Found problems: 247
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.
Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.
Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.
A function $f(S)$ assigns to each nine-element subset of $S$ of the set $\{1,2,\ldots, 20\}$ a whole number from $1$ to $20$. Prove that regardless of how the function $f$ is chosen, there will be a ten-element subset $T\subset\{1,2,\ldots, 20\}$ such that $f(T - \{k\})\neq k$ for all $k\in T$.
Let $A$ be the set $A = \{ 1,2, \ldots, n\}$. Determine the maximum number of elements of a subset $B\subset A$ such that for all elements $x,y$ from $B$, $x+y$ cannot be divisible by $x-y$.
[i]Mircea Lascu, Dorel Mihet[/i]
Consider a $3\times7$ grid of squares. Each square may be coloured green or white.
[list]
(a) Is it possible to find a colouring so that no subrectangle has all four corner squares of the same colour?
(b) Is it possible for a $4\times 6$ grid?
[/list]
[i]Subrectangles must have their corners at grid-points of the original diagram. The corner squares of a subrectangle must be different. The original diagram is a subrectangle of itself.[/i]
A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.
Consider the numbers arranged in the following way:
\[\begin{array}{ccccccc} 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 2 & 5 & 9 & 14 & 20 & \cdots & \cdots \\ 4 & 8 & 13 & 19 & \cdots & \cdots & \cdots \\ 7 & 12 & 18 & \cdots & \cdots & \cdots & \cdots \\ 11 & 17 & \cdots & \cdots & \cdots & \cdots & \cdots \\ 16 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\]
Find the row number and the column number in which the the number $20096$ occurs.
Let $a_0$ be a positive integer. Define the sequence $\{a_n\}_{n \geq 0}$ as follows: if \[ a_n = \sum_{i = 0}^jc_i10^i \] where $c_i \in \{0,1,2,\cdots,9\}$, then \[ a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. \] Is it possible to choose $a_0$ such that all terms in the sequence are distinct?
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.
Let $A$ be a set of $N$ residues $\pmod{N^2}$. Prove that there exists a set $B$ of $N$ residues $\pmod{N^2}$ such that the set $A+B=\{a+b \vert a \in A, b \in B \}$ contains at least half of all the residues $\pmod{N^2}$.
Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers
$a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.
Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$:
[b]a)[/b] $f(a)=0 \Leftrightarrow a=0$
[b]b)[/b] $f(ab)=f(a)f(b)$
[b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$.
Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$.
[i]Proposed by Masoud Shafaei[/i]
How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation
\[
e^z = \frac{z - 1}{z + 1} \, ?
\]
Let $R$ be the set of points $(x, y)$ such that $x$ and $y$ are positive, $x + y$ is at most 2013, and
\[
\lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set $R$. Recall that $\lfloor a \rfloor$ is the greatest integer that is less than or equal to $a$, and $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions:
$f(x+y) < f(x) + f(y)$
$f(f(x)) = \lfloor {x} \rfloor + 2$
Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.
In the beginning, there is a pair of positive integers $(m,n)$ written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs $(m,n)$ from the pairs $(7,79)$, $(17,71)$, $(10,101)$, $(21,251)$, $(50,405)$, can Alice guarantee to win when she makes the first move?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of above}
$
The numbers from 1 to $ 2013^2 $ are written row by row into a table consisting of $ 2013 \times 2013 $ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $ 1, 4, 9, \cdots, 2013^2 $ are simultaneously deleted.
How many cells remain?
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$.
[i]Proposed by Lewis Chen[/i]
Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)