Found problems: 190
2016 CMIMC, 8
Let $N$ be the number of triples of positive integers $(a,b,c)$ with $a\leq b\leq c\leq 100$ such that the polynomial \[P(x)=x^2+(a^2+4b^2+c^2+1)x+(4ab+4bc-2ca)\] has integer roots in $x$. Find the last three digits of $N$.
2016 CMIMC, 5
We define the $\emph{weight}$ of a path to be the sum of the numbers written on each edge of the path. Find the minimum weight among all paths in the graph below that visit each vertex precisely once:
[center][img]http://i.imgur.com/V99Eg9j.png[/img][/center]
2016 CMIMC, 1
Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.
2016 CMIMC, 4
For some integer $n > 0$, a square paper of side length $2^{n}$ is repeatedly folded in half, right-to-left then bottom-to-top, until a square of side length 1 is formed. A hole is then drilled into the square at a point $\tfrac{3}{16}$ from the top and left edges, and then the paper is completely unfolded. The holes in the unfolded paper form a rectangular array of unevenly spaced points; when connected with horizontal and vertical line segments, these points form a grid of squares and rectangles. Let $P$ be a point chosen randomly from \textit{inside} this grid. Suppose the largest $L$ such that, for all $n$, the probability that the four segments $P$ is bounded by form a square is at least $L$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.
2016 CMIMC, 1
For how many distinct ordered triples $(a,b,c)$ of boolean variables does the expression $a \lor (b \land c)$ evaluate to true?
2016 ASDAN Math Tournament, 3
Denote the dot product of two sequences $\langle x_1,\dots,x_n\rangle$ and $\langle y_1,\dots,y_n\rangle$ to be
$$x_1y_1+x_2y_2+\dots+x_ny_n.$$
Let $\langle a_1,\dots,a_n\rangle$ and $\langle b_1,\dots,b_n\rangle$ be two sequences of consecutive integers (i.e. for $1\leq i,i+1\leq n$, $a_i+1=a_{i+1}$ and similarly for $b_i$). Minnie permutes the two sequences so that their dot product, $m$, is minimized. Maximilian permutes the two sequences so that their dot product, $M$, is maximized. Given that $m=4410$ and $M=4865$, compute $n$, the number of terms in each sequence.
2016 ASDAN Math Tournament, 2
Two concentric circles have differing radii such that a chord of the outer circle which is tangent to the inner circle has length $18$. Compute the area inside the bigger circle which lies outside of the smaller circle.
2016 NIMO Summer Contest, 9
Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by David Altizio[/i]
2016 Macedonia JBMO TST, 2
Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.
2016 ASDAN Math Tournament, 6
In the diagram below, square $ABCD$ has side length $4$. Two congruent square $EGIK$ and $FHJL$ are drawn such that $AE=FB=BG=HC=CI=JD=DK=LA=1$ and $EF=GH=IJ=KL=2$. Compute the area of the region that lies in both $EGIK$ and $FHJL$.
2016 ASDAN Math Tournament, 1
Let $f(x)=(x-1)^3$. Find $f'(0)$.
2016 CMIMC, 10
Let $\mathcal{P}$ be the unique parabola in the $xy$-plane which is tangent to the $x$-axis at $(5,0)$ and to the $y$-axis at $(0,12)$. We say a line $\ell$ is $\mathcal{P}$-friendly if the $x$-axis, $y$-axis, and $\mathcal{P}$ divide $\ell$ into three segments, each of which has equal length. If the sum of the slopes of all $\mathcal{P}$-friendly lines can be written in the form $-\tfrac mn$ for $m$ and $n$ positive relatively prime integers, find $m+n$.
2016 CMIMC, 2
Let $S = \{1,2,3,4,5,6,7\}$. Compute the number of sets of subsets $T = \{A, B, C\}$ with $A, B, C \in S$ such that $A \cup B \cup C = S$, $(A \cap C) \cup (B \cap C) = \emptyset$, and no subset contains two consecutive integers.
2016 CMIMC, 8
Suppose $ABCD$ is a convex quadrilateral satisfying $AB=BC$, $AC=BD$, $\angle ABD = 80^\circ$, and $\angle CBD = 20^\circ$. What is $\angle BCD$ in degrees?
2016 ASDAN Math Tournament, 6
A container is filled with a total of $51$ red and white balls and has at least $1$ red ball and $1$ white ball. The probability of picking up $3$ red balls and $1$ white ball, without replacement, is equivalent to the probability of picking up $1$ red ball and $2$ white balls, without replacement. Compute the original number of red balls in the container.
2016 ASDAN Math Tournament, 15
Let $a$ be the least positive integer with $20$ positive divisors and $b$ be the least positive integer with $16$ positive divisors. What is $a+b$? (Note that for any integer $n$, both $1$ and $n$ are considered divisors of $n$.)
2016 ASDAN Math Tournament, 3
Real numbers $x,y,z$ form an arithmetic sequence satisfying
\begin{align*}
x+y+z&=6\\
xy+yz+zx&=10.
\end{align*}
What is the absolute value of their common difference?
2016 ASDAN Math Tournament, 10
Let $\mathcal{S}$ be the set of all possible $9$-digit numbers that use $1,2,3,\dots,9$ each exactly once as a digit. What is the probability that a randomly selected number $n$ from $\mathcal{S}$ is divisible by $27$?
2016 ASDAN Math Tournament, 1
You own two cats, Chocolate and Tea. Chocolate and Tea sleep for $C$ and $T$ hours a day respectively, where $C$ and $T$ are chosen independently and uniformly at random from the interval $[5,10]$. In a given day, what is the probability that Chocolate and Tea will together sleep for a total of at least $14$ hours?
2016 ASDAN Math Tournament, 2
Three unit circles are inscribed inside an equilateral triangle such that each circle is tangent to each of the other $2$ circles and to $2$ sides of the triangle. Compute the area of the triangle.
2016 ASDAN Math Tournament, 8
Let $f$ be a differentiable function such that $f'(0)=4$ and $f(0)=3$. Compute
$$\lim_{x\rightarrow\infty}\left(\frac{f\left(\frac{1}{x}\right)}{f(0)}\right)^x.$$
2016 ASDAN Math Tournament, 1
Bill is buying cans of soup. Cans come in $2$ shapes. Can $A$ is a rectangular prism shaped can with dimensions $20\times16\times10$, and can $B$ is a cylinder shaped can with radius $10$ and height $10$. Let $\alpha$ be the volume of the larger can, and $\beta$ be the volume of the smaller can. What is $\alpha-\beta$?
2016 NIMO Summer Contest, 15
Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $100m+n$.
[i]Proposed by Michael Ren[/i]
2016 CMIMC, 8
Brice is eating bowls of rice. He takes a random amount of time $t_1 \in (0,1)$ minutes to consume his first bowl, and every bowl thereafter takes $t_n = t_{n-1} + r_n$ minutes, where $t_{n-1}$ is the time it took him to eat his previous bowl and $r_n \in (0,1)$ is chosen uniformly and randomly. The probability that it takes Brice at least 12 minutes to eat 5 bowls of rice can be expressed as simplified fraction $\tfrac{m}{n}$. Compute $m+n$.
2016 ASDAN Math Tournament, 11
Let $ABC$ be a triangle with $AB=2$, $BC=3$, and $AC=4$. Consider all lines $XY$ such that $X$ lies on $AC$, $Y$ lies on $BC$, and $\triangle XYC$ has area equal to half that of $\triangle ABC$. What is the minimum possible length of $XY$?