Found problems: 5802
Let $p(n)$ be the greatest odd divisor of $n$. Prove that \[\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.\]
Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying
\[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\]
Show that for any real number $\alpha$, we have
\[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\]
Here $[\beta]$ denotes the greastest integer not larger than $\beta$.
Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$. Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.
Show that $\lceil (\sqrt{3}+1)^{2n})\rceil$ is divisible by $2^{n+1}.$
Fix an integer $n\geq4$. Let $C_n$ be the collection of all $n$–point configurations in the plane, every three points of which span a triangle of area strictly greater than $1.$ For each configuration $C\in C_n$ let $f(n,C)$ be the maximal size of a subconfiguration of $C$ subject to the condition that every pair of distinct points has distance strictly greater than $2.$ Determine the minimum value $f(n)$ which $f(n,C)$ achieves as $C$ runs through $C_n.$
[i]Radu Bumbăcea and Călin Popescu[/i]
Is it possible to find a set $A$ of eleven positive integers such that no six elements of $A$ have a sum which is divisible by $6$?
Is it possible to colour all integers greater than $1{}$ in three colours (each integer in one colour, all three colours must be used) so that the colour of the product of any two differently coloured numbers is different from the colour of each of the factors?
n \geq 6 is an integer. evaluate the minimum of f(n) s.t: any graph with n vertices and f(n) edge contains two cycle which are distinct( also they have no comon vertice)?
Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that :
\[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\]
[i]Proposed by Mohammad Ahmadi[/i]
Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties:
• $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero;
• for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.
Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds:
\[ f(f(n)) \leq \frac {n+f(n)} 2 . \]
Alphonse and Beryl play a game involving $n$ safes. Each safe can be opened by a unique key and each key opens a unique safe. Beryl randomly shuffles the $n$ keys, and after placing one key inside each safe, she locks all of the safes with her master key. Alphonse then selects $m$ of the safes (where $m < n$), and Beryl uses her master key to open just the safes that Alphonse selected. Alphonse collects all of the keys inside these $m$ safes and tries to use these keys to open up the other $n - m$ safes. If he can open a safe with one of the $m$ keys, he can then use the key in that safe to try to open any of the remaining safes, repeating the process until Alphonse successfully opens all of the safes, or cannot open any more. Let $P_m(n)$ be the probability that Alphonse can eventually open all $n$ safes starting from his initial selection of $m$ keys.
(a) Show that $P_2(3) = \frac23$.
(b) Prove that $P_1(n) = \frac1n$.
(c) For all integers $n \geq 2$, prove that $$P_2(n) = \frac2n \cdot P_1(n-1) + \frac{n-2}{n} \cdot P_2(n-1).$$
(d) Determine a formula for $P_2 (n)$.
Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction.
Prove that $h$ is remarkable if and only if it is prime.
(Recall that an common fraction has an integer numerator and a natural denominator.)
There are $n$ cities in Qingqiu Country. The distance between any two cities are different. The king of the country plans to number the cities and set up two-way air lines in such ways:
The first time, set up a two-way air line between city 1 and the city nearest to it.
The second time, set up a two-way air line between city 2 and the city second nearest to it.
...
The $n-1$th time, set up a two-way air line between city $n-1$ and the city farthest to it.
Prove: The king can number the cities in a proper way so that he can go to any other city from any city by plane.
Let $1 < x_1 < 2$ and, for $n = 1$, 2, $\dots$, define $x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2$. Prove that, for $n \ge 3$, $|x_n - \sqrt{2}| < 2^{-n}$.
Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$. Prove that there exists a triangle with vertices in $M$ and with area at most equal with $\frac 1{10}$.
[i]Yugoslavia[/i]
Given a positive integer $n$, there are $n$ boxes $B_1,...,B_n$. The following procedure can be used to add balls.
$$\text{(Procedure) Chosen two positive integers }n\geq i\geq j\geq 1\text{, we add one ball each to the boxes }B_k\text{ that }i\geq k\geq j.$$
For positive integers $x_1,...,x_n$ let $f(x_1,...,x_n)$ be the minimum amount of procedures to get all boxes have its amount of balls to be a multiple of 3, starting with $x_i$ balls for $B_i(i=1,...,n)$. Find the largest possible value of $f(x_1,...,x_n)$. (If $x_1,...,x_n$ are all multiples of 3, $f(x_1,...,x_n)=0$.)
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
Let $m \leq n$ be natural numbers. Starting with the product $t=m\cdot (m+1) \cdot (m+2) \cdot \cdots \cdot n$, let $T_{m, n}$ be the sum of products that can be obtained from deleting from $t$ pairs of consecutive integers (this includes $t$ itself). In the case where all the numbers are deleted, we assume the number $1$.
For example, $T_{2, 7} = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 \cdot 4 \cdot 5 + 2 \cdot 3 \cdot 4 \cdot 7 + 2 \cdot 3 \cdot 6 \cdot 7 + 2 \cdot 5 \cdot 6 \cdot 7 + 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 + 2 \cdot 5 + 2 \cdot 7 + 4 \cdot 7 + 6 \cdot 7 + 1 = 5040 + 120 + 168 + 252 + 420 + 840 + 6 + 10 + 14 + 20 + 28 + 42 + 1 = 6961$.
Taking $T_{n+1, n} = 1$.
Show that $T_{m, n+1}=T_{m, k-1} \cdot T_{k+2, n+1} + T_{m, k} \cdot T_{k+1, n+1}$ for all $1 \leq m \leq k \leq n$.
A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another.
(For example, $23=9+8+6.)$
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
[i]Ukraine[/i]