Found problems: 32
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]
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 $$\frac{d}{a+b}+\frac{a}{c+d}.$$
2012-2013 SDML (Middle School), 12
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
2012-2013 SDML (High School), 5
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product?
$\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$
2012-2013 SDML (Middle School), 9
If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team?
$\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$
2012-2013 SDML (Middle School), 6
What is the largest two-digit integer for which the product of its digits is $17$ more than their sum?