Found problems: 120
2012-2013 SDML (High School), 15
Let $\ell$ be a line in the plane. Two circles with respective radii $2$ and $4$ are tangent to $\ell$ on the same side so that their points of tangency are distance $9$ apart. The two common internal tangents to both circles are drawn. What is the area of the triangle formed by the line $\ell$ and the two internal tangents?
$\text{(A) }\frac{25}{3}\qquad\text{(B) }\frac{26}{3}\qquad\text{(C) }9\qquad\text{(D) }\frac{28}{3}\qquad\text{(E) }\frac{29}{3}$
2018-2019 SDML (High School), 3
How many three-digit positive integers $x$ are there with the property that $x$ and $2x$ have only even digits? (One such number is $x = 220$, since $2x = 440$ and each of $x$ and $2x$ has only even digits.)
$ \mathrm{(A) \ } 16 \qquad \mathrm{(B) \ } 18 \qquad \mathrm {(C) \ } 64 \qquad \mathrm{(D) \ } 100 \qquad \mathrm{(E) \ } 125$
2012-2013 SDML (Middle School), 14
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?
$\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$
2014-2015 SDML (High School), 7
Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have.
2014-2015 SDML (High School), 7
Let $a$, $b$, and $c$ be the roots of the polynomial $$x^3+4x^2-7x-1.$$ Which of the following has roots $ab$, $bc$, and $ac$?
$\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$
2011-2012 SDML (High School), 6
A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?
2011-2012 SDML (High School), 10
Let $X=\left\{1,2,3,4,5,6\right\}$. How many non-empty subsets of $X$ do not contain two consecutive integers?
$\text{(A) }16\qquad\text{(B) }18\qquad\text{(C) }20\qquad\text{(D) }21\qquad\text{(E) }24$
2014-2015 SDML (High School), 13
How many triangles formed by three vertices of a regular $17$-gon are obtuse?
$\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$
2014-2015 SDML (High School), 13
Six points are chosen on the unit circle such that the product of the distances from any other point on the unit circle is at most $2$. Find the area of the hexagon with these six points as vertices.
$\text{(A) }\frac{1}{2}\qquad\text{(B) }\frac{3}{2}\qquad\text{(C) }\frac{\sqrt{3}}{2}\qquad\text{(D) }\frac{3\sqrt{3}}{2}\qquad\text{(E) }\frac{3+\sqrt{3}}{2}$
2011-2012 SDML (High School), 1
The function $f$ is defined by $f\left(x\right)=x^2+3x$. Find the product of all solutions of the equation $f\left(2x-1\right)=6$.
2018-2019 SDML (High School), 14
A square array of dots with $7$ rows and $7$ columns is given. Each dot is colored either blue or red. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a line segment of the same color as the dots. If they are adjacent but of difference colors, they are then joined by a purple line segment. There are $20$ red line segments and $19$ blue line segments. Find the positive difference between the maximum and minimum number of red dots.
[asy]
size(4cm);
for (int i = 0; i <= 7; ++i) {
for (int j = 0; j <= 7; ++j) {
dot((i,j));
}
}
[/asy]
$ \mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 8$
2012-2013 SDML (High School), 12
The game tic-tac is played on a $3$ by $3$ square grid between players $X$ and $O$. They take turns, and on their turn a player writes their symbol onto one empty space of the grid. A player wins if they fill a row or column with three copies of their symbol; a player filling a main diagonal does [i]not[/i] end the game in a win for that player. If the grid is filled without determining the winner, the game is a draw. Assuming player $X$ goes first and the players draw the game, how many possibilities are there for the final state of the grid?
$\text{(A) }24\qquad\text{(B) }33\qquad\text{(C) }36\qquad\text{(D) }45\qquad\text{(E) }126$
2011-2012 SDML (High School), 5
In triangle $ABC$, $\angle{BAC}=15^{\circ}$. The circumcenter $O$ of triangle $ABC$ lies in its interior. Find $\angle{OBC}$.
[asy]
size(3cm,0);
dot((0,0));
draw(Circle((0,0),1));
draw(dir(70)--dir(220));
draw(dir(220)--dir(310));
draw(dir(310)--dir(70));
draw((0,0)--dir(220));
label("$A$",dir(70),NE);
label("$B$",dir(220),SW);
label("$C$",dir(310),SE);
label("$O$",(0,0),NE);
[/asy]
$\text{(A) }30^{\circ}\qquad\text{(B) }75^{\circ}\qquad\text{(C) }45^{\circ}\qquad\text{(D) }60^{\circ}\qquad\text{(E) }15^{\circ}$
2012-2013 SDML (High School), 6
Naoki's favorite positive integer $n$ is a two-digit number with distinct digits. It also has the property that when it is divided by $10$, $12$, and $14$, the remainder has a units digit of one. What is the value of $n$?
2012-2013 SDML (High School), 5
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?
2012-2013 SDML (High School), 1
What is the largest two-digit integer for which the product of its digits is $17$ more than their sum?
2012-2013 SDML (High School), 2
Jeremy has three cups. Cup $A$ has a cylindrical shape, cup $B$ has a conical shape, and cup $C$ has a hemispherical shape. The rim of the cup at the top is a unit circle for every cup, and each cup has the same volume. If the cups are ordered from least height to greatest height, what is the ordering of the cups?
2012-2013 SDML (High School), 4
Circle $\omega_1$ with center $O_1$ has radius $3$, and circle $\omega_2$ with center $O_2$ has radius $2$ and is internally tangent to $\omega_1$. The segment $AB$ is a chord of $\omega_1$ that is tangent to $\omega_2$ at $C$ with $\angle{O_1O_2C}=45^{\circ}$. Find the length of $AB$.
[asy]
pair O_1, O_2, A, B, C;
O_1 = origin;
O_2 = (-1,0);
A = (-1, 2.82842712475);
B = (2.82842712475,-1);
C = O_2+2*dir(45);
dot(O_1);
dot(O_2);
dot(A);
dot(B);
dot(C);
draw(circle(O_1,3));
draw(circle(O_2,2));
draw(O_1--O_2);
draw(O_2--C);
draw(A--B);
label("$O_1$",O_1,SE);
label("$O_2$",O_2,SW);
label("$A$",A,NW);
label("$B$",B,SE);
label("$C$",C,NE);
[/asy]
2018-2019 SDML (High School), 4
A beam of light shines from point $L$, reflects off a reflector at point $S$, and reaches point $D$ so that $\overline{SD}$ is perpendicular to $\overline{ML}$. Then $x$ is
[DIAGRAM NEEDED]
$ \mathrm{(A) \ } 13^\circ \qquad \mathrm{(B) \ } 26^\circ \qquad \mathrm {(C) \ } 32^\circ \qquad \mathrm{(D) \ } 58^\circ \qquad \mathrm{(E) \ } 64^\circ$
2012-2013 SDML (High School), 8
Let $a$, $b$, $c$, $d$ be real numbers. Suppose that $$\frac{a}{b+c}+\frac{b}{a+d}=\frac{3}{5},\qquad\frac{b}{c+d}+\frac{c}{a+b}=1,\qquad\frac{c}{a+d}+\frac{d}{b+c}=\frac{7}{5}.$$ Find the value of $$\frac{d}{a+b}+\frac{a}{c+d}.$$
2018-2019 SDML (High School), 2
When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is
$ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$
2014-2015 SDML (High School), 3
Let $a$ and $b$ be the roots of the equation $x^2-47x+289=0$. Compute $\sqrt{a}+\sqrt{b}$.
2011-2012 SDML (High School), 15
Let $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. The greatest common divisor of $b$ and $81$ is
$\text{(A) }1\qquad\text{(B) }3\qquad\text{(C) }9\qquad\text{(D) }27\qquad\text{(E) }81$
2011-2012 SDML (High School), 6
Luna and Sam have access to a windowsill with three plants. On the morning of January $1$, $2011$, the plants were sitting in the order of cactus, dieffenbachia, and orchid, from left to right. Every afternoon, when Luna waters the plants, she swaps the two plants sitting on the left and in the center. Every evening, when Sam waters the plants, he swaps the two plants sitting on the right and in the center. What was the order of the plants on the morning of January $1$, $2012$, $365$ days later, from left to right?
$\text{(A) cactus, orchid, dieffenbachia}\qquad\text{(B) dieffenbachia, cactus, orchid}$
$\text{(C) dieffenbachia, orchid, cactus}\qquad\text{(D) orchid, dieffenbachia, cactus}$
$\text{(E) orchid, cactus, dieffenbachia}$
2014-2015 SDML (High School), 15
How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by $120$?
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }12\qquad\text{(D) }24\qquad\text{(E) }32$