Found problems: 158
2017 ASDAN Math Tournament, 1
Suppose $(x+y)^2=25$ and $(x-y)^2=1$. Compute $xy$.
2017 CMIMC Individual Finals, 1
Jesse has ten squares, which are labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?
2017 ASDAN Math Tournament, 4
How many $6$-digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit.
2017 ASDAN Math Tournament, 10
The perimeter of an isosceles trapezoid is $24$. If each of the legs is two times the length of the shorter base and is two-thirds the length of the longer base, what is the area of the trapezoid?
2017 ASDAN Math Tournament, 6
The sum
$$\sum_{n=0}^{2016\cdot2017^2}2018^n$$
can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$.
2017 CMIMC Individual Finals, 2
Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$.
2017 CMIMC Individual Finals, 2
Define
\[f(h,t) =
\begin{cases}
8h & h = t \\
(h-t)^2 & h \neq t.
\end{cases}\]
Cody plays a game with a fair coin, where he begins by flipping it once. At each turn in the game, if he has flipped $h$ heads and $t$ tails and $h + t < 6$, he can choose either to stop and receive $f(h,t)$ dollars or he can flip the coin again; if $h + t = 6$ then the game ends and he receives $f(h,t)$ dollars. If Cody plays to maximize expectancy, how much money, in dollars, can he expect to win from this game?
2017 CMIMC Team, 6
George is taking a ten-question true-false exam, where the answer key has been selected uniformly at random; however, he doesn't know any of the answers! Luckily, a friend has helpfully hinted that no two consecutive questions have true as the correct answer. If George takes the exam and maximizes the expected number of questions he gets correct, how many of his answers are expected to be right?
2017 ASDAN Math Tournament, 7
For real numbers $x,y$ satisfying $x^2+y^2-4x-2y+4=0$, what is the greatest value of
$$16\cos^2\sqrt{x^2+y^2}+24\sin\sqrt{x^2+y^2}?$$
2017 CMIMC Team, 1
Find the integer $n$ such that
\[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual, $\lfloor\cdot\rfloor$ denotes the floor function.
2017 ASDAN Math Tournament, 1
In $\triangle ABC$, we have $\angle ABC=20^\circ$. In addition, $D$ is drawn on $\overline{AB}$ such that $AC=CD=BD$. Compute $\angle ACD$ in degrees.
2017 CMIMC Number Theory, 9
Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\]
2017 ASDAN Math Tournament, 9
Eddy owns $5$ different cats, and has $9$ fish to distribute among the cats. Each cat gets at least $1$ fish and at most $3$ fish. If the fish are indistinguishable, how many ways can Eddy distribute the $9$ fish among the $5$ cats?
2017 CMIMC Team, 7
Define $\{p_n\}_{n=0}^\infty\subset\mathbb N$ and $\{q_n\}_{n=0}^\infty\subset\mathbb N$ to be sequences of natural numbers as follows:
[list]
[*]$p_0=q_0=1$;
[*]For all $n\in\mathbb N$, $q_n$ is the smallest natural number such that there exists a natural number $p_n$ with $\gcd(p_n,q_n)=1$ satisfying \[\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.\]
[/list]
Find $q_3$.
2017 ASDAN Math Tournament, 8
Let $\triangle ABC$ be a right triangle with right angle $\angle ACB$. Square $DEFG$ is contained inside triangle $ABC$ such that $D$ lies on $AB$, $E$ lies on $BC$, $F$ lies on $AC$, $AD=AF$, and $GA=GD=GF$. Suppose that $CE=2$. If $M$ is the area of triangle $ABC$ and $N$ is the area of square $DEFG$, compute $M-N$.
2017 CMIMC Team, 9
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.
2017 ASDAN Math Tournament, 10
Alice lives on a continent with $6$ countries labeled $1$ through $6$. Each country randomly chooses one other country to allow entry from. Alice can travel to any country that allows entry from the country she is currently in, and can travel along a path through multiple countries in this manner. If Alice starts in county $1$, what is the expected number of countries that she can reach (including country $1$)?
2017 ASDAN Math Tournament, 4
An ant starts at corner $A$ of a square room $ABCD$ with side length $2\sqrt{2}$. In the middle of the room, there is a circular pillar of radius $1$ centered at the center of $ABCD$. What is the minimum distance it has to travel to get to corner $C$?
2017 Bundeswettbewerb Mathematik, 2
What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?
2017 Greece JBMO TST, Source
[url=https://artofproblemsolving.com/community/c675547][b]Greece JBMO TST 2017[/b][/url]
[url=http://artofproblemsolving.com/community/c6h1663730p10567608][b]Problem 1[/b][/url]. Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that
$$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$
Also, find the values of $a,b,c$ for which the equality happens.
[url=http://artofproblemsolving.com/community/c6h1663731p10567619][b]Problem 2[/b][/url]. Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$.
[url=http://artofproblemsolving.com/community/c6h1663732p10567627][b]Problem 3[/b][/url]. Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$.
[url=http://artofproblemsolving.com/community/c6h1663734p10567640][b]Problem 4[/b][/url]. Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue.
[list=1]
[*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there?
[*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color.
[*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red?
[/list]
2017 ASDAN Math Tournament, 2
Eric has $2$ boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up $\tfrac{1}{2}$ of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up $\tfrac{1}{3}$ of the apples in the second box. Suppose that Eric has $28$ apples in total. How many red apples does Eric have?
2017 CMIMC Team, 8
Alice and Bob have a fair coin with sides labeled $C$ and $M$, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two $C$'s then an $M$, they have $CCM$ recorded. They play the following game: Alice chooses a four-character string $\mathcal A$, then Bob chooses two distinct three-character strings $\mathcal B_1$ and $\mathcal B_2$ such that neither is a substring of $\mathcal A$. Bob wins if $\mathcal A$ shows up in the running record before either $\mathcal B_1$ or $\mathcal B_2$ do, and otherwise Alice wins. Given that Alice chooses $\mathcal A = CMMC$ and Bob plays optimally, compute the probability that Bob wins.
2017 Kazakhstan NMO, Problem 2
For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds
$$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$
2017 CMIMC Computer Science, 4
How many complete directed graphs with vertex set $V=\{1,2,3,4,5,6\}$ contain no $3$-cycles?
A graph is $\textit{directed}$ if all edges have a direction (e.g. from $u$ to $v$ rather than between $u$ and $v$), and $\textit{complete}$ if every pair of vertices has an edge between them. Further, a $\textit{3-cycle}$ in a directed graph is a triple $(u,v,w)$ of vertices such that there are edges from $u$ to $v$, $v$ to $w$, and $w$ to $u$.
2017 ASDAN Math Tournament, 3
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$. Find $ab$.