in \mathbb{N}.$$ Prove that the following inequality holds: $$\frac{x_1^2}{2x_1x_2-1}+\frac{x_2^2}{2x_2x_3-1}+\dotsc +\frac{x_{2018}^2}{2x_{2018}x_{2019}-1}+\frac{x_{2019}^2}{2x_{2019}x_{2020}-1}>\frac{2019^2}{x_{2019}^2+\frac{1}{x_{2019}^2}}.$$ [i]Proposed by Ivan Novak[/i]

Let $ABC$ be an equilateral triangle having sides of length 1, and let $P$ be a point in the interior of $\Delta ABC$ such that $\angle ABP = 15 ^\circ$. Find, with proof, the minimum possible value of $AP + BP + CP$. ([b]Comment:[/b] In fact this question is incorrect, unfortunately. A more reasonable problem: Prove that $AP + BP + CP \ge \sqrt{3}$.)

[b]6.[/b] Let $H_{n}(x)$ be the [i]n[/i]th Hermite polynomial. Find $ \lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$ For an arbitrary real y. [b](S.5)[/b] $H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$

Let \( A \) be a set of 2025 positive real numbers. For a subset \( T \subseteq A \), define \( M_T \) as the median of \( T \) when all elements of \( T \) are arranged in increasing order, with the convention that \( M_\emptyset = 0 \). Define \[ P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T. \] Find the smallest real number \( C \) such that for any set \( A \) of 2025 positive real numbers, the following inequality holds: \[ P(A) - Q(A) \leq C \cdot \max(A), \] where \(\max(A)\) denotes the largest element in \( A \).

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road (see diagram). [img]6763[/img] It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point $P$ which is $x$ miles ($0\le x < 12$) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses. Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time $w(x)$ that his journey (waiting plus travel time) could take. Find $w(2)$; find $w(4)$. For what value of $x$ will the time $w(x)$ be the longest? Sketch a graph of $y = w(x)$ for $0\le x < 12$.