Found problems: 5802
Prove: There exists a positive integer $n$ with $2021$ prime divisors, satisfying $n|2^n+1$.
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions:
i. $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$;
ii. $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$;
iii. There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k,\, x_j=x_l$;
a. Compute $|S_3(5)|$
b. Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$.
Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct).
[i]Evan Chen[/i]
Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_{5})$ of $(1, 2, 3, 4, 5)$. A [i]swap[/i] is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, and this allows Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$, and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]Out of all $\tbinom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?
[i]Proposed by Yang Liu[/i]
Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.
Given 98 points in a circle. Mary and Joseph play alternatively in the next way:
- Each one draw a segment joining two points that have not been joined before.
The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game.
Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$.
[i]Proposed by Amirhossein Gorzi[/i]
Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.
Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders:
(a) Each day, at least one task must be issued to a soldier.
(b) No soldier may have more than two task or receive more than one tasks in a single day.
(c) The lists of soldiers receiving tasks for two different days must not be the same.
(d) The first sergeant violating any of these orders will be jailed.
Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed?
[i]M. Kulikov[/i]
A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.
A sequence $a_1,a_2,\dots$ satisfy
$$
\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},
$$
for every $n\in\mathbb{N}$.
Let $c$ be a positive integer. Prove that, for every positive integer $n$,
$$
\frac{c^{a_n}-c^{a_{n-1}}}{n}
$$
is an integer.
Let $n$ be a positive integer. Let $d$ be an integer such that $d \ge n$ and $d$ is a divisor of $\frac{n(n + 1)}{2}$. Prove that the set $\{ 1, 2, \dots, n \}$ can be partitioned into disjoint subsets such that the sum of the numbers in each subset equals $d$.
Let $x_0, x_1, \dots , x_{n_0-1}$ be integers, and let $d_1, d_2, \dots, d_k$ be positive integers with $n_0 = d_1 > d_2 > \cdots > d_k$ and $\gcd (d_1, d_2, \dots , d_k) = 1$. For every integer $n \ge n_0$, define
\[
x_n = \left\lfloor{\frac{x_{n-d_1} + x_{n-d_2} + \cdots + x_{n-d_k}}{k}}\right\rfloor.
\]
Show that the sequence $\{x_n\}$ is eventually constant.
Let $\{a_n \}_{n=0}^{\infty}$ be a sequence given recrusively such that $a_0=1$ and $$a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}$$ for $n\geq 0$
Show that :
a) $a_n$ is a positive integer.
b) $a_n a_{n+1}-1$ is a square of an integer.
[i]Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).[/i]
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
Consider the sequence $u_0, u_1, u_2, ...$ defined by $u_0 = 0, u_1 = 1,$ and $u_n = 6u_{n - 1} + 7u_{n - 2}$ for $n \ge 2$. Show that there are no non-negative integers $a, b, c, n$ such that
$$ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.$$
Determine all integers $n\ge 2$ such that for any two infinite sequences of positive integers $a_1<a_2< \cdots $ and $b_1, b_2, \cdots$, such that $a_i\mid a_j$ for all $i<j$, there always exists a real number $c$ such that $$\lfloor{ca_i}\rfloor \equiv b_i \pmod {n}$$ for all $i\ge 1$.
[i]Proposed by Wong Jer Ren & Ivan Chan Kai Chin[/i]
[color=darkred]On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A [i]move[/i] consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.[/color]
a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: \[ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots\plus{}\sqrt[n_{k}]{\epsilon_{k}}}}\]is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$ to be this limit.
b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x\equal{}\sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$
A [i]triangulation[/i] of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a [i]Thaiangulation[/i] if all triangles in it have the same area.
Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)
[i]Proposed by Bulgaria[/i]