Found problems: 85335
Suppose a doubly infinite sequence of real numbers $. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .$ has the property that $$a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},$$ for all integers $n .$ Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \leq R$ for all $n$), then $a_n$ has the same value for all $n.$
Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds
\[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]
Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations
\[ x \equal{} y^3 \plus{} y \minus{} 8\]
\[y \equal{} z^3 \plus{} z \minus{} 8\]
\[ z \equal{} x^3 \plus{} x \minus{} 8.\]
Given is a triangle $ABC$ with the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on segment $AC$ such that $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively such that $KL \parallel BC$ and $KL$ is tangent to the inscribed circle of $\vartriangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
Alf, the alien from the $1980$s TV show, has a big appetite for the mineral apatite. However, he’s currently on a diet, so for each integer $k \ge 1,$ he can eat exactly $k$ pieces of apatite on day $k.$ Additionally, if he eats apatite on day $k,$ he cannot eat on any of days $k + 1, k + 2, \ldots, 2k - 1.$ Compute the maximum total number of pieces of apatite Alf could eat over days $1,2, \ldots,99,100.$
Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]
[/list]
The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect its sides at points $A_1$ and $C_1$, and the circumcircle of this triangle is at points $A_0$ and $C_0$, respectively. Lines $A_1C_1$ and $A_0C_0$ intersect at point P. Prove that the segment connecting $P$ to the center of the incircle of triangle $ABC$ is parallel to $AC$.
In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
$\textbf{(A) }328\qquad
\textbf{(B) }329\qquad
\textbf{(C) }330\qquad
\textbf{(D) }331\qquad
\textbf{(E) }332\qquad$
There are $100$ dwarfes with weight $1,2,...,100$. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
Given a nonconstant function $f : \mathbb{R}^+ \longrightarrow\mathbb{R}$ such that $f(xy) = f(x)f(y)$ for any $x, y > 0$, find functions $c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}$ that satisfy $c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)$ for all $x, y > 0$ and $c(x)+s(x) = f(x)$ for all $x > 0$.
Two circles touch internally at point $A$. A chord $ BC $ of the larger circle is drawn tangent to the smaller one at point $ D $. Prove that $ AD $ is the bisector of angle $ BAC $.
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most
\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]
In an old script found in ruins of Perspolis is written:
[code]
This script has been finished in a year whose 13th power is
258145266804692077858261512663
You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code]
Find the year the script is finished. Give a reason for your answer.
Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$.
[i](Proposed by Walther Janous)[/i]
Consider a right-angled triangle whose legs are $1$ cm long. Suppose that each point of the triangle was assigned a color from the set of Brown, Blue, Green and Orange colors. It proves that, whatever way this was done, there is at least one pair of points of the same color at a distance equal to or greater than $2-\sqrt 2$ cm from each other.
Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.
Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.
On sides $AB, AC, BC$ are points $M, X, Y$, respectively, such that $AX=MX$; $BY=MY$. $K$, $L$ are midpoints of $AY$ and $BX$. $O$ is circumcenter of $ABC$, $O_1$, $O_2$ are symmetric with $O$ with respect to $K$ and $L$. Prove that $X, Y, O_1, O_2$ are concyclic.