Found problems: 85335
Define $f(x)$ to be the nearest integer to $x$, with the greater integer chosen if two integers are tied for being the nearest. For example, $f(2.3) = 2$, $f(2.5) = 3$, and $f(2.7) = 3$. Define $[A]$ to be the area of region $A$. Define region $R_n$, for each positive integer $n$, to be the region on the Cartesian plane which satisfies the inequality $f(|x|) + f(|y|) < n$. We pick an arbitrary point $O$ on the perimeter of $R_n$, and mark every two units around the perimeter with another point. Region $S_{nO}$ is defined by connecting these points in order.
[b]a)[/b] Prove that the perimeter of $R_n$ is always congruent to $4 \pmod{8}$.
[b]b)[/b] Prove that $[S_{nO}]$ is constant for any $O$.
[b]c)[/b] Prove that $[R_n] + [S_{nO}] = (2n-1)^2$.
[i]Proposed by Lewis Chen[/i]
Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.
Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$, such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?
A teacher gives a multiple choice test to $15$ students and that each student answered each question. Each question had $5$ choices, but remarkably, no pair of students had more than $2$ answers in common. What is the maximum number of questions that could have been on the quiz?
When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive?
$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$
There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving [i]exactly[/i] $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?
Let $(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n$ and $a_0=1$ and $a_1=2$ prove that all the terms of the sequence are integers
Points $D$ and $E$ lie on the lines $BC$ and $AC$ respectively so that $B$ is between $C$ and $D$, $C$ is between $A$ and $E$, $BC = BD$ and $\angle BAD = \angle CDE$. It is known that the ratio of the perimeters of the triangles $ABC$ and $ADE$ is $2$. Find the ratio of the areas of these triangles.
Problem 8. For every non-negative integer n, define an n-variable function K_n (x_1,x_2,…,x_n ) as follows:
K_0=1
K_1 (x_1 )=〖x_1〗^2
K_(n+2) (x_1,x_2,…,x_(n+2) )=〖x_(n+2)〗^2.K_(n+1) (x_1,x_2,…,x_(n+1) )+(x_(n+2)+x_(n+1))K_n (x_1,x_2,…,x_n )
Prove that:
K_n (x_1,x_2,…,x_n )=K_n (x_n,…〖,x〗_2,x_1 )
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.
[i]Thanasin Nampaisarn, Thailand[/i]
Evaluate the following limit
$$\lim_{x\to 0} (1 + 2x + 3x^2 + 4x^3 +...)^{1/x}$$
Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?
Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.
A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$
[i]Proposed by Ivan Chan Kai Chin[/i]
Give the conditions under which the equation $$ \sqrt{x - a} + \sqrt{x - b} = \sqrt{x - c }$$ has roots, assuming that the numbers $ a $, $ b $, $ c $ are pairs of differences
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that any real numbers $x{}$ and $y{}$ satisfy \[f(xf(x)+f(y))=f(f(x^2))+y.\]
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$.
[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0));
label("$B$", (0, 0), SW);
label("$A$", (12, 0), ESE);
label("$C$", (2.4, 3.6), SE);
label("$D$", (0, 5), N);[/asy]
What is the area of quadrilateral $ABCD$?
$\textbf{(A) }12\qquad\textbf{(B) }24\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }36$
There are direct routes from every city of a certain country to every other city. The prices are known in advance. Two tourists (they do not necessary start from one city) have decided to visit all the cities, using only direct travel lines. The first always chooses the cheapest ticket to the city, he has never been before (if there are several -- he chooses arbitrary destination among the cheapests). The second -- the most expensive (they do not return to the first city). Prove that the first will spend not more money for the tickets, than the second.
For a given natural number $n$, two players randomly (uniformly distributed) select a common number $0 \le j \le n$, and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that
\[ \sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), \quad (n \to \infty),\]
where $\gamma$ is the Euler constant.
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\]
holds for all $x,y \in \mathbb{R}$.
Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$.
a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$.
b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.
In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$, one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$: if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$, then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$. Show that there is at least one island which is connected to any other island by a bridge.
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
An $\textit{abundant number}$ is a natural number, the sum of whose proper divisors is greater than the number itself. For instance, $12$ is an abundant number: \[1+2+3+4+6=16>12.\] However, $8$ is not an abundant number: \[1+2+4=7<8.\] Which one of the following natural numbers is an abundant number?
$\begin{array}{c@{\hspace{14em}}c@{\hspace{14em}}c} \textbf{(A) }14&\textbf{(B) }28&\textbf{(C) }56\end{array}$