Found problems: 85335
Let $n$ be a positive integer and let $a_1, a_2, \cdots a_k$ be all numbers less than $n$ and coprime to $n$ in increasing order. Find the set of values the function $f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1)$.
In a convex hexagon $AC'BA'CB'$, every two opposite sides are equal. Let $A_1$ denote the point of intersection of $BC$ with the perpendicular bisector of $AA'$. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$
Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If
\[
S = \sum_{i = 1}^\infty r_i,
\]
then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$.
[i]Proposed by Aaron Lin[/i]
A circle is inscribed in a square of side $m$, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches:
$\textbf{(A)}\ \frac{\pi m^2}{2}\qquad
\textbf{(B)}\ \frac{3\pi m^2}{8}\qquad
\textbf{(C)}\ \frac{\pi m^2}{3}\qquad
\textbf{(D)}\ \frac{\pi m^2}{4}\qquad
\textbf{(E)}\ \frac{\pi m^2}{8}$
Around the acute-angled triangle $ABC$ ($AC>CB$) a circle is circumscribed, and the point $N$ is midpoint of the arc $ACB$ of this circle. Let the points $A_1$ and $B_1$ be the feet of perpendiculars on the straight line $NC$, drawn from points $A$ and $B$ respectively (segment $NC$ lies inside the segment $A_1B_1$). Altitude $A_1A_2$ of triangle $A_1AC$ and altitude $B_1B_2$ of triangle $B_1BC$ intersect at a point $K$ . Prove that $\angle A_1KN=\angle B_1KM$, where $M$ is midpoint of the segment $A_2B_2$ .
Anna's number is obtained by writing down $20$ consecutive positive integers, one after another in arbitrary order. Bob's number is obtained in the same way, but with $21$ consecutive positive integers. Can they obtain the same number?
$\frac{a^3}{b^3}$+$\frac{a^3+1}{b^3+1}$+...+$\frac{a^3+2015}{b^3+2015}$=2016
b - positive integer, b can't be 0
a - real
Find $\frac{a^3}{b^3}$*$\frac{a^3+1}{b^3+1}$*...*$\frac{a^3+2015}{b^3+2015}$
Let $ M=\{1,2,...,2013\} $ and let $ \Gamma $ be a circle. For every nonempty subset $ B $ of the set $ M $, denote by $ S(B) $ sum of elements of the set $ B $, and define $ S(\varnothing)=0 $ ( $ \varnothing $ is the empty set ). Is it possible to join every subset $ B $ of $ M $ with some point $ A $ on the circle $ \Gamma $ so that following conditions are fulfilled:
$ 1 $. Different subsets are joined with different points;
$ 2 $. All joined points are vertices of a regular polygon;
$ 3 $. If $ A_1,A_2,...,A_k $ are some of the joined points, $ k>2 $ , such that $ A_1A_2...A_k $ is a regular $ k-gon $, then $ 2014 $ divides $ S(B_1)+S(B_2)+...+S(B_k) $ ?
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following:
$$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$
How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers?
$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$
Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?
Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.
[i]Proposed by Evan Chen[/i]
Let $ABCDEF$ be a convex hexagon. Segments $AE$ and $BF$ intersect at $X$ and segments $BD$ and $CE$ intersect in $Y.$ It's known that $$ \angle XBC = \angle XDE = \angle YAB = \angle YEF = 80^\circ \text{ and } \angle XCB = \angle XED = \angle YBA = \angle YFE = \angle 70^\circ.$$ Let $P$ and $Q$ be such points on line $XY$ that segments $PX$ and $AF$ intersect, segments $QY$ and $CD$ intersect and $\angle APF = \angle CQD = 30 ^\circ.$ Estimate the sum: \[ \frac{BX}{BF} + \frac{BY}{BD} + \frac{EX}{EA} + \frac{EY}{EC} + \frac{PX}{PY} + \frac{QY}{QX} \]
[i]Proposed by Gogi Khimshiashvili, Georgia [/i]
Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple.
Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.
I have a very good solution of this but I want to see others.
Let the midpoint$ M$ of the side$ AB$ of an inscribed quardiletar, $ABCD$.Let$ P $the point of intersection of $MC$ with $BD$. Let the parallel from the point $C$ to the$ AP$ which intersects the $BD$ at$ S$. If $CAD$ angle=$PAB$ angle= $\frac{BMC}{2}$ angle, prove that $BP=SD$.
Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$
Find the value of $t$ such that the maximum value of $OP$ is 3.
Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$
[i]Proposed by Alex Schreiber, Germany[/i]
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$
Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).
Let $P(x)$ be a nonzero polynomial with integer coefficients. Let $a_{0}=0$ and for $i \ge 0$ define $a_{i+1}=P(a_{i})$. Show that $\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}$ for all $m, n \in \mathbb{N}$.
(a) Let $x$ and $y$ be integers. Prove that $x = y$ if $x^n \equiv y^n$ mod $n$ for all positive integers $n$.
(b) For which pairs of integers $(x, y)$ are there infinitely many positive integers $n$ such that $x^n \equiv y^n$ mod $n$?
Find the largest positive integer $n$ such that $n^3 + 4n^2 - 15n - 18$ is the cube of an integer.
Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where
a) $f(n)=n\ln{n}$;
b) $f(n)=n$.
[i]David Yang.[/i]