Found problems: 85335
Triangle $\vartriangle ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. $\vartriangle ABC$ has incircle $\gamma$ and circumcircle $\omega$. $\gamma$ has center at $I$. Line $AI$ is extended to hit $\omega$ at $P$. What is the area of quadrilateral $ABPC$?
Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\in S$, there exist $f_1, \cdots, f_k \in F$ (not necessarily different), such that $f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))$.
Find the least positive integer $m$, such that if $F$ satisfies property $P(2019)$, then it also satisfies property $P(m)$.
Quadrilateral $ABCD$ is inscribed in a circle of radius $6$. If $\angle BDA = 40^o$ and $AD = 6$, what is the measure of $\angle BAD$ in degrees?
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school?
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 7.3\qquad\textbf{(C)}\ 7.7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 8.3 $
Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint.
what i have is [hide="this"]we may assume that the union of the $A_{i}$s is $S$. [/hide]
A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that
this figure can be placed onto a Cartesian plane so that it covers at least $n+1$
points with integer coordinates.
Let $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ a sequence of positive real numbers, such that:
\[x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*\]
a) Prove that the sequences $(x_n+y_n)_{n\ge 1}$ and $(x_ny_n)_{n\ge 1}$ have limit.
b) Prove that the sequences $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ have limit and that their limits are equal.
Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$.
a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$.
b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$.
[i]Proposed by Vincent Huang[/i]
Let $A=2^7(7^{14}+1)+2^6\cdot 7^{11}\cdot 10^2+2^6\cdot 7^7\cdot 10^{4}+2^4\cdot 7^3\cdot 10^6$. Prove that the number $A$ ends with $14$ zeros.
[i](I. Gorodnin)[/i]
A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.
The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$.
[i]Proposed by Belgium[/i]
Al wishes to label the faces of his cube with the integers $2009,2010,$ and $2011,$ with one integer per face, such that adjacent faces (faces that share an edge) have integers that differ by at most $1.$ Determine the number of distinct ways in which he can label the cube, given that two configurations that can be rotated on to each other are considered the same, and that we disregard the orientation in which each number is written on to the cube.
Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$, and $m^k$ is divisible by $k^m$.
[list=a]
[*] Prove that $n^k$ is divisible by $k^n$.
[*] Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1.
[/list]
Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$. The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$? (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)
A circle passing through $A, B$ and the orthocenter of triangle $ABC$ meets sides $AC, BC$ at their inner points. Prove that $60^o < \angle C < 90^o$ .
(A. Blinkov)
This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base.
[img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$.
Prove
\[ AN \cdot NC = CD \cdot BN. \]
$A, B$ and $C$ are points in the plane with integer coordinates.
The lengths of the sides of triangle $ABC$ are integer numbers.
Prove that the perimeter of the triangle is an even number.
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that
[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]
[i]Proposed by Jaedon Whyte[/i]
Given two (identical) polygonal domains in the Euclidean plane, it is not possible in general to superpose the two using only translations and rotations. Prove that this can however be achieved by splitting one of the domains into a finite number of polygonal subdomains which then fit together, via translations and rotations in the plane, to recover the other domain.
A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.
Positive integer $m$ is such that the sum of decimal digits of $8^m$ equals 8. Can the last digit of $8^m$ be equal 6?
(Author: V. Senderov)
(compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)
How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the above}
$
A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.