Found problems: 252
What is the value of $ x$ if $ |x \minus{} 1| \equal{} |x \minus{} 2|$?
$ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad
\textbf{(B)}\ \frac {1}{2}\qquad
\textbf{(C)}\ 1\qquad
\textbf{(D)}\ \frac {3}{2}\qquad
\textbf{(E)}\ 2$
Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$. Then prove \[ |x| \geq \frac{1}{|a|+|b|+|c|-1}. \]
Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of
\[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\]
[i]Proposed by snap7822[/i]
The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.
Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.
Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations?
\begin{align*}
x^2+3y&=9\\
(|x|+|y|-4)^2&=1\\
\end{align*}
$\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$
The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to
$\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$
Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.
Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.
[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle.
[b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication:
$$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$
In every cell of a $ 60 \times 60$ board is written a real number, whose absolute value is less or equal than $ 1$. The sum of all numbers on the board equals $ 600$.
Prove that there is a $ 12 \times 12$ square in the board such that the absolute value of the sum of all numbers on it is less or equal than $ 24$.
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$?
$\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$
Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.
The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation:
$ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad
\textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad
\textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\
\textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad
\textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?
$n$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 139
\qquad\textbf{(E)}\ \text{None of the above}
$
Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root.
[i]K. Kokhas & F. Petrov[/i]
The numbers $a, b, c$ satisfy the condition $| a - b | = 2 | b - c | = 3 | c - a |$. Prove that $a = b = c$.
Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0,a_1,a_2,a_3$ are constant. Then
$\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$
$\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$
Prove that for any three real numbers $x,y,z$ the following inequality holds:
$$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself).
Show that $f(1)=0$.
Alice creates the graphs $y=|x-a|$ and $y=c-|x-b|$ , where $a,b,c\in\mathbb{R^+}$. She observes that these two graphs and $x$ axis divides the positive side of the plane ($x,y>0$) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
[hide=There might be a translation error] In the original statement,it says $XOY$ plane,instead of positive side of the plane. I think these 2 are the same,but I might be wrong [/hide]
The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building.
How long will it be until Atlantis becomes completely submerged?
[i]Proposed by Ethan Tan[/i]