Found problems: 85335
Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula
$$a_n =3(n^2 + n) + 7$$
Prove that this sequence has the following properties:
1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $,
2( No term of the sequence is the cube of an integer.
Let $n \ge 2$ be a positive integer and $\mathcal{F}$ the set of functions $f:\{1,2,\ldots,n\} \to \{1,2,\ldots,n\}$ that satisfy $f(k) \le f(k+1) \le f(k)+1,$ for all $k \in \{1,2,\ldots,n-1\}.$
a) Find the cardinal of the set $\mathcal{F}.$
b) Find the total number of fixed points of the functions in $\mathcal{F}.$
Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$
Two of the Geometry box tools are placed on the table as shown. Determine the angle $\angle ABC$
[img]https://2.bp.blogspot.com/--DWVwVQJgMM/XU1OK08PSUI/AAAAAAAAKfs/dgZeYwiYOrQJE4eKQT5s13GQdBEHPqy9QCK4BGAYYCw/s1600/prmo%2B16%2BChandigarh%2Bp7.png[/img]
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$
[i]Proposed by Kapil Pause[/i]
Let $ a,b,c$ be real positive numbers such that absolute difference between any two of them is less than $ 2$. Prove that: $ a \plus{} b \plus{} c < \sqrt {ab \plus{} 1} \plus{} \sqrt {ac \plus{} 1} \plus{} \sqrt {bc \plus{} 1}$
The $m \times n$ chessboard is colored by black and white. In one step, two neighbouring squares are selected (squares with a common side) and their color changes according to the follwing way:
- white becomes black,
- black become red,
- Red becomes white.
For which $m$ and $n$, these steps can change the colors of all the initial squares from white to black and from black to white?
As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour. Ermenson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Ermenson?
$ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 $
We define two types of operation on polynomial of third degree:
a) switch places of the coefficients of polynomial(including zero coefficients), ex:
$ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$
b) replace the polynomial $P(x)$ with $P(x+1)$
If limitless amount of operations is allowed,
is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?
In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$
If a polynomial of degree n has integer values when evaluated in each of $k,k+1,\ldots,k+n$, where $k$ is an integer, prove that the polynomial has integer values when evaluated at each integer $x$.
Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of
\[
\left|1-\sum_{i \in X} a_{i}\right|
\]
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that
\[
\sum_{i \in X} b_{i}=1.
\]
The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares of the same colour.
[i]Note: Any rectangle is assumed here to have sides contained in the lines of the grid.[/i]
[i](Bulgaria - Nikolay Beluhov)[/i]
A semicircle of radius $2$ is inscribed inside of a rectangle, as shown in the diagram below. The diameter of the semicircle coincides with the bottom side of the rectangle, and the semicircle is tangent to the rectangle at all points of intersection. Compute the length of the diagonal of the rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/c/7/81fcfb759188eae7dcb82fa5d58fb9525d85de.png[/img]
Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.)
[i]Kornpholkrit Weraarchakul[/i]
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
Given sequence $\{a_n\}$ satisfying:
$$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$
It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.
Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that
a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$$
b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$$
Let $\vartriangle ABC$ be an equilateral triangle with side length $M$ such that points $E_1$ and $E_2$ lie on side $AB$, $F_1$ and $F_2$ lie on side $BC$, and $G1$ and $G2$ lie on side $AC$, such that $$m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2}$$ and the area of polygon $E_1E_2F_1F_2G_1G_2$ equals the combined areas of $\vartriangle AE_1G_2$, $\vartriangle BF_1E_2$, and $\vartriangle CG_1F_2$. Find the ratio $\frac{m}{M}$.
[img]https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png[/img]
For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Prove that there exists a positive integer $k$, which does not have the digit $9$ in its decimal representation, such that:
$$S(2^{24^{2017}}k)=S(k)$$
A classroom consists of a $5\times5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?
$ABC$ is a triangle with $\hat{A}<\hat{C}$, and $D$ is the point on $BC$ such that $B\hat{A}D=A\hat{C}B$. The perpendicular bisectors of $AD$ and $AC$ intersect in the point $E$. Prove that $B\hat{A}E=90^\circ$.
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)
[i](5 points)[/i]
Triangle $\triangle ABC$ has side lengths $\overline{AB} = 13, \overline{BC} = 14,$ and $\overline{AC} = 15$. Suppose $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $P$ be a point on $\overline{MN}$, such that if the circumcircles of triangles $\triangle BMP$ and $\triangle CNP$ intersect at a second point $Q$ distinct from $P$, then $PQ$ is parallel to $AB$. The value of $AP^2$ can be expressed as a common fraction of the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.