For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4. [i]Proposed by Gerhard Wöginger, Austria[/i]

Let $E$ be a family of subsets of $\{1,2,\ldots,n\}$ with the property that for each $A\subset \{1,2,\ldots,n\}$ there exist $B\in F$ such that $\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2$. (where $A \bigtriangleup B = (A\setminus B) \cup (B\setminus A)$ is the symmetric difference). Denote by $f(n,d)$ the minimum cardinality of such a family. a) Prove that if $n$ is even then $f(n,0)\leq n$. b) Prove that if $n-d$ is even then $f(n,d)\leq \lceil \frac n{d+1}\rceil$. c) Prove that if $n$ is even then $f(n,0) = n$

The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?

There are $2022$ black balls numbered $1$ to $2022$ and $2022$ white balls numbered $1$ to $2022$ as well. There are also $1011$ black boxes and white boxes each. In each box we put two balls that are the same color as the the box. Prove that no matter how the balls are distributed, we can always pick one ball from each box such that the $2022$ balls we chose have all the numbers from $1$ to $2022$.

$25$ men sit around a circular table. Every hour there is a vote, and each must respond [i]yes [/i]or [i]no[/i]. Each man behaves as follows: on the $n^{\text{th}}$, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.

We have finite white and finite black points that for each 4 oints there is a line that white points and black points are at different sides of this line.Prove there is a line that all white points and black points are at different side of this line.

Given is a unit cube in space. Find the maximal integer $n$ such that there are $n$ points, satisfying the following conditions: (a) All points lie on the surface of the cube; (b) No face contains all these points; (c) The $n$ points are the vertices of a polygon.

One hundred tennis players took part in a tournament where they played with each other exactly one game, with no draws. At the end of the tournament a table (ranking) is formed depending on the number of victories. It is known that one tennis player finished the tournament on $k$-th place and is the only one with that number of victories, and he has beaten every tennis player who is placed above him in the table and lost to anyone ranked weaker than him on the table. Find the smallest value of $k$.

In a country there are $n\geq 2$ cities. Any two cities has exactly one two-way airway. The government wants to license several airlines to take charge of these airways with such following conditions: i) Every airway can be licensed to exactly one airline. ii) By choosing one arbitrary airline, we can move from a city to any other cities, using only flights from this airline. What is the maximum number of airlines that the government can license to satisfy all of these conditions?

There are $212$ points inside or on a given unit circle. Prove that there are at least $2001$ pairs of points having distances at most $1$.

A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$. [asy] draw((30,0)--(-70,0), Arrow); draw((30,0)--(-20,-40)); draw((-20,-40)--(80,-40), Arrow); draw((0,-60)--(-40,20), dashed, Arrow); draw((0,-60)--(0,15), dashed); draw((0,15)--(40,-65),dashed, Arrow); [/asy]

A number of unit discs are given inside a square of side $100$ such that (i) no two of the discs have a common interior point, and (ii) every segment of length $10$, lying entirely within the square, meets at least one disc. Prove that there are at least $400$ discs in the square.

Prove that total number of ones which is showed in all nonrestricted partitions of natural number $n$ is equal to sum of numbers of distinct elements in that partitions.

There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]

The numbers $1,2,...,2n-1,2n$ are divided into two disjoint sets, $a_1 < a_2 < ... < a_n$ and $b_1 > b_2 > ... > b_n$. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2.$$

Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon. (proposed by J. Tabov)

Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.

There is a board with $n$ rows and $12$ columns. Each cell of the board contains a $1$ or a $0$. The board has the following properties: [list=i] [*]All rows are distinct. [*]Each row contains exactly $4$ cells with $1$. [*]For every $3$ rows, there is a column that intersects them in $3$ cells with $0$. [/list] Find the largest $n$ for which a board with these properties exists.

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that [b]i.)[/b] no three points of $ S$ are collinear, and [b]ii.)[/b] for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} \plus{} \sqrt {2 \cdot n} \]

[b]p1.[/b] Suppose $x \ne 1$ or $10$ and logarithms are computed to the base $10$. Define $y= 10^{\frac{1}{1-\log x}}$ and $z = ^{\frac{1}{1-\log y}}$ . Prove that $x= 10^{\frac{1}{1-\log z}}$ [b]p2.[/b] If $n$ is an odd number and $x_1, x_2, x_3,..., x_n$ is an arbitrary arrangement of the integers $1, 2,3,..., n$, prove that the product $$(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)$$ is an even number (possibly negative or zero). [b]p3.[/b] Prove that $\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}$ for all integers $n = 1,2,3,...$ [b]p4.[/b] Prove that if three angles of a convex polygon are each $60^o$, then the polygon must be an equilateral triangle. [b]p5.[/b] Find all solutions, real and complex, of $$4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0$$ [b]p6.[/b] A man is $\frac38$ of the way across a narrow railroad bridge when he hears a train approaching at $60$ miles per hour. No matter which way he runs he can [u]just [/u] escape being hit by the train. How fast can he run? Prove your assertion. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares. [i]Proposed by Holden Mui[/i]

[b]a)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green. [b]b)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]