Found problems: 85335
Let $n\geqslant 3$ be an integer. Ion draws a regular $n$-gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the $n$-gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.
A rectangular floor measures $ a$ by $ b$ feet, where $ a$ and $ b$ are positive integers with $ b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $ 1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $ (a,b)$?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
Can it happen that $6$ parallelepipeds, no two of which have common points, are placed in space so that there is a point outside of them from which no vertex of a parallelepiped is visible? (The parallelepipeds are not transparent.)
(V Proizvolov)
What is the smallest prime number dividing the sum $ 3^{11} \plus{} 5^{13}$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 3^{11} \plus{} 5^{13}\qquad\textbf{(E)}\ \text{none of these}$
What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?
A circle centered at $O{}$ passes through the vertices $B{}$ and $C{}$ of the acute-angles triangle $ABC$ and intersects the sides $AC{}$ and $AB{}$ at $D{}$ and $E{}$ respectively. The segments $CE$ and $BD$ intersect at $U{}.$ The ray $OU$ intersects the circumcircle of $ABC$ at $P{}.$ Prove that the incenters of the triangles $PEC$ and $PBD$ coincide.
Find the smallest positive integer that ends in $56$, is a multiple of $56$, and has the sum of its digits equal to $56$.
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
Let $p$ and $q$ be mutually prime natural numbers greater than $1$. Starting with the permutation $(1, 2, . . . , n)$, in one move we can switch the places of two numbers if their difference is $p$ or $q$. Prove that with such moves we can get any another permutation if and only if $n \ge p + q - 1$.
2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
How many natural numbers $n$ satisfy the following conditions:
i) $219<=n<=2019$,
ii) there exist integers $x, y$, so that $1<=x<n<y$, and $y$ is divisible by all natural numbers from $1$ to $n$ with the exception of the numbers $x$ and $x + 1$ with which $y$ is not divisible by.
Two circles $\alpha{}$ and $\beta{}$ with centers $A{}$ and $B{}$ respectively intersect at points $C{}$ and $D{}$. The segment $AB{}$ intersects $\alpha{}$ and $\beta{}$ at points $K{}$ and $L{}$ respectively. The ray $DK$ intersects the circle $\beta{}$ for the second time at the point $N{}$, and the ray $DL$ intersects the circle $\alpha{}$ for the second time at the point $M{}$. Prove that the intersection point of the diagonals of the quadrangle $KLMN$ coincides with the incenter of the triangle $ABC$.
[i]Konstantin Knop[/i]
Prove that for all positive real numbers $a,b,c$ the following inequality holds:
\[(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7\]
and determine all cases of equality.
[i]Lucian Petrescu[/i]
$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.
Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.
Let P be a point inside triangle ABC such that 3<ABP = 3<ACP = <ABC + <ACB. Prove that AB/(AC + PB) = AC/(AB + PC).
The sequence of positive integers $a_1, a_2, \ldots, a_{2025}$ is defined as follows: $a_1=2^{2024}+1$ and $a_{n+1}$ is the greatest prime factor of $a_n^2-1$ for $1 \leq n \leq 2024$. Find the value of $a_{2024}+a_{2025}$.
Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
[asy]
fill((3,2.5)--(3,4.5)--(5.3,4.5)--(5.3,2.5)--cycle,mediumgray);
draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0));
draw((3,0)--(3,4.5));
draw((0,4.5)--(5.3,4.5));
draw((5.3,7)--(5.3,2.5));
draw((7,2.5)--(3,2.5));
[/asy]
$\textbf{(A) }A\qquad\textbf{(B) }B \qquad\textbf{(C) }C \qquad\textbf{(D) }D\qquad\textbf{(E) }E$
$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$
Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$.
a) Prove that for each $ c\in C$, $ A_c$ is infinite.
b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$.
c) Prove that each element of $ A_c$ is odd or even.
Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$.
d) $ |c| > 2$.
e) $ c\in \mathbb Q\backslash\mathbb Z$.
f) $ c$ is a non-algebraic number
g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.
Let $\sum_{n=0}^{\infty} \frac{x^n (x-1)^{2n}}{n!}=\sum_{n=0}^{\infty} a_{n}x^{n}$. Show that no three consecutive $a_n$ can be equal to $0$.
Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$.
The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that
\[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]
Let $x,y$ be real numbers. Define a sequence $\{a_n \}$ through the recursive formula
\[ a_0=x,a_1=y,a_{n+1}=\frac{a_na_{n-1}+1}{a_n+a_{n-1}},\]
Find $a_n$.