Found problems: 120
2014-2015 SDML (High School), 5
The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.
2011-2012 SDML (High School), 2
A man who is $2$ meters tall is standing $5$ meters away from a lamppost that is $6$ meters high. How long is the man's shadow cast by the lamppost, in meters?
$\text{(A) }2\qquad\text{(B) }\frac{7}{3}\qquad\text{(C) }\frac{5}{2}\qquad\text{(D) }4\qquad\text{(E) }\frac{5}{3}$
2012-2013 SDML (High School), 3
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?
2018-2019 SDML (High School), 6
For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer?
$ \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 20 \qquad \mathrm {(C) \ } 39 \qquad \mathrm{(D) \ } 49 \qquad \mathrm{(E) \ } 59$
2014-2015 SDML (High School), 8
Triangles $ABC$ and $BDC$ are such that $\angle{ABC}=\angle{BDC}=90^{\circ}$ and $\angle{DBC}=\angle{CAB}$. Let $Q$ be a point on $\overline{BD}$ such that $\overline{QC}\perp\overline{AD}$. Suppose that $BD=15$. Then $DQ$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2014-2015 SDML (High School), 4
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
2012-2013 SDML (High School), 9
Sammy and Tammy run laps around a circular track that has a radius of $1$ kilometer. They begin and end at the same point and at the same time. Sammy runs $3$ laps clockwise while Tammy runs $4$ laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly $1$ kilometer?
$\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$
2018-2019 SDML (High School), 7
In a game of Shipbattle, Willis secretly places his aircraft carrier somewhere in a $9 \times 9$ grid, represented by five consecutive squares. Two example positions are shown below.
[asy]
size(5cm);
fill((2,7)--(7,7)--(7,8)--(2,8)--cycle);
fill((5,1)--(6,1)--(6,6)--(5,6)--cycle);
for (int i = 0; i <= 9; ++i)
{
draw((i,0)--(i,9));
draw((0,i)--(9,i));
}
[/asy]
Phyllis then takes shots at the grid, one square at a time, trying to hit Willis's aircraft carrier. What is the minimum number of shots that Phyllis must take to ensure that she hits the aircraft carrier at least once?
2011-2012 SDML (High School), 13
The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$
2014-2015 SDML (Middle School), 4
If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?
2012-2013 SDML (High School), 6
A convex quadrilateral $ABCD$ is constructed out of metal rods with negligible thickness. The side lengths are $AB=BC=CD=5$ and $DA=3$. The figure is then deformed, with the angles between consecutive rods allowed to change but the rods themselves staying the same length. The resulting figure is a convex polygon for which $\angle{ABC}$ is as large as possible. What is the area of this figure?
$\text{(A) }6\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }10\qquad\text{(E) }12$
2012-2013 SDML (High School), 4
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
2014-2015 SDML (High School), 1
Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni?
$\text{(A) }200\qquad\text{(B) }300\qquad\text{(C) }400\qquad\text{(D) }500\qquad\text{(E) }600$
2014-2015 SDML (High School), 3
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }10\qquad\text{(B) }11\qquad\text{(C) }12\qquad\text{(D) }13\qquad\text{(E) }14$
2014-2015 SDML (High School), 2
A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?
2018-2019 SDML (High School), 10
If $s$ and $d$ are positive integers such that $\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d},$ then the smallest possible value of $s + d$ is
$ \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 8 \qquad \mathrm {(C) \ } 10 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 96$
2011-2012 SDML (High School), 9
The graph of the equation $x^3-2x^2y+xy^2-2y^3=0$ is the same as the graph of
$\text{(A) }x^2+y^2=0\qquad\text{(B) }x=y\qquad\text{(C) }y=2x^2-x\qquad\text{(D) }x=y^3\qquad\text{(E) }x=2y$
2011-2012 SDML (High School), 8
In a certain base $b$ (different from $10$), $57_b^2=2721_b$. What is $17_b^2$ in this base?
$\text{(A) }201_b\qquad\text{(B) }261_b\qquad\text{(C) }281_b\qquad\text{(D) }289_b\qquad\text{(E) }341_b$
2014-2015 SDML (High School), 1
If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?
2018-2019 SDML (High School), 3
In the diagram below, $\angle B = 43^\circ$ and $\angle D = 102^\circ$. Find $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F$.
[NEEDS DIAGRAM]
2018-2019 SDML (High School), 7
Given $A = \left\{1,2,3,5,8,13,21,34,55\right\}$, how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$?
$ \mathrm{(A) \ } 34 \qquad \mathrm{(B) \ } 35 \qquad \mathrm {(C) \ } 43\qquad \mathrm{(D) \ } 51 \qquad \mathrm{(E) \ } 55$
2014-2015 SDML (High School), 5
Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$?
$\text{(A) }47\qquad\text{(B) }137\qquad\text{(C) }227\qquad\text{(D) }677\qquad\text{(E) }1127$
2014-2015 SDML (High School), 14
What is the greatest integer $n$ such that $$n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}?$$
$\text{(A) }31\qquad\text{(B) }59\qquad\text{(C) }74\qquad\text{(D) }88\qquad\text{(E) }112$
2012-2013 SDML (High School), 7
Consider the shape shown below, formed by gluing together the sides of seven congruent regular hexagons. The area of this shape is partitioned into $21$ quadrilaterals, all of whose side lengths are equal to the side length of the hexagon and each of which contains a $60^{\circ}$ angle. In how many ways can this partitioning be done? (The quadrilaterals may contain an internal boundary of the seven hexagons.)
[asy]
draw(origin--origin+dir(0)--origin+dir(0)+dir(60)--origin+dir(0)+dir(60)+dir(0)--origin+dir(0)+dir(60)+dir(0)+dir(60)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)+dir(0)--origin+dir(0)+dir(60)+dir(0)+dir(60)+dir(120)+dir(60)+dir(120)+dir(180)+dir(120)+dir(180)+dir(240)+dir(180)+dir(240)+dir(300)+dir(240)+dir(300)+dir(0)+dir(300)--cycle);
draw(2*dir(60)+dir(120)+dir(0)--2*dir(60)+dir(120)+2*dir(0),dashed);
draw(2*dir(60)+dir(120)+dir(60)--2*dir(60)+dir(120)+2*dir(60),dashed);
draw(2*dir(60)+dir(120)+dir(120)--2*dir(60)+dir(120)+2*dir(120),dashed);
draw(2*dir(60)+dir(120)+dir(180)--2*dir(60)+dir(120)+2*dir(180),dashed);
draw(2*dir(60)+dir(120)+dir(240)--2*dir(60)+dir(120)+2*dir(240),dashed);
draw(2*dir(60)+dir(120)+dir(300)--2*dir(60)+dir(120)+2*dir(300),dashed);
draw(dir(60)+dir(120)--dir(60)+dir(120)+dir(0)--dir(60)+dir(120)+dir(0)+dir(60)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)+dir(240)--dir(60)+dir(120)+dir(0)+dir(60)+dir(120)+dir(180)+dir(240)+dir(300),dashed);
[/asy]
2012-2013 SDML (Middle School), 7
Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?