Found problems: 252
[b]a)[/b] Let $ \bold {u,v,w,} $ be three coplanar vectors of absolute value $ 1. $ Show that there exist $ \varepsilon_1 ,\varepsilon_2, \varepsilon_3\in \{ \pm 1\} $ such that
$$ \big| \varepsilon_1\bold u +\varepsilon_2\bold v +\varepsilon_3\bold w \big|\le 1. $$
[b]b)[/b] Give an example of three vectors such that the inequality above does not work for any sclaras from $ \{ \pm 1\} . $
For $n(n\geq3)$ positive intengers $a_1,a_2,\cdots,a_n$. Put the numbers on a circle. In each operation, calculate difference between two adjacent numbers and take its absolute value. Put the $n$ numbers we get on another ciecle (do not change their order). Find all $n$, satisfying that no matter how $a_1,a_2,\cdots,a_n$ are given, all numbers on the circle are equal after limited operations.
Prove that for every convex function $ f(x)$ defined on the interval $ \minus{}1\leq x \leq 1$ and having absolute value at most $ 1$,
there is a linear function $ h(x)$ such that \[ \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}.\] [L. Fejes-Toth]
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 $a_1-a_{2n}=n$.
Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon.
[i]Daniel Jinga[/i]
Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash. "That seems fair to me," confirms Hannah encouragingly.
While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it. There is a monic polynomial of degree $n$, with real coefficients. The first two terms after $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$, but the rest of the polynomial is cut off where Michael's page is ripped. Wendy barely makes out a little of Michael's scribbling, showing that $a_{n-1}=-a_{n-2}$. Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial. Wendy knows neither the value of $n$, nor the value of $a_{n-1}$, but still she finds a [greatest] lower bound for the answer to the problem. Find the absolute value of that lower bound.
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
The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions:
$f_0(x) = |x|$ for all $x \in R$
$f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$.
For each positive integer $n$, solve the equation $f_n(x)=1$.
Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that
\[
\left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\]
for every positive integer $n$.
Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$
be a polynomial with real coefficients and $a_{2016} \neq 0$
satisfies
$|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$
Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also)
edited: complex roots
Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions?
1. $f(0,0)=1$;
2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$;
3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$.
Justify your answer.
Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of the above}
$
Find all real pairs $x,y$ such that
\begin{align*}
x-|y+1|&=1, \\
x^2+y&=10.
\end{align*}
The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle.
(a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$?
(b) What is the maximum area of this triangle?
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying:
$$ a^2+b^2+c^2-ab-bc-ca=0. $$
Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.
A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer.
(a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$.
(b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$
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 $|x_1+x_2+\cdots+x_{2006}|$.
Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.
Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.
Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n,\ n=1, 2, \cdots$. Then $S_{17}+S_{33}+S_{50}$ equals:
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -1 \qquad\textbf{(E)}\ -2$