Found problems: 23
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?
$\text{(A) }2\qquad\text{(B) }\frac{7}{3}\qquad\text{(C) }\frac{5}{2}\qquad\text{(D) }4\qquad\text{(E) }\frac{5}{3}$
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
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$
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?
$\text{(A) }201_b\qquad\text{(B) }261_b\qquad\text{(C) }281_b\qquad\text{(D) }289_b\qquad\text{(E) }341_b$
2011-2012 SDML (High School), 8
The distinct positive integers $a$ and $b$ have the property that $$\frac{a+b}{2},\quad\sqrt{ab},\quad\frac{2}{\frac{1}{a}+\frac{1}{b}}$$ are all positive integers. Find the smallest possible value of $\left|a-b\right|$.
2011-2012 SDML (High School), 12
Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?
$\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$
2011-2012 SDML (High School), 4
What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$?
$\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$
2011-2012 SDML (High School), 4
In triangle $ABC$, $AB=3$, $AC=5$, and $BC=4$. Let $P$ be a point inside triangle $ABC$, and let $D$, $E$, and $F$ be the projections of $P$ onto sides $BC$, $AC$, and $AB$, respectively. If $PD:PE:PF=1:1:2$, then find the area of triangle $DEF$. (Express your answer as a reduced fraction.)
(will insert image here later)
2011-2012 SDML (High School), 5
What is the greatest number of regions into which four planes can divide three-dimensional space?
2011-2012 SDML (High School), 11
Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$
2011-2012 SDML (High School), 3
The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)
2011-2012 SDML (High School), 7
Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. Find the maximum value of $x^4y+xy^4$.
2011-2012 SDML (High School), 3
Two standard six-sided dice are tossed. What is the probability that the sum of the numbers is greater than $7$?
$\text{(A) }1\qquad\text{(B) }\frac{5}{12}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{4}{9}\qquad\text{(E) }\frac{7}{36}$
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?
$\text{(A) }16\qquad\text{(B) }18\qquad\text{(C) }20\qquad\text{(D) }21\qquad\text{(E) }24$
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 $f\left(2x-1\right)=6$.
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]
$\text{(A) }30^{\circ}\qquad\text{(B) }75^{\circ}\qquad\text{(C) }45^{\circ}\qquad\text{(D) }60^{\circ}\qquad\text{(E) }15^{\circ}$
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?
$\text{(A) cactus, orchid, dieffenbachia}\qquad\text{(B) dieffenbachia, cactus, orchid}$
$\text{(C) dieffenbachia, orchid, cactus}\qquad\text{(D) orchid, dieffenbachia, cactus}$
$\text{(E) orchid, cactus, dieffenbachia}$
2011-2012 SDML (High School), 7
The line that is tangent to the circle $x^2+y^2=25$ at the point $\left(3,4\right)$ intersects the $x$-axis at $\left(k,0\right)$. What is $k$?
$\text{(A) }\frac{25}{4}\qquad\text{(B) }\frac{19}{3}\qquad\text{(C) }25\qquad\text{(D) }\frac{25}{3}\qquad\text{(E) }-\frac{7}{3}$
2011-2012 SDML (High School), 2
The $120$ permutations of the word BORIS are arranged in alphabetical order, from BIORS to SROIB. What is the $60$th word in this list?
2011-2012 SDML (High School), 14
How many numbers among $1,2,\ldots,2012$ have a positive divisor that is a cube other than $1$?
$\text{(A) }346\qquad\text{(B) }336\qquad\text{(C) }347\qquad\text{(D) }251\qquad\text{(E) }393$
2011-2012 SDML (High School), 1
If $\left(0.67\right)^x=0.5$, then find the value of $16\cdot\left(0.67\right)^{3x}$.
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }16\qquad\text{(D) }64\qquad\text{(E) }128$