Found problems: 233
For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.
Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.
Let $x$ and $y$ be real numbers with $x>y$ such that $x^2y^2+x^2+y^2+2xy=40$ and $xy+x+y=8$. Find the value of $x$.
Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
34. Find the sum of the ages of everyone who wrote a problem for this year’s HMMT November contest. If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - |X - Y|)$
35. Find the total number of occurrences of the digits $0, 1 \ldots , 9$ in the entire guts round (the official copy). If your
answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - \frac{|X-Y|}{2})$
36. Find the number of positive integers less than $1000000$ which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\text{max}(0, 20 - 80|1 -
\frac{X}{Y}|)$ rounded to the nearest integer.
How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.)
Let $ABCD$ be a square of side length $13$. Let $E$ and $F$ be points on rays $AB$ and $AD$ respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$. If $EF$ intersects $BC$ at $X$ and $BX = 6$, determine $DF$.
Let $ABCD$ be a parallelogram with $AB=8$, $AD=11$, and $\angle BAD=60^\circ$. Let $X$ be on segment $CD$ with $CX/XD=1/3$ and $Y$ be on segment $AD$ with $AY/YD=1/2$. Let $Z$ be on segment $AB$ such that $AX$, $BY$, and $DZ$ are concurrent. Determine the area of triangle $XYZ$.
Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.
Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.
Sherry and Val are playing a game. Sherry has a deck containing $2011$ red cards and $2012$ black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins $1$ dollar; otherwise, she loses $1$ dollar. In addition, Val must guess red exactly $2011$ times. If Val plays optimally, what is her expected profit from this game?
Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then [i]neither[/i] of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?
DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.
Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$.
Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$?
Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.
Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.
How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?
Let $ABCD$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle AOB = \angle COD = 135^\circ$, $BC=1$. Let $B^\prime$ and $C^\prime$ be the reflections of $A$ across $BO$ and $CO$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $AB^\prime C^\prime$ and $BCD$, respectively. If $M$ is the midpoint of $OH_1$, and $O^\prime$ is the reflection of $O$ about the midpoint of $MH_2$, compute $OO^\prime$.
1. If five fair coins are flipped simultaneously, what is the probability that at least three of them show heads?
2. How many perfect squares divide $10^{10}$?
3. Evaluate $\frac{2016!^2}{2015!2017!}$ . Here $n!$ denotes $1 \times 2 \times \ldots \times n$.
Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?
You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle?
[asy]
import graph; size(5cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((12,0)--(12,1)--(11,1)--(11,0)--cycle);
fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black);
draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle);
draw((15,0)--(15,1)--(16,1)--(16,0)--cycle);
fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black);
[/asy]
Find the value of
\[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]