Found problems: 1340
Let f$(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x) $ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. (Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2\rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.)
Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$?
Remark: Two cells are called connected if they have a common edge.
If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$.
Show that the sequence
$\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$
contains infinitely many odd numbers.
Let $x>1$ be a real number that is not an integer. Denote $\{x\}$ as its decimal part and $\lfloor x\rfloor$ the floor function. Prove that
$$ \left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}$$
Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?
Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows:
[LIST]
[*] $a_0=k$ [/*]
[*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*]
[/LIST]
Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence.
[i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.
Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that
\[0<|xy-zw|<C\alpha ^{-4}\]
where $\alpha =\frac{|X|}{n}$.
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.
Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)
A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?
Show that every integer greater than $1$ can be written as a sum of two square-free integers.
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$
The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
[list=a]
[*]Determine all real numbers $x{}$ satisfying $\lfloor x\rfloor^2-x=-0.99$.
[*]Prove that if $a\leqslant -1$, the equation $\lfloor x\rfloor^2-x=a$ does not have real solutions.
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On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.
a) Can $n$ be $6$ ?
b) Can $n$ be $7$ ?
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
Let $n$ and $k$ be positive integers, where $n > 1$ is odd. Suppose $n$ voters are to elect one of the $k$ cadidates from a set $A$ according to the rule of "majoritarian compromise" described below. After each voter ranks the candidates in a column according to his/her preferences, these columns are concatenated to form a $k$ x $n$ voting matrix. We denote the number of ccurences of $a \in A$ in the $i$-th row of the voting matrix by $a_{i}$ . Let $l_{a}$ stand for the minimum integer $l$ for which $\sum^{l}_{i=1}{a_{i}}> \frac{n}{2}$.
Setting $l'= min \{l_{a} | a \in A\}$, we will regard the voting matrices which make the set $\{a \in A | l_{a} = l' \}$ as admissible. For each such matrix, the single candidate in this set will get elected according to majoritarian compromise. Moreover, if $w_{1} \geq w_{2} \geq ... \geq w_{k} \geq 0$ are given, for each admissible voting matrix, $\sum^{k}_{i=1}{w_{i}a_{i}}$ is called the total weighted score of $a \in A$. We will say that the system $(w_{1},w_{2}, . . . , w_{k})$ of weights represents majoritarian compromise if the total score of the elected candidate is maximum among the scores of all candidates.
(a) Determine whether there is a system of weights representing majoritarian compromise if $k = 3$.
(b) Show that such a system of weights does not exist for $k > 3$.
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 $f(n,0) = n$
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as
\[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \]
where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.
Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.
There are $n$ coins in a row, $n\geq 2$. If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$. No move is allowed if all $n$ coins are tails.
Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $ \lfloor\frac {2^m}{3}\rfloor $ moves