Found problems: 85335
The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and
$$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$
Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(ab) = f(a)f(b)$, and
(2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.
Let $n$ be positive integer and let $a_1, a_2,...,a_n$ be real numbers. Prove that there exist positive integers $m, k$ $<=n$ , $|$ $(a_1+a_2+...+a_m)$ $-$ $(a_{m+1}+a_{m+2}+...+a_n)$ $|$ $<=$ $|$ $a_k$ $|$
Let $a, b, n$ be positive integers with $\gcd(a, b)=1$. Prove that \[\sum_{k}\left\{ \frac{ak+b}{n}\right\}=\frac{n-1}{2},\] where $k$ runs through a complete system of residues modulo $m$.
Consider a cylinder and a cone with a common base such that the volume of the
part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$
There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.
Solve the system of equations
$$\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}$$
Graph $G$ becomes planar when any vertex is removed. Prove that its vertices can be properly colored with 5 colors. (Using the four-color theorem without proof is not allowed!)
[i]Proposed by D. Karpov[/i]
In the diagram below, point $A$ lies on the circle centered at $O$. $AB$ is tangent to circle $O$ with $\overline{AB} = 6$. Point $C$ is $\frac{2\pi}{3}$ radians away from point $A$ on the circle, with $BC$ intersecting circle $O$ at point $D$. The length of $BD$ is $3$. Compute the radius of the circle.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/baa528c776eb50455f31ae50a3ec28efc291e8.png[/img]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:
$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$
[i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]
Suppose $a$, $b$, and $c$ are numbers satisfying the three equations:
$$a + 2b = 20,$$
$$b + 2c = 2,$$
$$c + 2a = 3.$$
Find $9a + 9b + 9c$.
Find the interior angle between two sides of a regular octagon (degrees).
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
How many multiples of $12$ divide $12!$ and have exactly $12$ divisors?
[i]Proposed by Adam Bertelli[/i]
Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $|f(k)| \leq k$ for all positive integers $k$ and there is a prime number $p>2024$ which satisfies both of the following conditions:
$1)$ For all $a \in \mathbb{N}$ we have $af(a+p) = af(a)+pf(a),$
$2)$ For all $a \in \mathbb{N}$ we have $p|a^{\frac{p+1}{2}}-f(a).$
[i]Authored by Nikola Velov[/i]
3. Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints.
Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .
An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.
We have $10$ identical tiles as shown. The tiles can be rotated, but not flipper over. A $7 \times 7$ board should be covered with these tiles so that exactly one unit square is covered by two tiles and all other fields by one tile. Designate all unit sqaures that can be covered with two tiles.
[img]https://cdn.artofproblemsolving.com/attachments/d/5/6602a5c9e99126bd656f997dee3657348d98b5.png[/img]
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
Let $ABCD$ be a parallelogram. Let $W, X, Y,$ and $Z$ be points on sides $AB, BC, CD,$ and $DA$, respectively, such that the incenters of triangles $AWZ, BXW, CYX,$ and $DZY$ form a parallelogram. Prove that $WXYZ$ is a parallelogram.
Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions:
Any three points are not collinear.
No point is found inside the circle whose diameter has ends at any two points of $S_n$.
[b]NOTE: [/b] The points on the circumference are not considered to be inside the circle.
For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.