Found problems: 85335
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $XYZ$.
[i]Proposed by Michael Ren[/i]
[u]Set 1
[/u]
[b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.)
[b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten?
[b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$.
PS. You should use hide for answers.
Six points are chosen on the unit circle such that the product of the distances from any other point on the unit circle is at most $2$. Find the area of the hexagon with these six points as vertices.
$\text{(A) }\frac{1}{2}\qquad\text{(B) }\frac{3}{2}\qquad\text{(C) }\frac{\sqrt{3}}{2}\qquad\text{(D) }\frac{3\sqrt{3}}{2}\qquad\text{(E) }\frac{3+\sqrt{3}}{2}$
In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.
[i]Proposed by Oleksiy Klurman, Ukraine[/i]
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the $3$-by-$3$ array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over?
$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 112 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\
148 \qquad\textbf{(E)}\ 160$
Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$?
A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.
Find all pairs of real numbers $(a,b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$.
$(a)$ Prove that $t$ is parallel to $AC$.
$(b)$ Prove that the lines $r,s,t$ are concurrent.
Let $n\geq 2$ be an integer. Determine the number of terms in the polynomial $$\prod_{1\leq i< j\leq n}(x_i+x_j)$$ whose coefficients are odd integers.
Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$.
Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40 \end{cases}$$
$ 2.46\times 8.163\times (5.17+4.829) $ is closest to:
$ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $
Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.
Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$.
$$xy+yz = 30$$
$$yz+zx = 36$$
$$zx+xy = 42$$
[i]Proposed by Nathan Xiong[/i]
prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure.
[b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common.
we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.
Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers.
(a) Prove that there exist infinitely many sets of $4$ consecutive good numbers.
(b) Find all sets of $5$ consecutive good numbers.
[i]Proposed by Michael Ma[/i]
Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$.
[i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]
[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $
[b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.
Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$.
At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine:
a) the cosine of the angle between the planes $MNP$ and $ABC$.
b) the distance from $D$ to the plane $MNP$