Found problems: 85335
Kara rolls a six-sided die, and if on that first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled?
[i]Proposed by Andrew Wu[/i]
Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that
\[
\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad
\angle CFB = 2 \angle ACB.
\]
Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum
\[
\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.
\]
The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done.
[i]Yudi Satria, Jakarta[/i]
Let the circles $\omega$ and $\omega'$ intersect in points $A$ and $B$. The tangent to circle $\omega$ at $A$ intersects $\omega'$ at $C$ and the tangent to circle $\omega'$ at $A$ intersects $\omega$ at $D$. Suppose that the internal bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $E$ and $F$, respectively, and the external bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $X$ and $Y$, respectively. Prove that the perpendicular bisector of $XY$ is tangent to the circumcircle of triangle $BEF$.
[i]Proposed by Mahdi Etesami Fard[/i]
A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.
There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point.
a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least
$\lfloor \frac{(a-1)(a-2)}{10} \rfloor$
intersection points.
b) Find all $a$ for which the evaluation in a) is the best possible.
Find all integer solutions of the equation $14^x - 3^y = 2015$.
(Malik Talbi)
Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.
$f(x)$ is a function defined on $\mathbb{R}$, satisfying: $f(10+x)=f(10-x),f(20-x)=-f(20+x)$. Then, $f(x)$ is
$\text{(A)}$even function, but not periodic function
$\text{(B)}$even function, and periodic function
$\text{(C)}$odd function, but not periodic function
$\text{(D)}$odd function, and periodic function
Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear
In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up.
Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game.
Determine whether one of the players can force a victory.
[img]https://cdn.artofproblemsolving.com/attachments/6/e/5e80f60f110a81a74268fded7fd75a71e07d3a.png[/img]
[u]Round 1[/u]
[b]p1.[/b] Tom and Jerry stole a chain of $7$ sausages and are now trying to divide the bounty. They take turns biting the sausages at one of the connections. When one of them breaks a connection, he may eat any single sausages that may fall out. Tom takes the first bite. Each of them is trying his best to eat more sausages than his opponent. Who will succeed?
[b]p2. [/b]The King of the Mountain Dwarves wants to light his underground throne room by placing several torches so that the whole room is lit. The king, being very miserly, wants to use as few torches as possible. What is the least number of torches he could use? (You should show why he can't do it with a smaller number of torches.)
This is the shape of the throne room:
[img]https://cdn.artofproblemsolving.com/attachments/b/2/719daafd91fc9a11b8e147bb24cb66b7a684e9.png[/img]
Also, the walls in all rooms are lined with velvet and do not reflect the light. For example, the picture on the right shows how another room in the castle is partially lit.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/0f6971274e8c2ff3f2d0fa484b567ff3d631fb.png[/img]
[b]p3.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests.
One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table."
"But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor.
Now Pooh can tell how many knights are at the table. Can you?
[b]p4.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$.
[b]p5.[/b] There are $40$ piles of stones with an equal number of stones in each. Two players, Ann and Bob, can select any two piles of stones and combine them into one bigger pile, as long as this pile would not contain more than half of all the stones on the table. A player who can’t make a move loses. Ann goes first. Who wins?
[u]Round 2[/u]
[b]p6.[/b] In a galaxy far, far away, there is a United Galactic Senate with $100$ Senators. Each Senator has no more than three enemies. Tired of their arguments, the Senators want to split into two parties so that each Senator has no more than one enemy in his own party. Prove that they can do this. (Note: If $A$ is an enemy of $B$, then $B$ is an enemy of $A$.)
[b]p7.[/b] Harry has a $2012$ by $2012$ chessboard and checkers numbered from $1$ to $2012 \times 2012$. Can he place all the checkers on the chessboard in such a way that whatever row and column Professor Snape picks, Harry will be able to choose three checkers from this row and this column such that the product of the numbers on two of the checkers will be equal to the number on the third?
[img]https://cdn.artofproblemsolving.com/attachments/b/3/a87d559b340ceefee485f41c8fe44ae9a59113.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $\eta \in [0, 1]$ be a relative measure of material absorbence. $\eta$ values for materials combined together are additive. $\eta$ for a napkin is $10$ times that of a sheet of paper, and a cardboard roll has $\eta = 0.75$. Justin can create a makeshift cup with $\eta = 1$ using $50$ napkins and nothing else. How many sheets of paper would he need to add to a cardboard roll to create a makeshift cup with $\eta = 1$?
The five numbers $17$, $98$, $39$, $54$, and $n$ have a mean equal to $n$. Find $n$.
Another round, another diagram...
[asy]
unitsize(1cm);
defaultpen(linewidth(0.45));
real[][] arr = {
{0,0,0,0},
{0,0,0,0},
{0,0,0,0},
{0,0,0,0}};
for (int i=0; i<4; ++i){
for (int j=0; j<4; ++j){
if(arr[3-j][i] != 0){
label((string) arr[3-j][i], (i+0.5, j+0.5));
}
}
}
label("$+$", (-0.5, 4.5), dir(-45));
label("$-$", (4.5, -0.5), dir(135));
label("\Large 13", (-0.5, 3.5));
label("\Large 28", (-0.5, 2.5));
label("\Large 23", (-0.5, 0.5));
label("\Large 7", (4.5, 3.5));
label("\Large 8", (4.5, 1.5));
label("\Large 8", (4.5, 0.5));
label("\Large 12", (3.5, -0.5));
label("\Large 12", (2.5,-0.5));
label("\Large 7", (0.5, -0.5));
label("\Large 23", (3.5, 4.5 ));
label("\Large 25", (2.5,4.5));
label("\Large 28", (1.5,4.5));
label("\Large 13", (0.5,4.5));
for(int i = 1; i <= 3; ++i){
draw((i, 0)--(i, 4));
draw((0, i)--(4, i));
}
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle, linewidth(1.5));
draw((-0.8,-0.8)--(0,0), linewidth(1.5));
draw((4,4 )--(4.8,4.8), linewidth(1.5));
[/asy]
Use [code]\begin{asy}
\end{asy}[/code]environment to render the diagram correctly in a latex document. Remember to write [code]\usepackage{asymptote}[/code] in the preamble.
And of course, replace the 0's in the array at the beginning of the code with the numbers you wish to fill it in with.
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by [i]Nikola Velov, Macedonia[/i]
Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?
Let \( A \) and \( B \) be positive integers, and let \( S \) be a set of positive integers with the following properties:
(1) For every non-negative integer $k$, $\text{ } A^k \in S$;
(2) If a positive integer $ n \in S$, then every positive divisor of $ n$ is in $S$;
(3) If $m ,n \in S$ and $m,n$ are coprime, then $mn \in S$;
(4) If $n \in S$, then $An + B \in S$.
Prove that all positive integers coprime to \( B \) are in \( S \).
Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$
a) Prove that $A,P,Q$ are collinear.
b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.
Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.
Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.
Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
Let $n$ be a positive integer. The numbers $1, 2, \dots, 2n+1$ are arranged in a circle in that order, and some of them are [i] marked[/i].
We define, for each $k$ such that $1\leq k \leq 2n+1$ , the interval $I_k$ to be the closed circular interval starting at $k$ and ending in $k+n$ (taking remainders mod(2n+1)). We call in interval [i]magical[/i] if it contains strictly more than half of all the marked elements.
Prove that the following two statements are equivalent:
1. At least $n+1$ of the intervals $I_1, I_2, \dots, I_{2n+1}$ are magical
2. The number of marked numbers is odd
Let $\omega_1$ and $\omega_2$ be two circles that intersect at two points: $A$ and $B$. Let $C$ and $E$ be on $\omega_1$, and $D$ and $F$ be on $\omega_2$ such that $CD$ and $EF$ meet at $B$ and the three lines $CE$, $DF$, and $AB$ concur at a point $P$ that is closer to $B$ than $A$. Let $\Omega$ denote the circumcircle of $\triangle DEF$. Now, let the line through $A$ perpendicular to $AB$ hit $EB$ at $G$, $GD$ hit $\Omega$ at $J$, and $DA$ hit $\Omega$ again at $I$. A point $Q$ on $IE$ satisfies that $CQ=JQ$. If $QJ=36$, $EI=21$, and $CI=16$, then the radius of $\Omega$ can be written as $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of a prime, and $\gcd(a, c) = 1$. Find $a+b+c$.
[i]Proposed by Kevin Zhao[/i]