Found problems: 357
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.
There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h \plus{} 1$, $ h \plus{} 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]
Qing initially writes the ordered pair $(1,0)$ on a blackboard. Each minute, if the pair $(a,b)$ is on the board, she erases it and replaces it with one of the pairs $(2a-b,a)$, $(2a+b+2,a)$ or $(a+2b+2,b)$. Eventually, the board reads $(2014,k)$ for some nonnegative integer $k$. How many possible values of $k$ are there?
[i]Proposed by Evan Chen[/i]
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
For any positive integer $n$ consider all representations $n = a_1 + \cdots+ a_k$, where $a_1 > a_2 > \cdots > a_k > 0$ are integers such that for all $i \in \{1, 2, \cdots , k - 1\}$, the number $a_i$ is divisible by $a_{i+1}$. Find the longest such representation of the number $1992.$
Let $S$ be a set of $n$ points in the coordinate plane. Say that a pair of points is [i]aligned[/i] if the two points have the same $x$-coordinate or $y$-coordinate. Prove that $S$ can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most $n^{3/2}$ unordered pairs of distinct points in $S$ are aligned but not in the same subset.
The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$.
(1) Find the smallest integer $k$, such that if the minimum number in $A,B\in S$ respectively are both $\geq k$, then $A\rightarrow B$ implies $B\rightarrow A$.
(2) For the $k$ obtained in (1), how many tuples can we pick from $S$, such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$.
What is the units digit of $ 13^{2003}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.
Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$.
Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$.
Time allowed for this problem was 1 hour.
In a country, there are some cities and the city named [i]Ben Song[/i] is capital. Each cities are connected with others by some two-way roads. One day, the King want to choose $n$ cities to add up with [i]Ben Song[/i] city to establish an [i]expanded capital[/i] such that the two following condition are satisfied:
(i) With every two cities in [i]expanded capital[/i], we can always find a road connecting them and this road just belongs to the cities of [i]expanded capital[/i].
(ii) There are exactly $k$ cities which do not belong to [i]expanded capital[/i] have the direct road to at least one city of [i]expanded capital[/i].
Prove that there are at most $\binom{n+k}{k}$ options to expand the capital for the King.
Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$.
For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows.
(i) $a_1=<a>$
(ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$,
if $a_n=0$, then $a_{n+1}=0$.
(1) For $a=\sqrt{2}$, find $a_n$.
(2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$.
(3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$.
[i]2011 Tokyo University entrance exam/Science, Problem 2[/i]
Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.
Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.
Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$.
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.
[i]V. Dolnikov[/i]
[b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].
On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.
Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to the point $ C$ so that $ \frac {BC}{AB} \equal{} \lambda$.
Determine all values of $ \lambda$ such that, for any point $ M$ on the line and for any initial position of the $ n$ fleas, there exists a sequence of moves that will take them all to the position right of $ M$.
Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).
Prove that for any integer $n$, there exists a unique polynomial $Q$ with coefficients in $\{0,1,\ldots,9\}$ such that $Q(-2) = Q(-5) = n$.
The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.
Prove that there exists a triangle which can be cut into 2005 congruent triangles.
A magician should determine the area of a hidden convex $ 2008$-gon $ A_{1}A_{2}\cdots A_{2008}$. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in $ 2006$ steps.