Olympiad problems

2016 ASDAN Math Tournament, 3

Let $H$ be the orthocenter of triangle $ABC$, and $D$ be the foot of $A$ onto $BC$. Given that $DB=3$, $DH=2$, and $DC=6$, calculate $HA$.

2016 ASDAN Math Tournament, 10

Tags: 2016 , Guts Round

A point $P$ and a segment $AB$ with length $20$ are randomly drawn on a plane. Suppose that the probability that a randomly selected line passing through $P$ intersects segment $AB$ is $\tfrac{1}{2}$. Next, randomly choose point $Q$ on segment $AB$. What is the probability with respect to choosing $Q$ that a circle centered at $Q$ passing through $P$ contains both $A$ and $B$ in its interior?

2016 CMIMC, 3

Tags: 2016 , algebra , CMIMC

Let $\ell$ be a real number satisfying the equation $\tfrac{(1+\ell)^2}{1+\ell^2}=\tfrac{13}{37}$. Then \[\frac{(1+\ell)^3}{1+\ell^3}=\frac mn,\] where $m$ and $n$ are positive coprime integers. Find $m+n$.

2016 CMIMC, 2

Tags: CMIMC , geometry , 2016

Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?

2016 ASDAN Math Tournament, 10

Tags: 2016 , Algebra Test

Let $a_1,a_2,\dots$ be a sequence of real numbers satisfying $$\frac{a_{n+1}}{a_n}-\frac{a_{n+2}}{a_n}-\frac{a_{n+1}a_{n+2}}{a_n^2}=\frac{na_{n+2}a_{n+1}}{a_n}.$$ Given that $a_1=-1$ and $a_2=-\tfrac{1}{2}$, find the value of $\tfrac{a_9}{a_{20}}$.

2016 ASDAN Math Tournament, 3

Tags: 2016 , Guts Round

A number $n$ is $\textit{almost prime}$ if any of $n-2$, $n-1$, $n$, $n+1$, or $n+2$ is prime. Compute the smallest positive integer that is not $\textit{almost prime}$.

2016 ASDAN Math Tournament, 20

Tags: 2016 , Guts Round

Let $ABC$ be a triangle such that $AB=9$, $BC=6$, and $AC=10$. $2$ points $D_1,D_2$ are labeled on $BC$ such that $BC$ is subdivided into $3$ equal segments; $4$ points $E_1,E_2,\dots,E_4$ are labeled on $AC$ such that $AC$ is subdivided into $5$ equal segments; and $8$ points $F_1,F_2,\dots,F_8$ are labeled on $AB$ such that $AB$ is subdivided into $9$ equal segments. All possible cevians are drawn from $A$ to each $D_i$; from $B$ to each $E_j$; and from $C$ to each $F_k$. At how many points in the interior of $\triangle ABC$ do at least $2$ cevians intersect?

2016 ASDAN Math Tournament, 9

Tags: 2016 , Algebra Test

Let $P(x)$ be a monic cubic polynomial. The line $y=0$ and $y=m$ intersect $P(x)$ at points $A,C,E$ and $B,D<F$ from left to right for a positive real number $m$. If $AB=\sqrt{7}$, $CD=\sqrt{15}$, and $EF=\sqrt{10}$, what is the value of $m$?

2016 ASDAN Math Tournament, 1

Tags: 2016 , Algebra Tiebreaker

Let $x$ and $y$ be positive real numbers such that $x+y=\tfrac{1}{x}+\tfrac{1}{y}=5$. Compute $x^2+y^2$.

2016 CMIMC, 7

Tags: 2016 , CMIMC , algebra

Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations \begin{align*}(a+b)(c+d)&=143,\\(a+c)(b+d)&=150,\\(a+d)(b+c)&=169.\end{align*} Compute the smallest possible value of $a^2+b^2+c^2+d^2$.

2016 ASDAN Math Tournament, 4

Tags: 2016 , Geometry Test

Let $ABCD$ be a rectangle with $AB=9$ and $BC=3$. Suppose that $D$ is reflected across $AC$ to a point $E$. Compute the area of trapezoid $AEBC$.

2016 ASDAN Math Tournament, 4

Tags: 2016 , Guts Round

Eddy is traveling to England and needs to exchange USD to GBP (US dollars to British pounds). The current exchange rate is $1.3$ USD for $1$ GBP. He exchanges $x$ USD to GBP and while in England, uses $\tfrac{x}{2}$ GBP. When he returns, the value of the British pound has changed so that $1$ GBP equals $\alpha$ USD. After exchanging all his remaining GBP, he notes that he has $\tfrac{x}{2}$ USD left. What is $\alpha$?

2016 CMIMC, 1

Tags: CMIMC , 2016 , combinatorics

The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?

2016 ASDAN Math Tournament, 7

Tags: 2016 , Discrete Math Test

Heesu, Xingyou, and Bill are in a class with $9$ other children. The teacher randomly arranges the children in a circle for story time. However, Heesu, Xingyou, and Bill want to sit near each other. Compute the probability that all $3$ children are seated within a consecutive group of $5$ seats.

2016 ASDAN Math Tournament, 27

Tags: 2016 , Guts Round

Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\tfrac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\tfrac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)? Let $C$ be the actual answer and $A$ be the answer you will submit. Your score will be given by $\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}$.