Found problems: 5923
The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
[asy]unitsize(3mm);
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marker m=marker(scale(5)*d,Fill);
path f1=(0,0);
path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1);
path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^
(-2,2)--(-3,3);
draw(f1,m);
draw(shift(5,0)*f2,m);
draw(shift(5,0)*g2);
draw(shift(12,0)*f3,m);
draw(shift(12,0)*g3);
draw(shift(21,0)*f4,m);
draw(shift(21,0)*g4);
label("$F_1$",(0,-4));
label("$F_2$",(5,-4));
label("$F_3$",(12,-4));
label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$
Consider the functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f{}$ is continous. For any real numbers $a<b<c$ there exists a sequence $(x_n)_{n\geqslant 1}$ which converges to $b{}$ and for which the limit of $g(x_n)$ as $n{}$ tends to infinity exists and satisfies \[f(a)<\lim_{n\to\infty}g(x_n)<f(c).\][list=a]
[*]Give an example of a pair of such functions $f,g$ for which $g{}$ is discontinous at every point.
[*]Prove that if $g{}$ is monotonous, then $f=g.$
[/list]
The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?
Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies
\[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \]
Prove that $\lim_{n\to \infty}c_n$ exists and find its value.
[i]Proposed by Sadovnichy-Grigorian-Konyagin[/i]
$(a_n)$ is a sequence with $a_1=1$ and $|a_n| = |a_{n-1}+2|$ for every positive integer $n\geq 2$. What is the minimum possible value of $\sum_{i = 1}^{2000}a_{i}$?
$ \textbf{(A)}\ -4000
\qquad\textbf{(B)}\ -3000
\qquad\textbf{(C)}\ -2000
\qquad\textbf{(D)}\ -1000
\qquad\textbf{(E)}\ \text{None}
$
Find the number of strictly increasing sequences of nonnegative integers with the first term $0$ and the last term $15$, and among any two consecutive terms, exactly one of them is even.
Lê Anh Vinh
Let $a_0 = 1998$ and $a_{n+1} =\frac{a_n^2}{a_n +1}$ for each nonnegative integer $n$.
Prove that $[a_n] = 1994- n$ for $0 \le n \le 1000$
Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.
Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
In the number arrangement
\[\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}\]
what is the number that will appear directly below the number $2010$?
Given the series $ 2\plus{}1\plus{}\frac {1}{2}\plus{}\frac {1}{4}\plus{}...$ and the following five statements:
(1) the sum increases without limit
(2) the sum decreases without limit
(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small
(4) the difference between the sum and 4 can be made less than any positive quantity no matter how small
(5) the sum approaches a limit
Of these statments, the correct ones are:
$\textbf{(A)}\ \text{Only }3 \text{ and }4\qquad
\textbf{(B)}\ \text{Only }5 \qquad
\textbf{(C)}\ \text{Only }2\text{ and }4 \qquad
\textbf{(D)}\ \text{Only }2,3\text{ and }4 \qquad
\textbf{(E)}\ \text{Only }4\text{ and }5$
Let $a_0,a_1,\dots$ be a sequence of positive reals such that
$$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.
Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.
[u]Super Mario 64![/u]
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and fi nds himself in a room with $4$ doors. This room is the fi rst in a sequence of $2$ indistinugishable rooms. In each room, $1$ door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other $3$ doors lead to the fi rst room.
[b]p9.[/b] Suppose that in every room, Mario randomly picks a door to walk through. What is the expected number of doors (not including Mario's initial entrance to the fi rst room) through which Mario will pass before he reaches Bowser's level?
[b]p10.[/b] Suppose that instead there are $6$ rooms with $4$ doors. In each room, $1$ door leads to the next room in the sequence (or, for the last room, Bowser's level), while the other $3$ doors lead to the first room. Now what is the expected number of doors through which Mario will pass before he reaches Bowser's level?
[b]p11.[/b] In general, if there are $d$ doors in every room (but still only $1$ correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level?
Let $a_1,a_2,a_3,\ldots$ be the sequence of real numbers defined by $a_1=1$ and
$$a_m=\frac1{a_1^2+a_2^2+\ldots+a_{m-1}^2}\qquad\text{for }m\ge2.$$
Determine whether there exists a positive integer $N$ such that
$$a_1+a_2+\ldots+a_N>2017^{2017}.$$
We define the following sequences:
• Sequence $A$ has $a_n = n$.
• Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise.
• Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$
.• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise.
• Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$
Prove that the terms of sequence E are exactly the perfect cubes.
Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
Original in Hungarian; translated with Google translate; polished by myself.
Prove that, the signs $\varepsilon_n = \pm 1$ can be chosen such that the function $f(s) = \sum_{n = 1}^\infty\frac{\varepsilon_n}{n^s}\colon \{s\in\Bbb C:\operatorname{Re}s > 1\}\to \Bbb C$ converges to every complex value at every point $\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}$ (i.e. for every $\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}$ and every $z \in \Bbb C$, there exists a sequence $s_n \to \xi$, $\operatorname{Re}s_n > 1$, for which $f(s_n) \to z$).
Two sequences $(a_n)$ and $(b_n)$ satisfy that $a_0=1,a_1=4,a_2=49$, and
$\begin{cases}
a_{n+1}=7a_n+6b_n-3\\
b_{n+1}=8a_n+7b_n-4\\
\end{cases}$ for $n=0,1,2,\cdots,$.
Prove that $a_n$ is a perfect square for $n=0,1,2,\cdots,$.
Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows:
$C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.
Let $P$ be a convex $1986$-gon in the plane. Let $A,D$ be interior points of two distinct sides of P and let $B,C$ be two distinct interior points of the line segment $AD$. Starting with an arbitrary point $Q_1$ on the boundary of $P$, define recursively a sequence of points $Q_n$ as follows:
given $Q_n$ extend the directed line segment $Q_nB$ to meet the boundary of $P$ in a point $R_n$ and then extend $R_nC$ to meet the boundary of $P$ again in a point, which is defined to be $Q_{n+1}$. Prove that for all $n$ large enough the points $Q_n$ are on one of the sides of $P$ containing $A$ or $D$.
Let $d(n)$ denote the number of positive divisors of $n$. For any given integer $a \geq 3$, define a sequence $\{a_i\}_{i=0}^\infty$ satisfying
[list]
[*] $a_{0}=a$, and
[*] $a_{n+1}=a_{n}+(-1)^{n} d(a_{n})$ for each integer $n \geq 0$.
[/list]
For example, if $a=275$, the sequence would be \[275, \overline{281,279,285,277,279,273}.\]
Prove that for each positive integer $k$ there exists a positive integer $N$ such that if such a sequence has period $2k$ and all terms of the sequence are greater than $N$ then all terms of the sequence have the same parity.
[i]Proposed by Navid[/i]
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$