Found problems: 190
2016 CMIMC, 3
Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$, and $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Let point $P$ be the unique point in the plane $ABC$ such that $\triangle PBM\sim\triangle PNC$. What is $AP$?
2016 CMIMC, 1
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\left\lceil \sqrt{s} \right\rceil \mid s \in S\}$. Find the number of sets $T$ such that $\vert f(T) \vert = 2$ and $f(f(T)) = \{2\}$.
2016 CMIMC, 10
For all positive integers $m\geq 1$, denote by $\mathcal{G}_m$ the set of simple graphs with exactly $m$ edges. Find the number of pairs of integers $(m,n)$ with $1<2n\leq m\leq 100$ such that there exists a simple graph $G\in\mathcal{G}_m$ satisfying the following property: it is possible to label the edges of $G$ with labels $E_1$, $E_2$, $\ldots$, $E_m$ such that for all $i\neq j$, edges $E_i$ and $E_j$ are incident if and only if either $|i-j|\leq n$ or $|i-j|\geq m-n$.
$\textit{Note: }$ A graph is said to be $\textit{simple}$ if it has no self-loops or multiple edges. In other words, no edge connects a vertex to itself, and the number of edges connecting two distinct vertices is either $0$ or $1$.
2016 Macedonia JBMO TST, 3
We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest number of different numbers that can be used to cover the square?
2016 CMIMC, 7
There are eight people, each with their own horse. The horses are arbitrarily arranged in a line from left to right, while the people are lined up in random order to the left of all the horses. One at a time, each person moves rightwards in an attempt to reach their horse. If they encounter a mounted horse on their way to their horse, the mounted horse shouts angrily at the person, who then scurries home immediately. Otherwise, they get to their horse safely and mount it. The expected number of people who have scurried home after all eight people have attempted to reach their horse can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.
2016 ASDAN Math Tournament, 6
Suppose we have $3$ baskets and $4$ distinguishable balls. Each ball is placed into a randomly selected basket. Compute the probability that the basket with the most balls has at least $3$ balls.
2017 NIMO Problems, 8
For each nonnegative integer $n$, we define a set $H_n$ of points in the plane as follows:
[list]
[*]$H_0$ is the unit square $\{(x,y) \mid 0 \le x, y \le 1\}$.
[*]For each $n \ge 1$, we construct $H_n$ from $H_{n-1}$ as follows. Note that $H_{n-1}$ is the union of finitely many square regions $R_1, \ldots, R_k$. For each $i$, divide $R_i$ into four congruent square quadrants. If $n$ is odd, then the upper-right and lower-left quadrants of each $R_i$ make up $H_n$. If $n$ is even, then the upper-left and lower-right quadrants of each $R_i$ make up $H_n$.
[/list]
The figures $H_0$, $H_1$, $H_2$, and $H_3$ are shown below.
[asy]
pair[]sq(int n){pair[]a;
if(n == 0)a.push((.5,.5));
else for(pair k:sq(n-1)) { pair l=1/2^(n+1)*(1,(-1)^(1+(n%2)));a.push(k+l);a.push(k-l); }
return a;}
void hh(int n,real k){
pair[] S=sq(n);real r=1/2^(n+1);
for(pair p:S)filldraw(shift(p+(k,0))*((r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle));
label("$H_"+string(n)+"$",(k+.5,-.3));}
size(7cm);
for(int i=0;i<=3;++i)hh(i,1.6*i);
[/asy]
Suppose that the point $P = (x,y)$ lies in $H_n$ for all $n \ge 0$. The greatest possible value of $xy$ is $\tfrac{m}{n}$, for relatively prime positive integers $m, n$. Compute $100m+n$.
[i]Proposed by Michael Tang[/i]
2016 ASDAN Math Tournament, 13
Suppose $\{a_n\}_{n=1}^\infty$ is a sequence. The partial sums $\{s_n\}_{n=1}^\infty$ are defined by
$$s_n=\sum_{i=1}^na_i.$$
The Cesàro sums are then defined as $\{A_n\}_{n=1}^\infty$, where
$$A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i.$$
Let $a_n=(-1)^{n+1}$. What is the limit of the Cesàro sums of $\{a_n\}_{n=1}^\infty$ as $n$ goes to infinity?
2016 ASDAN Math Tournament, 26
The Euclidean Algorithm on inputs $a$ and $b$ is a way to find the greatest common divisor $\gcd(a,b)$. Suppose WLOG that $a>b$. On each step of the Euclidan Algorithm, we solve the equation $a=bq+r$ for integers $q,r$ such that $0\leq r<b$, and repeat on $b$ and $r$. Thus $\gcd(a,b)=\gcd(b,r)$, and we repeat. If $r=0$, we are done. For example, $\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5$, because $100=15\cdot6+10$, $15=10\cdot1+5$, and $10=5\cdot2+0$. Thus, the Euclidean Algorithm here takes $3$ steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs $a,b$ where $0<a,b<10^{2016}$?
Let $C$ be the actual answer and $A$ be the answer you submit. If $\tfrac{|A-C|}{C}>\tfrac{1}{2}$, then your score will be $0$. Otherwise, your score will be given by $\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}$.
2016 CMIMC, 7
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
2016 CMIMC, 3
We have 7 buckets labelled 0-6. Initially bucket 0 is empty, while bucket $n$ (for each $1 \leq n \leq 6$) contains the list $[1,2, \ldots, n]$. Consider the following program: choose a subset $S$ of $[1,2,\ldots,6]$ uniformly at random, and replace the contents of bucket $|S|$ with $S$. Let $\tfrac{p}{q}$ be the probability that bucket 5 still contains $[1,2, \ldots, 5]$ after two executions of this program, where $p,q$ are positive coprime integers. Find $p$.
2016 CMIMC, 6
In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB=700$ and $PD=821$, what is $AC$?
2016 CMIMC, 7
Given the list \[A=[9,12,1,20,17,4,10,7,15,8,13,14],\] we would like to sort it in increasing order. To accomplish this, we will perform the following operation repeatedly: remove an element, then insert it at any position in the list, shifting elements if necessary. What is the minimum number of applications of this operation necessary to sort $A$?
2016 Kosovo Team Selection Test, 2
Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .
2016 CMIMC, 3
At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.
2016 ASDAN Math Tournament, 2
Simplify the expression
$$\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}.$$
2016 CMIMC, 4
A line with negative slope passing through the point $(18,8)$ intersects the $x$ and $y$ axes at $(a,0)$ and $(0,b)$, respectively. What is the smallest possible value of $a+b$?
2016 CMIMC, 2
Six people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability that all the coins are now heads can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$.
2016 ASDAN Math Tournament, 8
A circle with center $O$ is drawn in the first quadrant of the 2D Cartesian plane (the quadrant with both positive $x$ and $y$ values) such that it lies tangent to the $x$ and $y$-axes. A line is drawn with slope $m>1$ and passing through the origin; the line intersects the circle at two points $A$ and $B$, with $A$ closer to the origin than $B$. Suppose that $ABO$ is an equilateral triangle. Compute $m$.
2016 ASDAN Math Tournament, 8
There are $n$ integers $a$ such that $0\leq a<91$ and $a$ is a solution to $x^3+8x^2-x+83\equiv0\pmod{91}$. What is $n$?
2016 ASDAN Math Tournament, 4
Three roots of the quartic polynomial $f(x)=x^4+ax^3+bx+c$ are $-1$, $3$, and $5$. What is $a+b-c$?
2016 ASDAN Math Tournament, 3
Compute
$$\int_0^\pi\frac{1-\sin x}{1+\sin x}dx.$$
2016 ASDAN Math Tournament, 14
In the diagram to the right, squares are drawn on the side of the triangle with side lengths $5$, $6$, and $7$ as shown below. The corners of adjacent squares are then connected. What is the area of the resulting hexagon?
2016 ASDAN Math Tournament, 4
Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.
2016 CMIMC, 5
Recall that in any row of Pascal's Triangle, the first and last elements of the row are $1$ and each other element in the row is the sum of the two elements above it from the previous row. With this in mind, define the $\textit{Pascal Squared Triangle}$ as follows:
[list]
[*] In the $n^{\text{th}}$ row, where $n\geq 1$, the first and last elements of the row equal $n^2$;
[*] Each other element is the sum of the two elements directly above it.
[/list]
The first few rows of the Pascal Squared Triangle are shown below.
\[\begin{array}{c@{\hspace{7em}}
c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{2pt}}
c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{2pt}}
c@{\hspace{2pt}}c} \vspace{4pt}
\text{Row 1: } & & & & & & 1 & & & & & \\\vspace{4pt}
\text{Row 2: } & & & & & 4 & & 4 & & & & \\\vspace{4pt}
\text{Row 3: } & & & & 9 & & 8 & & 9 & & & \\\vspace{4pt}
\text{Row 4: } & & &16& &17& &17& & 16& & \\\vspace{4pt}
\text{Row 5: } & &25 & &33& &34 & &33 & &25 &
\end{array}\]
Let $S_n$ denote the sum of the entries in the $n^{\text{th}}$ row. For how many integers $1\leq n\leq 10^6$ is $S_n$ divisible by $13$?