Found problems: 85335
For the positive integers $a, b, x_1$ we construct the sequence of numbers $(x_n)_{n=1}^{\infty}$ such that $x_n = ax_{n-1} + b$ for each $n \ge 2$. Specify the conditions for the given numbers $a, b$ and $x_1$ which are necessary and sufficient for all indexes $m, n$ to apply the implication $m | n \Rightarrow x_m | x_n$.
(Jaromír Šimša)
Let $p$ be an odd prime. Prove that for every integer $k$, there exist integers $a, b$ such that $p|a^2 + b^2 - k$.
a) Factorize $xy - x - y + 1$.
b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.
Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]
Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.
For an isosceles triangle $ABC$ where $AB=AC$, it is possible to construct, using only compass and straightedge, an isosceles triangle $PQR$ where $PQ=PR$ such that triangle $PQR$ is similar to triangle $ABC$, point $P$ is in the interior of line segment $AC$, point $Q$ is in the interior of line segment $AB$, and point $R$ is in the interior of line segment $BC$. Describe one method of performing such a construction. Your method should work on every isosceles triangle $ABC$, except that you may choose an upper limit or lower limit on the size of angle $BAC$.
[asy]
defaultpen(linewidth(0.7));
pair a= (79,164),b=(19,22),c=(138,22),p=(109,91),q=(38,67),r=(78,22);
pair point = ((p.x+q.x+r.x)/3,(p.y+q.y+r.y)/3);
draw(a--b--c--cycle);
draw(p--q--r--cycle);
label("$A$",a,dir(point--a));
label("$B$",b,dir(point--b));
label("$C$",c,dir(point--c));
label("$P$",p,dir(point--p));
label("$Q$",q,dir(point--q));
label("$R$",r,dir(point--r));[/asy]
We want to construct an isosceles triangle using a compass and a straightedge. We are given two of the following four data: the length of the base of the triangle $(a),$ the length of the leg of the triangle $(b),$ the radius of the inscribed circle $(r),$ and the radius of the circumscribed circle $(R).$ In which of the six possible cases will we definitely be able to construct the triangle?
[i]Proposed by György Rubóczky, Budapest[/i]
The fraction $\frac{3}{10}$ can be written as the sum of two positive fractions with numerator $1$ as follows: $\frac{3}{10} =\frac{1}{5}+\frac{1}{10}$ and also $\frac{3}{10}=\frac{1}{4}+\frac{1}{20}$. There are the only two ways in which this can be done. In how many ways can $\frac{3}{1984}$ be written as the sum of two positive fractions with numerator $1$?
Is there a positive integer $n,$ not divisible by $3$, such that $\frac{3}{n}$ can be written as the sum of two positive fractions with numerator $1$ in exactly $1984$ ways?
Let $a,b,c > 0$ be real numbers with the sum equal to $3$. Show that:
$$\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab} \ge 3$$
Let a be a non-zero integer, and $n \ge 3$ another integer. Prove that the following polynomial is irreducible in the ring of integer polynomials (i.e. it cannot be written as a product of two non-constant integer polynomials):
$$f(x) = x^n + ax^{n-1} + ax^{n-2} +... + ax -1$$
Let $p>5$ be a prime number and $A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}$ be the set of all quadratic residues modulo $p$, excluding zero. Prove that there doesn't exist any natural $a,c$ satisfying $(ac,p)=1$ such that set $B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}$ and set $A$ are disjoint modulo $p$.
[i]This problem was proposed by Amir Hossein Pooya.[/i]
Let $n\ge 1$ be a positive integer and $x_1,x_2\ldots ,x_n$ be real numbers such that $|x_{k+1}-x_k|\le 1$ for $k=1,2,\ldots ,n-1$. Prove that
\[\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}\]
[i]Gh. Eckstein[/i]
Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.
Investigate the existence and uniqueness of the solution of the problem $yu_x = xu_y, u|_{x=1} =\cos y$ in a neighbourhood of the point $(1, y_0)$.
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\dfrac{a(a+1)}2+\dfrac{b(b+1)}2=\dfrac{c(c+1)}2.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ $\textit{legs}$ of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of $\textit{at least}$ six distinct PTTNs.
What is the value of the product $$\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?$$
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) } 50$
(Game) At the beginning of the game, the organisers place paper disks on the table, grouped into piles which may contain various numbers of disks. The two players take turns. On a player’s turn, their opponent selects two piles (one if there is only one pile left), and the player must remove some number of disks from one of the piles selected. This means that at least one disk has to be removed, and removing all disks in the pile is also permitted. The player removing the last disk from the table wins.
[i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]
Prove the following:
a) If $ABCA'B'C'$ is a right prism and $M \in (BC), N \in (CA), P \in (AB)$ such that $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent, then the prism $ABCA'B'C'$ is regular.
b) If $ABCA'B'C'$ is a regular prism and $\frac{AA'}{AB}=\frac{\sqrt6}{4}$ , then there are $M \in (BC), N \in (CA), P \in (AB)$ so that the lines $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent.
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.
When planning a trip from Temuco to the extreme north of the country, a truck driver notices that to cross the Atacama desert you must cross a distance of $800$ km between two stations consecutive service. Your truck can only store $50$ liters of benzene, and has a yield of $10$ km per liter. The trucker can leave gasoline stored in cans on the side of the road in different points along the way. For example, with an initial total charge of $50$ liters you can travel $100$ km, leave $30$ liters stored at the point you reached, and return to the starting point (with zero load) to refuel. The trucker decides to start the trip and arrives at the first service station with a zero load of fuel.
a) Can the trucker cross the desert if at this service station the total supply is $140$ liters?
b) Can the trucker cross the desert if the total supply of gasoline at the station is $180$ liters?
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]
Prove that after deleting the perfect squares from the list of positive integers the number we find in the $n^{th}$ position is equal to $n+\{\sqrt{n}\},$ where $\{\sqrt{n}\}$ denotes the integer closest to $\sqrt{n}.$
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.