Found problems: 85335
The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap AC$ and $J = HH_2 \cap H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.
A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).
In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$
Let $ABC$ be an acute-angled triangle. The circle $\Gamma$ with $BC$ as diameter intersects $AB$ and $AC$ again at $P$ and $Q$, respectively. Determine $\angle BAC$ given that the orthocenter of triangle $APQ$ lies on $\Gamma$.
Equilateral triangle $\vartriangle ABC$ is inscribed in circle $\Omega$, which has a radius of $1$. Let the midpoint of $BC$ be $M$. Line $AM$ intersects $\Omega$ again at point $D$. Let $\omega$ be the circle with diameter $MD$. Compute the radius of the circle that is tangent to BC on the same side of $BC$ as $\omega$, internally tangent to $\Omega$, and externally tangent to $\omega$.
We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.
Let $ABC$ be an acute triangle with $AB<AC$ and let $O$ be its circumcenter. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let $E$ be the intersection of the line $AB$ with the perpendicular line to $AO$ through $D$. Let $F$ be the intersection of the perpendicular line to $OC$ through $C$ with the line parallel to $AC$ and passing through $E$. Finally, let the lines $CE$ and $DF$ intersect in $G$. Show that $AG$ and $BF$ are parallel.
Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.
Find, with proof, all nonconstant polynomials $P(x)$ with real coefficients such that, for all nonzero
real numbers $z$ with $P(z)\neq 0$ and $P(\frac{1}{z}) \neq 0$ we have $$\frac{1}{P(z)}+\frac{1}{P(\frac{1} {z})}=z+\frac{1}{z}.$$
Define a sequence of positive integers $\{a_n\}$ through the recursive formula:
$a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?
Let $x_1,x_2,x_3, \dots$ be a sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots.$$ Establish necessary and sufficient conditions on $x_1$ and $x_2$ for $x_n$ to be an integer for infinitely many values of $n.$
Given that $x$, $y$ are positive integers with $x(x+1)|y(y+1)$, but neither $x$ nor $x+1$ divides either of $y$ or $y+1$, and $x^2 + y^2$ as small as possible, find $x^2 + y^2$.
Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.
In a $n\times n$ table ($n>1$) $k$ unit squares are marked.One wants to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it.For what maximum $k$ is it always possible?
Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that
\[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\]
for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$
Note: This is variant A6' of the three variants given for this problem.
A unit square is cut into four pieces that can be arranged to make an isosceles triangle as shown below. What is the perimeter of the triangle? Express your answer in simplest radical form.
[asy]
filldraw((0, 3)--(-1, 3)--(-2, 2)--(-1, 1)--cycle,lightgrey);
filldraw((0, 3)--(1, 3)--(2, 2)--(1, 1)--cycle,lightgrey);
filldraw((0, 4)--(-1, 3)--(1, 3)--cycle,grey);
draw((-1, 1)--(0,0)--(1, 1));
filldraw((4,1)--(3,2)--(2,0)--(3,0)--cycle,lightgrey);
filldraw((4,1)--(5,2)--(6,0)--(5,0)--cycle,lightgrey);
filldraw((4,1)--(3,0)--(5,0)--cycle,grey);
draw((3,2)--(4,4)--(5,2));
[/asy]
Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$?
$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$
Let $V$ be the region in the Cartesian plane consisting of all points $(x,y)$ satisfying the simultaneous conditions
$$|x|\le y\le|x|+3\text{ and }y\le4.$$Find the centroid of $V$.
The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:
Determine all integers $a$ and $b$ such that
\[(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}\]
is a perfect square.
[u]Round 5[/u]
[b]p13.[/b] For a square pyramid whose base has side length $9$, a square is formed by connecting the centroids of the four triangular faces. What is the area of the square formed by the centroids?
[b]p14.[/b] Farley picks a real number p uniformly at random in the range $\left( \frac13, \frac23 \right)$. She then creates a special coin that lands on heads with probability $p$ and tails with probability $1 - p$. She flips this coin, and it lands on heads. What is the probability that $p > \frac12$?
[b]p15.[/b] Let $ABCD$ be a quadrilateral with $\angle A = \angle C = 90^o$. Extend $AB$ and $CD$ to meet at point $P$. Given that $P B = 3$, $BA = 21$, and $P C = 1$, find $BD^2$
[u]Round 6[/u]
[b]p16.[/b] Three congruent, mutually tangent semicircles are inscribed in a larger semicircle, as shown in the diagram below. If the larger semicircle has a radius of $30$ units, what is the radius of one of the smaller semicircles?
[img]https://cdn.artofproblemsolving.com/attachments/5/e/1b73791e95dc4ed6342f0151f3f63e1b31ae3c.png[/img]
[b]p17.[/b] In isosceles trapezoid $ABCD$ with $BC \parallel AD$, the distances from $A$ and $B$ to line $CD$ are $3$ and $9$, respectively. If the distance between the two bases of trapezoid $ABCD$ is $5$, find the area of quadrilateral $ABCD$.
[b]p18.[/b] How many ways are there to tile the “$E$” shape below with dominos? A domino covers two adjacent squares.
[img]https://cdn.artofproblemsolving.com/attachments/b/b/82bdb8d8df8bc3d00b9aef9eb39e55358c4bc6.png[/img]
[u]Round 7[/u]
[b]p19.[/b] In isoceles triangle $ABC$, $AC = BC$ and $\angle ACB = 20^o$. Let $\Omega$ be the circumcircle of triangle $ABC$ with center $O$, and let $M$ be the midpoint of segment $BC$. Ray $\overrightarrow{OM}$ intersects $\Omega$ at $D$. Let $\omega$ be the circle with diameter $OD$. $AD$ intersects $\omega$ again at a point $X$ not equal to $D$. Given $OD = 2$, find the area of triangle $OXD$.
[b]p20.[/b] Find the smallest odd prime factor of $2023^{2029} + 2026^{2029} - 1$.
[b]p21.[/b] Achyuta, Alan, Andrew, Anish, and Ava are playing in the EMCC games. Each person starts with a paper with their name taped on their back. A person is eliminated from the game when anybody rips their paper off of their back. The game ends when one person remains. The remaining person then rips their paper off of their own back. At the end of the game, each person collects the papers that they ripped off. How many distinct ways can the papers be distributed at the end of the game?
[u]Round 8[/u]
[b]p22.[/b] Anthony has three random number generators, labelled $A$, $B$ and $C$.
$\bullet$ Generator$ A$ returns a random number from the set $\{12, 24, 36, 48, 60\}$.
$\bullet$ Generator $B$ returns a random number from the set $ \{15, 30, 45, 60\}$.
$\bullet$ Generator $C$ returns a random number from the set $\{20, 40, 60\}$.
He uses generator $A$, $B$, and then $C$ in succession, and then repeats this process indefinitely. Anthony keeps a running total of the sum of all previously generated numbers, writing down the new total every time he uses a generator. After he uses each machine $10 $ times, what is the average number of multiples of $60$ that Anthony will have written down?
[b]p23.[/b] A laser is shot from one of the corners of a perfectly reflective room shaped like an equilateral triangle. The laser is reflected 2497 times without shining into a corner of the room, but after the 2497th reflection, it shines directly into the corner it started from. How many different angles could the laser have been initially pointed?
[b]p24.[/b] We call a k-digit number blissful if the number of positive integers $n$ such that $n^n$ ends in that $k$-digit number happens to be nonzero and finite. What is the smallest value of $k$ such that there exists a blissful $k$-digit number?
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3131523p28369592]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\leq x\leq 2020$. Find the remainder when $N$ is divided by $1000$.
Laurie loves multiplying numbers in her head. One day she decides to multiply two $2$-digit numbers $x$ and $y$ such that $x\leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
Let $A$ be a set contains $2000$ distinct integers and $B$ be a set contains $2016$ distinct integers. $K$ is the numbers of pairs $(m,n)$ satisfying \[ \begin{cases} m\in A, n\in B\\ |m-n|\leq 1000 \end{cases} \] Find the maximum value of $K$