Found problems: 85335
Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.
Let $ABCD$ be a tetrahedron such that $AB = CD = \sqrt{41}$, $AC = BD = \sqrt{80}$, and $BC = AD = \sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt{n}}{p}$, when $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$.
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
$\textbf{(A)}\ 105 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 135 \qquad\textbf{(D)}\
150 \qquad\textbf{(E)}\ 165$
Determine all bounded closed subsets $F$ of the plane with the following property: $F$ consists of at least two points and always contains two points $A$ and $B$ as well as at least one of the two semicircular arcs over the segment $AB$.
Definitions: A subset of the $F$ of the plane is said to be closed if: For every point $P$ of the plane that is not an element of $F$ , there is a (non-degenerate) disc with center $P$ that has no elements of $F$.
Find all functions $f: \mathbb Q \to \mathbb Q$ such that $ f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x)$ for all $x,y \in \mathbb Q$.
Suppose $ABCD$ is a convex quadrilateral with $\angle{ABD}=105^\circ, \angle{ADB}=15^\circ, AC=7,$ and $BC=CD=5.$ Compute the sum of all possible values of $BD.$
Given a board $3\times3$ and $9$ cards with some numbers (known to the players). Two players, in turn, put those cards on the board. The first wins if the sum of the numbers in the first and the third row is greater than in the first and the third column. Prove that it doesn't matter what numbers are on the cards, but if the first plays the best way, the second can not win.
A bowling contest consists of several series. Mary got 185 points in her previous series and thereby increased her average score per series from 176 to 177 points. How many points would Mary need in her next series to increase her average to 178?
$ \text{(A)}\ 184 \qquad \text{(B)}\ 185 \qquad \text{(C)}\ 186 \qquad \text{(D)}\ 187 \qquad \text{(E)}\ 188$
[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday?
[b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$.
[b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$?
Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$.
[b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there?
[b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is $2$ miles, which is $10,560$ feet, and Bella covers $2 \tfrac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella?
$\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520$
Let $ABC$ be an isosceles triangle with base $AC$. Points $D$ and $E$ are chosen on the sides $AC$ and $BC$, respectively, such that $CD = DE$. Let $H, J,$ and $K$ be the midpoints of $DE, AE,$ and $BD$, respectively. The circumcircle of triangle $DHK$ intersects $AD$ at point $F$, whereas the circumcircle of triangle $HEJ$ intersects $BE$ at $G$. The line through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH \cap GF =$ {$M$}. Prove that $J, M,$ and $K$ are collinear points.
Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c, 2^c, \cdots , n^c$ are integers. Prove that $c$ is an integer.
Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide?
[i]Boris Frenkin[/i]
(a) Do there exist 2009 distinct positive integers such that their sum is divisible by each of the given numbers?
(b) Do there exist 2009 distinct positive integers such that their sum is divisible by the sum of any two of the given numbers?
In convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and \[ AB+DE = BC+EF = CD+FA. \] The midpoints of sides $AB$, $BC$, $DE$, $EF$ are $A_1$, $B_1$, $D_1$, $E_1$, and segments $A_1D_1$ and $B_1E_1$ meet at $O$. Prove that $\angle D_1OE_1 = \frac12 \angle DEF$.
\(ABCD\) is a convex quadrilateral, \(AC\) bisects the angle \(\angle BAD\). Points \(E\) and \(F\) are on the sides \(BC\) and \(CD\) respectively such that \(EF \parallel BD\). Extend \(FA\) and \(EA\) to points \(P\) and \(Q\) respectively, such that the circle \(\omega_1\) passing through points \(A\), \(B\), \(P\) and the circle \(\omega_2\) passing through points \(A\), \(D\), \(Q\) are both tangent to line \(AC\). Prove that the points \(B\), \(P\), \(Q\), \(D\) are concyclic.
The incircle $ (I)$ of a given scalene triangle $ ABC$ touches its sides $ BC$, $ CA$, $ AB$ at $ A_1$, $ B_1$, $ C_1$, respectively. Denote $ \omega_B$, $ \omega_C$ the incircles of quadrilaterals $ BA_1IC_1$ and $ CA_1IB_1$, respectively. Prove that the internal common tangent of $ \omega_B$ and $ \omega_C$ different from $ IA_1$ passes through $ A$.
On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k\geq 0$ lines. How many different values of $k$ are possible?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
In the Bank of Shower, a bored customer lays $n$ coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up.
Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$. Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$, $f(n) > 0$.
Let $N$ be the smallest positive integer so that \[
M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right)
\]is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$.
[i]Proposed by Edward Wan and Brandon Wang[/i]
1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i].
Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.
Call a set of positive integers $\textit{good}$ if there is a partition of it into two sets $S$ and $T$, such that there do not exist three elements $a, b, c \in S$ such that $a^b = c$ and such that there do not exist three elements $a, b, c \in T$ such that $a^b = c$ ($a$ and $b$ need not be distinct). Find the smallest positive integer $n$ such that the set $\{2, 3, 4, \dots, n\}$ is \textit{not} good.
Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.
Define a function $f:\mathbb{Z}_{\ge 0} \to\mathbb{Z}$ with $f\left(0\right)=1$ and \[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right)\]
for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits.
[i]Proposed by Tristan Shin[/i]
Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.
There is a unique positive real number $x$ such that the three numbers $\log_8(2x),\log_4x,$ and $\log_2x,$ in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Given a graph with $n \geq 4$ vertices. It is known that for any two of vertices there is a vertex connected with none of these two vertices.
Find the greatest possible number of the edges in the graph.