Found problems: 85335
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .
There are $8$ cities in a certain kingdom. The king wants to have a system of roads constructed so that one can go along those roads from any city to any other one without going through more than one intermediate city and so that no more than $k$ roads go out of any city. For what values of $k$ is this possible?
(D. Fomin, Leningrad)
Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$ holds for all positive integers $n$.
Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles?
Can a square be dissected into $15$ congruent polygons which are not rectangles?
Let $p>5$ be a prime number. For any integer $x$, define
\[{f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}\]
Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is divisible by $p^3$.
Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$, with $n\geq 1$, is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$. Prove that $\alpha$ is an integer.
Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children in $42$. What is the sum of the ages of the oldest three?
$ \mathrm{(A) \ } 51 \qquad \mathrm{(B) \ } 54 \qquad \mathrm {(C) \ } 57 \qquad \mathrm{(D) \ } 60 \qquad \mathrm{(E) \ } 63$
There are $100$ boxes, each containing either a red cube or a blue cube. Alex has a sum of money initially, and places bets on the colour of the cube in each box in turn. The bet can be anywhere from $0$ up to everything he has at the time. After the bet has been placed, the box is opened. If Alex loses, his bet will be taken away. If he wins, he will get his bet back, plus a sum equal to the bet. Then he moves onto the next box, until he has bet on the last one, or until he runs out of money. What is the maximum factor by which he can guarantee to increase his amount of money, if he knows that the exact number of blue cubes is
[list][b](a)[/b] $1$;
[b](b)[/b] some integer $k$, $1 < k \leq 100$.[/list]
Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total?
[img]https://cdn.artofproblemsolving.com/attachments/0/8/e46929e7ec3fff9be4892ef954ae299e0cb8c7.png[/img]
A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
Let $ABC$ be an acute triangle. Define $A_1$ the midpoint of the largest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ be the feet of the perpendiculars from $A_1$ on the lines $AB$ and $AC$ , respectively. Define $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, and $C_3$ analogously. Show that the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ are concurrent.
How many values of $x$ satisfy the equation $$(x^2 - 9x + 19)^{x^2 + 16x + 60 }= 1?$$
Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that:
$(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$.
Prove $a_n< \frac{1}{n-1}$.
$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.
For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy $$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$
[i]Proposed by Kyle Lee[/i]
Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.
Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.
Let $H_n=\sum_{r=1}^{n} \frac{1}{r}$. Show that
$$n-(n-1)n^{-1\slash (n-1)}>H_n>n(n+1)^{1\slash n}-n$$
for $n>2$.
Call a positive integer $n$ [i]good[/i] if there are $n$ integers, positive or negative, and not necessarily distinct, such that their sum and products are both equal to $n$. Show that the integers of the form $4k+1$ and $4l$ are good.
A non-zero digit is chosen in such a way that the probability of choosing digit $ d$ is $ \log_{10}(d\plus{}1) \minus{} \log_{10} d$. The probability that the digit $ 2$ is chosen is exactly $ \frac12$ the probability that the digit chosen is in the set
$ \textbf{(A)}\ \{2,3\} \qquad \textbf{(B)}\ \{3,4\} \qquad \textbf{(C)}\ \{4,5,6,7,8\} \qquad \textbf{(D)}\ \{5,6,7,8,9\} \qquad \textbf{(E)}\ \{4,5,6,7,8,9\}$
How many pairs of integers $(m,n)$ are there such that $mn+n+14=\left (m-1 \right)^2$?
$
\textbf{a)}\ 16
\qquad\textbf{b)}\ 12
\qquad\textbf{c)}\ 8
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 2
$