Found problems: 85335
Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$.
[i]Proposed by Ryan Alweiss[/i]
a)Prove that for every n,natural number exist natural numbers a and b such that
$(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$
b)Using first equation prove that for every n exist m such that
$(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$
There are $40$ members of jury, that want to choose problem for contest. There are list with $30$ problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved $26$ problems, and every two members solved different sets of problems.
Prove, that they can find problem for contest.
Exhibit a $13$-digit integer $N$ that is an integer multiple of $2^{13}$ and whose digits consist of only $8$s and $9$s.
Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
\[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\]
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$
What is the value of $2-(-2)^{-2}$?
$ \textbf{(A) } -2
\qquad\textbf{(B) } \dfrac{1}{16}
\qquad\textbf{(C) } \dfrac{7}{4}
\qquad\textbf{(D) } \dfrac{9}{4}
\qquad\textbf{(E) } 6
$
Call a number [i]precious[/i] if it is the sum of two distinct powers of two. Find all precious numbers $n$ such that $n^2$ is also precious.
[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation.
$$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$
[b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.
There is a square of checkered paper measuring $102 \times 102$ squares and a connected figure of unknown shape, consisting of 101 cells. What is the largest number of such figures that can be cut from this square with a guarantee? A figure made up of cells is called [i]connected [/i] if any two its cells can be connected by a chain of its cells in which any two adjacent cells have a common side.
Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Prove that inside any acute-angled triangle, there exists a point $P$ such that the feet of the perpendiculars dropped from $P$ to the sides of the triangle are the vertices of an equilateral triangle.
(NB Vassiliev)
One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$.
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.
Consider the real function defined by $f(x) =\frac{1}{|x + 3| + |x + 1| + |x - 2| + |x -5|}$ for all $x \in R$.
a) Determine its maximum.
b) Graphic representation.
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have :
$$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$
Who has the winning strategy.
Proposed by [i]Alireza Haghi[/i]
The figure below shows two parallel lines, $ \ell$ and $ m$, that are distance $ 12$ apart:
[asy]unitsize(7);
draw((-7, 0) -- (12, 0));
draw((-7, 12) -- (12, 12));
real r = 169 / 48;
draw(circle((0, r), r));
draw(circle((5, 12 - r), r));
pair A = (0, 0);
pair B = (5, 12);
dot(A);
dot(B);
label("$A$", A, plain.S);
label("$B$", B, plain.N);
label("$\ell$", (12, 0), plain.E);
label("$m$", (12, 12), plain.E);[/asy]
A circle is tangent to line $ \ell$ at point $ A$. Another circle is tangent to line $ m$ at point $ B$. The two circles are congruent and tangent to each other as shown. The distance between $ A$ and $ B$ is $ 13$. What is the radius of each circle?
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
What is the minimum value of \[\sqrt {x^2 - 4x + 7 - 2\sqrt 2} + \sqrt {x^2 - 8x + 27 - 6\sqrt 2}\] where $x$ is a real number?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3\sqrt 2
\qquad\textbf{(C)}\ 1 + \sqrt 2
\qquad\textbf{(D)}\ 2\sqrt 2
\qquad\textbf{(E)}\ \text{None of above}
$
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $ 1$. The polygons meet at a point $ A$ in such a way that the sum of the three interior angles at $ A$ is $ 360^\circ$. Thus the three polygons form a new polygon with $ A$ as an interior point. What is the largest possible perimeter that this polygon can have?
$ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 21\qquad \textbf{(E)}\ 24$
Compute $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}.$
Let $x, y$ be positive real numbers with $x + y + xy= 3$. Prove that$$x + y\ge 2.$$ When does equality holds?
(K. Czakler, GRG 21, Vienna)