Found problems: 85335
Alice throws two standard dice, with $A$ being the number on her first die and $B$ being the number on her second die. She then draws the line $Ax + By = 2013$. Boris also throws two standard dice, with $C$ being the number on his first die and $D$ being the number on his second die. He then draws the line $Cx + Dy = 2014$. Compute the probability that these two lines are parallel.
Let $ABC$ be a triangle. Let $X$ be the point on side $AB$ such that $\angle{BXC} = 60^{\circ}$. Let $P$ be the point on segment $CX$ such that $BP\bot AC$. Given that $AB = 6, AC = 7,$ and $BP = 4,$ compute $CP$.
Prove that there exists, for $n \geq 4$, a set $S$ of $3n$ equal circles in space that can be partitioned into three subsets $s_5, s_4$, and $s_3$, each containing $n$ circles, such that each circle in $s_r$ touches exactly $r$ circles in $S.$
Two circles $ k_1 $ and $ k_2 $ are given with centers $ P $ and $ R $ respectively, touching externally at point $ A $. Let $ p $ be their common tangent line which does not pass trough $ A $ and touch $ k_1 $ at $ B $ and $ k_2 $ at $ C $. $ PR $ cuts $ BC $ at point $ E $ and $ k_2 $ at $ A $ and $ D $. If $ AB=2AC $ find $ \frac{BC}{DE} $.
A fair coin is repeatedly flipped until $2019$ consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently.
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*}
$\textbf{(A) } 1 \qquad
\textbf{(B) } 2 \qquad
\textbf{(C) } 3 \qquad
\textbf{(D) } 4 \qquad
\textbf{(E) } 8 $
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
Consider the functions $f$ defined on the set of integers such that
\[f(x)=f(x^2+x+1)\]
for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.
Fill each white square in with a number so that each of the $27$ three-digit numbers whose digits are all $1$, $2$, or $3$ is used exactly once. For each pair of white squares sharing a side, the two numbers must have equal digits in exactly two of the three positions (ones, tens, hundreds). Some numbers have been given to you.
You do not need to prove that your answer is the only one possible; you merely need to nd an answer that satis es the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justi cation acceptable.)
[asy]
unitsize(16);
int[][] a = {
{999, 999, 999, 000, 000, 212, 000, 000},
{000, 888, 000, 213, 888, 000, 888, 123},
{000, 888, 000, 000, 000, 000, 131, 000},
{000, 888, 121, 888, 000, 113, 888, 000},
{000, 000, 000, 000, 312, 999, 999, 999}};
for (int i = 0; i < 8; ++i) {
for (int j = 0; j < 5; ++j) {
if (a[j][i] != 999) draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] == 888) fill((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(8pt));
}
}
[/asy]
[img]https://cdn.artofproblemsolving.com/attachments/f/b/bd9d0902922cd34e6e1b089373e515df698a9f.png[/img]
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
For every pair of positive integers $(n,m)$ with $n<m$, denote $s(n,m)$ be the number of positive integers such that the number is in the range $[n,m]$ and the number is coprime with $m$. Find all positive integers $m\ge 2$ such that $m$ satisfy these condition:
i) $\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}$ for all $n=1,2,...,m-1$;
ii) $2022^m+1$ is divisible by $m^2$
There are $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visitors decide to take a trip around the empire, visiting a different city each week in some random order. $3$ of these cities are inhabited by mathematicians, who will talk to all visitors about the laws of statistics. A visitor with this knowledge has probability $0$ of buying a lottery ticket, else they have probability $0.5$ of buying one. What is the expected number of visitors who will play the Martingale Lottery?
Given is a rhombus $ABCD$ of side 1. On the sides $BC$ and $CD$ we are given the points $M$ and $N$ respectively, such that $MC+CN+MN=2$ and $2\angle MAN = \angle BAD$. Find the measures of the angles of the rhombus.
[i]Cristinel Mortici[/i]
Let $a$ and $b$ be two positive real numbers such that $a^{2007} = a + 1$ and $b^{4014} = b + 3a$. Determine whether $a > b$ or $b > a$.
Find all the positive integers $x$ and $y$ that satisfy the equation $x(x - y) = 8y - 7$
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.
[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$.
[b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials.
[b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace:
$\bullet$ two candies in the box with one chocolate bar,
$\bullet$ two muffins in the box with one chocolate bar,
$\bullet$ two chocolate bars in the box with one candy and one muffin,
$\bullet$ one candy and one chocolate bar in the box with one muffin,
$\bullet$ one muffin and one chocolate bar in the box with one candy.
Is it possible that after some time it remains only one object in the box?
[b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points?
[b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
In $\triangle ABC, \angle BAC$ is a right angle. $BP$ and $CQ$ are bisectors of $\angle B$ and $\angle C$ respectively, which intersect $AC$ and $AB$ at $P$ and $Q$ respectively. Two perpendicular segments $PM$ and $QN$ are drawn on $BC$ from $P$ and $Q$ respectively. Find the value of $\angle MAN$ with proof.
If $ a,b,c>0$ and $ a\plus{}b\plus{}c\equal{}3$ prove that:
$ \frac{{a^3 \left( {a \plus{} b} \right)}}{{a^2 \plus{} ab \plus{} b^2 }} \plus{} \frac{{b^3 \left( {b \plus{} c} \right)}}{{b^2 \plus{} bc \plus{} c^2 }} \plus{} \frac{{c^3 \left( {c \plus{} a} \right)}}{{c^2 \plus{} ac \plus{} a^2 }} \ge 2$
P.s.
I dont know if it has been posted before.
Let $\alpha=1+a_1x+a_2x^2+\ldots$ be a formal power series with coefficients in the field of two elements. Let
$$a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}$$(For example, $a_{36}=1$ since $36=100100_2$)
Prove that $\alpha^3+x\alpha+1=0$.
Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 75 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ \text{None}$
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.
Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$. Prove that if $y_1 < y_2 < \cdots < y_n$, then \[ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. \]
Let $\Gamma$ be the circle of an acute triangle $ABC$ and let $H$ be its orthocenter. The circle $\omega$ with diameter $AH$ cuts $\Gamma$ at point $D$ ($D\ne A$). Let $M$ be the midpoint of the smaller arc $BC$ of $\Gamma$ . Let $N$ be the midpoint of the largest arc $BC$ of the circumcircle of the triangle $BHC$. Prove that there is a circle that passes through the points $D, H, M$ and $N$.
Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$