Found problems: 252
2020 Macedonian Nationаl Olympiad, 2
Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that
$|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$,
with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.
2018 Israel National Olympiad, 3
Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.
2019 CCA Math Bonanza, I10
What is the minimum possible value of \[\left|x\right|-\left|x-1\right|+\left|x+2\right|-\left|x-3\right|+\left|x+4\right|-\cdots-\left|x-2019\right|\] over all real $x$?
[i]2019 CCA Math Bonanza Individual Round #10[/i]
2006 AIME Problems, 15
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.
2008 ITest, 41
Suppose that \[x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009.\]
Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\displaystyle\sum_{n=1}^{2008}x_n$.
2025 Kosovo National Mathematical Olympiad`, P4
Show that for any real numbers $a$ and $b$ different from $0$, the inequality
$$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$
holds. When is equality achieved?
2024 CCA Math Bonanza, I9
Find the median value of $m$ over all integers $m$ where $|m^2 + 8m - 65|$ is a perfect power. A perfect power is any integer at least $2$ which can be written as $a^b$, where $a$, $b$ are integers and $b \ge 2$.
[i]Individual #9[/i]
2022 JBMO Shortlist, A6
Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 + |(a - b)(b - c)(c - a)|.$$
2013 Bogdan Stan, 2
Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties:
$ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $
$ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $
Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $
[i]Cristinel Mortici[/i]
2005 National High School Mathematics League, 14
Nine balls numbered $1,2,\cdots,9$ are put on nine poines that divide the circle into nine equal parts. The sum of absolute values of the difference between the number of two adjacent balls is $S$. Find the probablity of $S$ takes its minumum value.
Note: If one way of putting balls can be the same as another one by rotating or specular-reflecting, then they are considered the same way.
2011 AMC 12/AHSME, 18
Suppose that $|x+y|+|x-y|=2$. What is the maximum possible value of $x^2-6x+y^2$?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9
$
2017 AMC 8, 21
Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a+b+c=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$?
$\textbf{(A) }0\qquad\textbf{(B) }1\text{ and }-1\qquad\textbf{(C) }2\text{ and }-2\qquad\textbf{(D) }0,2,\text{ and }-2\qquad\textbf{(E) }0,1,\text{ and }-1$
2011 District Olympiad, 2
[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:
$$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$
2018 AMC 10, 12
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*}
$\textbf{(A) } 1 \qquad
\textbf{(B) } 2 \qquad
\textbf{(C) } 3 \qquad
\textbf{(D) } 4 \qquad
\textbf{(E) } 8 $
2008 Harvard-MIT Mathematics Tournament, 3
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
2009 Today's Calculation Of Integral, 423
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.
1991 AMC 12/AHSME, 2
$|3 - \pi| =$
$ \textbf{(A)}\ \frac{1}{7}\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 3 - \pi\qquad\textbf{(D)}\ 3 + \pi\qquad\textbf{(E)}\ \pi - 3 $
2009 District Olympiad, 4
[b]a)[/b] Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle.
[b]b)[/b] Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.
1966 Czech and Slovak Olympiad III A, 1
Consider a system of inequalities \begin{align*}y-x&\ge|x+1|-|x-1|, \\ |y&-x|-y+x\ge2.\end{align*} Draw solutions of each inequality in the plane separately and highlight solution of the system.
2011 Dutch BxMO TST, 3
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.
2019 LIMIT Category B, Problem 6
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
$\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$
$\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$
2012 Today's Calculation Of Integral, 846
For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.
2013 Peru IMO TST, 2
Let $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$
1968 AMC 12/AHSME, 34
With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin$\dagger$ by which it was originally defeated. The number voting for the bill on the re-vote was $\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?
$\textbf{(A)}\ 75 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 20$
$\dagger$ In this context, margin of defeat (passage) is defined as the number of nays minus the number of ayes (nays-ayes).
2015 Danube Mathematical Competition, 2
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.