Found problems: 85335
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
Let $f : N \to N$ be a function such that the set $\{k | f(k) < k\}$ is finite.
Prove that the set $\{k | g(f(k)) \le k\}$ is infinite for all functions $g : N \to N$.
If $x$ and $y$ are real numbers with $(x+y)^4=x-y$, what is the maximum possible value of $y$?
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.)
Boris Frenkin
When we write the number $n > 2$ as the sum of some integers consecutive positives (at least two addends), we say that we have an [i]elegant decomposition[/i] of $n$. Two [i]elegant decompositions[/i] will be different if any of them contains some term that does not contains the other. How many different elegant decompositions does the number $3^{2004}$ have?
Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.
Let $ABC$ be a triangle with $AB = 13, BC = 14,$ and $CA = 15.$ Pick points $Q$ and $R$ on $AC$ and $AB$ such that $\angle CBQ = \angle BCR = 90^\circ.$ There exist two points $P_1\neq P_2$ in the plane of $ABC$ such that $\triangle P_1 QR, \triangle P_2QR,$ and $\triangle ABC$ are similar (with vertices in order). Compute the sum of the distances from $P_1$ to $BC$ and $P_2$ to $BC.$
Find the sum of all distinct positive divisors of the number $104060401$.
For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$
Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places.
Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy
\begin{align*}
\vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}}
\end{align*}
$(b)$ Find all real numbers $x$ that satisfy following equation $:$
\begin{align*}
\frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6}
\end{align*}
In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular.
a)Prove that midpoints M,N,P,Q of AD,AB,BC,CD form a rectangular
b)If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO
Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\).
[i]Proposed by Karthik Vedula[/i]
For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$.
Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process.
(1) Find $a,\ b,\ c,\ d$.
(2) Let $A'$ be the image of the point $A$ by the process. Find all values of $k$ and the coordinates of $A$ such that the tangent line of $C$ at $A$ and the tangent line of $C$ at $A'$ formed by the process are perpendicular at the origin.
Answer the following questions.
(1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$.
(2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$.
Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$.
(3) Find $ f_n(x)$ for all $ n$.
(4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.
For every ordered pair of integers $(i,j)$, not necessarily positive, we wish to select a point $P_{i,j}$ in the Cartesian plane whose coordinates lie inside the unit square defined by
\[ i < x < i+1, \qquad j < y < j+1. \]
Find all real numbers $c > 0$ for which it's possible to choose these points such that for all integers $i$ and $j$, the (possibly concave or degenerate) quadrilateral $P_{i,j} P_{i+1,j} P_{i+1,j+1} P_{i,j+1}$ has perimeter strictly less than $c$.
[i]Karthik Vedula[/i]
For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?
In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any.
(a) $A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}$ and $B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}$
(b) $A=\mathbb{Q}$ and $B=\mathbb{Q}\cup \{\pi \} $
In (c) and (d) we say $(x,y)>(z,t)$ where $x,y,z,t \in \mathbb{R}$ , whenever $x>z$ or $x=z$ and $y>t$.
(c) $A=\mathbb{R}$ and $B=\mathbb{R}^2$
(d) $X=\{2^{-x}| x\in \mathbb{N}\}$ , then $A=X \times (X\cup \{0\})$ and $B=(X \cup \{ 0 \}) \times X$
(e) If $A,B \subset \mathbb{R}$ , such that there exists a surjective non-decreasing function from $A$ to $B$ and a surjective non-decreasing function from $B$ to $A$ , does there exist a surjective strictly increasing function from $B$ to $A$?
Time allowed for this problem was 2 hours.
A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).
A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.
Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.