Found problems: 252
2001 Estonia National Olympiad, 4
It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.
PEN G Problems, 4
Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]
2009 AIME Problems, 14
The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$.
1969 Czech and Slovak Olympiad III A, 4
Determine all complex numbers $z$ such that \[\Bigl|z-\bigl|z+|z|\bigr|\Bigr|-|z|\sqrt3\ge0\] and draw the set of all such $z$ in complex plane.
2011 Junior Balkan Team Selection Tests - Romania, 4
Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.
1941 Moscow Mathematical Olympiad, 079
Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.
1990 AMC 8, 7
When three different numbers from the set $ \{-3,-2,-1, 4, 5\} $ are multiplied, the largest possible product is
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $
2014 Math Prize For Girls Problems, 20
How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation
\[
e^z = \frac{z - 1}{z + 1} \, ?
\]
2011 All-Russian Olympiad, 1
Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?
ICMC 6, 1
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]
2015 Romania National Olympiad, 1
Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality:
$$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$
2021 Polish Junior MO First Round, 3
The numbers $a, b, c$ satisfy the condition $| a - b | = 2 | b - c | = 3 | c - a |$. Prove that $a = b = c$.
2012 ELMO Shortlist, 8
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
2018 Greece National Olympiad, 3
Let $n,m$ be positive integers such that $n<m$ and $a_1, a_2, ..., a_m$ be different real numbers.
(a) Find all polynomials $P$ with real coefficients and degree at most $n$ such that:
$|P(a_i)-P(a_j)|=|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$.
(b) If $n,m\ge 2$ does there exist a polynomial $Q$ with real coefficients and degree $n$ such that:
$|Q(a_i)-Q(a_j)|<|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$
Edit: See #3
2005 Today's Calculation Of Integral, 9
Calculate the following indefinite integrals.
[1] $\int (x^2+4x-3)^2(x+2)dx$
[2] $\int \frac{\ln x}{x(\ln x+1)}dx$
[3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$
[4] $\int \frac{dx}{\sin x\cos ^ 2 x}$
[5] $\int \sqrt{1-3x}\ dx$
2004 National Olympiad First Round, 35
We are placing $n$ integers whose sum is $94$ over a circle such that each number is equal to the absolute value of the difference of (clockwise) next two numbers. What is the largest $n$ that makes such placing possible?
$
\textbf{(A)}\ 188
\qquad\textbf{(B)}\ 186
\qquad\textbf{(C)}\ 141
\qquad\textbf{(D)}\ 100
\qquad\textbf{(E)}\ 47
$
2014 Contests, 1
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?
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 $
2004 Junior Balkan Team Selection Tests - Romania, 3
Let $A$ be a $8\times 8$ array with entries from the set $\{-1,1\}$ such that any $2\times 2$ sub-square of the array has the absolute value of the sum of its element equal with 2. Prove that the array must have at least two identical lines.
2008 Romanian Master of Mathematics, 2
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
2007 Germany Team Selection Test, 2
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$
2010 Tuymaada Olympiad, 3
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 $f(1)=0$.
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.
2011 Bosnia Herzegovina Team Selection Test, 3
Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number
\[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\]
is a perfect square.
2022 Iran MO (3rd Round), 3
Prove that for natural number $n$ it's possible to find complex numbers $\omega_1,\omega_2,\cdots,\omega_n$ on the unit circle that
$$\left\lvert\sum_{j=1}^{n}\omega_j\right\rvert=\left\lvert\sum_{j=1}^{n}\omega_j^2\right\rvert=n-1$$
iff $n=2$ occurs.