Found problems: 85335
[b](a)[/b] Prove that for all positive integers $m,n$ we have
\[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\]
[b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that
\[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\]
Has a limit.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(f(x)+y)(x-f(y)) = f(x)^2-f(y^2).$$
There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted:
(i) Remove an equal number of balls from both boxes.
(ii) Increase the number of balls in one of the boxes by a factor $k$.
Is it possible to remove all of the balls from both boxes with just these two actions,
1. if $k = 2$?
2. if $k = 3$?
sppose that $\sigma_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\sigma_k(n)=\sum_{d|n}d^k$. $\rho_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\rho_k\ast \sigma_k=\delta$. find a formula for $\rho_k$.($\frac{100}{6}$ points)
$p$ is a prime and $m$ is a non-negative integer $< p-1$.
Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.
Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$,
$n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.
In a class of seven students, a poll is conducted. The poll asks
$~$(a) Were you born after $2021$?
$~$(b) Were you born after $2019$?
$~$(c) Were you born in $2020$?
Five students respond "yes" to question (a), and four students respond "no" to question (b). If everyone truthfully answers all questions, how many students responded "yes" to (c)?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\] Find the smallest possible value of $a^2 + b^2 + c^2$.
[i]Binh Luan and Nhan Xet, Vietnam[/i]
When $(a-b)^n$, $n\geq 2$, $ab\neq 0$, is expanded by the binomial theorem, it is found that , when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals:
$\textbf{(A) }\tfrac12k(k-1)\qquad
\textbf{(B) }\tfrac12k(k+1)\qquad
\textbf{(C) }2k-1\qquad
\textbf{(D) }2k\qquad
\textbf{(E) }2k+1$
Is rational or irrational,the number
$$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$
Michael is trying to drive a bus from his home, $(0,0)$, to school, located at $(6,6)$. There are horizontal and vertical roads at every line $x=0,1,\ldots,6$ and $y=0,1,\ldots,6$. The city has placed $6$ roadblocks on lattice point intersections $(x,y)$ with $0\leq x,y \leq 6$. Michael notices that the only path he can take that only goes up and to the right is directly up from $(0,0)$ to $(0,6)$, and then right to $(6,6)$. How many sets of $6$ locations could the city have blocked?
Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively.
Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if
$$g_a : g_b : g_c = g_x : g_y : g_z, $$
then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.
$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum
$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$
may achieve and find all $a,b,c$ for which equality occurs.\\ \\
[i](Andrei Bâra)[/i]
Let $f:[0,1]\to\mathbb R$ be a continuous function. Define a sequence of functions $f_n:[0,1]\to\mathbb R$ in the following way:
$$f_0(x)=f(x),\qquad f_{n+1}(x)=\int^x_0f_n(t)\text dt,\qquad n=0,1,2,\ldots.$$Prove that if $f_n(1)=0$ for all $n$, then $f(x)\equiv0$.
Prove that for arbitrary positive numbers the following inequality holds
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]
Let $\{a_n\}_{n\in \mathbb{N}}$ a sequence of non zero real numbers.
For $m \geq 1$, we define:
\[ X_m = \left\{X \subseteq \{0, 1,\dots, m - 1\}: \left|\sum_{x\in X} a_x \right| > \dfrac{1}{m}\right\}. \]
Show that
\[\lim_{n\to\infty}\frac{|X_n|}{2^n} = 1.\]
A triangular colony area is divided into four fields of varying size as shown in the figure below shows. The only other thing we know is that the distances $AF$, $FD$, $BF$ and $FE$ have the lengths $5$, $2$, $4$ and $2$ respectively (in $10$s of m). When the lots are distributed, Joar gets to choose first. Which lot should he choose to get the one with the largest area?
[img]https://cdn.artofproblemsolving.com/attachments/9/5/073e2699d54c8ee3a4dd7d23b69c2a894fd93e.png[/img]
Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ?
Justify your answer.
Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities?
The numbers $1,2,3,\dots,1000$ are written on the board. Patya and Vassya are playing a game. They take turn alternatively erasing a number from the board. Patya begins. If after a turn all numbers (maybe one) on the board be divisible by a natural number greater than $1$ the player who last played loses. If after some number of steps the only remaining number on the board be $1$ then they call it a draw. Determine the result of the game if they both play their best.
Let ABCDA'B'C'D' be a rectangular parallelipiped, where ABCD is the lower face and A, B, C and D' are below A', B', C' and D', respectively. The parallelipiped is divided into eight parts by three planes parallel to its faces. For each vertex P, let V P denote the volume of the part containing P. Given that V A= 40, V C = 300 , V B' = 360 and V C'= 90, find the volume of ABCDA'B'C'D'.
$N$ boxes are arranged in a circle and are numbered $1,2,3,.....N$ In a clockwise direction. A ball is assigned a number from${1,2,3,....N}$ and is placed in one of the boxes.A round consist of the following; if the current number on the ball is $n$, the ball is moved $n$ boxes in the clockwise direction and the number on the ball is changed to $n+1$ if $n<N$ and to $1$ if $n=N$. Is it possible to choose $N$, the initial number on the ball, and the first position of the ball in such a way that the ball gets back to the same box with the same number on it for the first time after exactly $2020$ rounds
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[
2a^n+3b^n=4c^n.
\]
Let $N = 2014! + 2015! + 2016! + \dots + 9999!$. How many zeros are at the end of the decimal representation of $N$?
[i]Proposed by Evan Chen[/i]
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?