Found problems: 252
2015 AMC 10, 12
Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$?
$ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $
1971 AMC 12/AHSME, 20
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
$\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$
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
2014 Vietnam National Olympiad, 3
Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.
2010 AMC 10, 13
What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$?
$ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$
2013 Tuymaada Olympiad, 6
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]
2000 Moldova Team Selection Test, 8
Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$.
Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.
2016 China Western Mathematical Olympiad, 3
Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have:
$$|a_1|+|a_i|\leq2$$
2014 Harvard-MIT Mathematics Tournament, 23
Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.
2009 Ukraine National Mathematical Olympiad, 2
Find all functions $f : \mathbb Z \to \mathbb Z$ such that
\[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\]
[b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$
2018 Bosnia and Herzegovina Team Selection Test, 4
Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$
2018 Macedonia National Olympiad, Problem 2
Let $n$ be a natural number and $C$ a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, ..., x_{n}, 1$ such that the absolute value of the difference between any two adjacent terms is equal to $C$.
2005 Finnish National High School Mathematics Competition, 3
Solve the group of equations: \[\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}\]
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.
1989 Flanders Math Olympiad, 3
Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2
(k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot
\alpha}| = 4\]
1941 Moscow Mathematical Olympiad, 079
Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.
2007 Pre-Preparation Course Examination, 1
Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.
2007 Today's Calculation Of Integral, 237
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$
2005 Putnam, B3
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
2013 Online Math Open Problems, 18
Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
(You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.)
[i]Ray Li[/i]
1954 Moscow Mathematical Olympiad, 285
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 $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.
PEN H Problems, 72
Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.
2011 Putnam, B1
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]
2012 Today's Calculation Of Integral, 852
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows.
(1) $g_n(x)=(1+x)^n$
(2) $g_n(x)=\sin n\pi x$
(3) $g_n(x)=e^{nx}$
1970 Miklós Schweitzer, 4
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\]
J. Suranyi