Olympiad problems

2012-2013 SDML (High School), 3

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

2012-2013 SDML (High School), 9

Tags: SDML High School , 2012-13 SDML Round 2

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? $\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$

2012-2013 SDML (High School), 6

Tags: geometry , SDML High School , 2012-13 SDML Round 2

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

Tags: geometry , 3D geometry , SDML High School , SDML Middle School , 2012-13 SDML Round 2

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), 7

Tags: geometry , SDML High School , 2012-13 SDML Round 2

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

Tags: SDML High School , 2012-13 SDML Round 2

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), 13

Tags: algebra , polynomial , SDML High School , 2012-13 SDML Round 2

A polynomial $P$ is called [i]level[/i] if it has integer coefficients and satisfies $P\left(0\right)=P\left(2\right)=P\left(5\right)=P\left(6\right)=30$. What is the largest positive integer $d$ such that for any level polynomial $P$, $d$ is a divisor of $P\left(n\right)$ for all integers $n$? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }6\qquad\text{(E) }10$

2012-2013 SDML (High School), 8

Tags: algebra , polynomial , SDML High School , 2012-13 SDML Round 2

A polynomial $P$ with degree exactly $3$ satisfies $P\left(0\right)=1$, $P\left(1\right)=3$, and $P\left(3\right)=10$. Which of these cannot be the value of $P\left(2\right)$? $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2012-2013 SDML (High School), 14

Tags: number theory , greatest common divisor , arithmetic sequence , SDML High School , 2012-13 SDML Round 2

A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2012-2013 SDML (Middle School), 15

Tags: geometry , SDML High School , 2012-13 SDML Round 2

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

2012-2013 SDML (Middle School), 11

Tags: geometry , 3D geometry , NEEDS DIAGRAM , SDML Middle School , 2012-13 SDML Round 2

Six different-sized cubes are glued together, one on top of the other. The bottom cube has edge length $8$. Each of the other cubes has four vertices at the midpoints of the edges of the cube below it as shown. The entire solid is then dipped in red paint. What is the total area of the red-painted surface on the solid? (will insert image here later) $\text{(A) }630\qquad\text{(B) }632\qquad\text{(C) }648\qquad\text{(D) }694\qquad\text{(E) }756$

2012-2013 SDML (Middle School), 3

Tags: factorial , SDML High School , 2012-13 SDML Round 2

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

2012-2013 SDML (Middle School), 10

Tags: number theory , prime numbers , SDML High School , 2012-13 SDML Round 2

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 (High School), 7

Tags: SDML High School , 2012-13 SDML Round 2

A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly $60$ seconds, all of the living cells simultaneously split into two identical copies of itself, and this continues to happen every $60$ seconds thereafter. After performing the experiment for awhile, the scientist realizes the population of the culture will be unbounded and quickly shuts down the experiment before the cells take over the world. What is the smallest number of cells that the experiment could have started with? $\text{(A) }30\qquad\text{(B) }31\qquad\text{(C) }60\qquad\text{(D) }61\qquad\text{(E) }62$

2012-2013 SDML (High School), 1

Tags: SDML High School , 2012-13 SDML Round 2

Let $\bullet$ be the operation such that $a\bullet b=10a-b$. What is the value of $\left(\left(\left(2\bullet0\right)\bullet1\right)\bullet3\right)$? $\text{(A) }1969\qquad\text{(B) }1987\qquad\text{(C) }1993\qquad\text{(D) }2007\qquad\text{(E) }2013$

2012-2013 SDML (High School), 3

Tags: SDML High School , 2012-13 SDML Round 2

Let $b=\log_53$. What is $\log_b\left(\log_35\right)$? $\text{(A) }-1\qquad\text{(B) }-\frac{3}{5}\qquad\text{(C) }0\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }1$

2012-2013 SDML (High School), 11

Tags: trigonometry , SDML High School , 2012-13 SDML Round 2

Suppose that $\cos\left(3x\right)+3\cos\left(x\right)=-2$. What is the value of $\cos\left(2x\right)$? $\text{(A) }-\frac{1}{2}\qquad\text{(B) }-\frac{1}{\sqrt[3]{2}}\qquad\text{(C) }\frac{1}{\sqrt[3]{2}}\qquad\text{(D) }\sqrt[3]{2}-1\qquad\text{(E) }\frac{1}{2}$

2012-2013 SDML (High School), 2

Tags: SDML High School , 2012-13 SDML Round 2

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 (High School), 10

Tags: geometry , SDML High School , 2012-13 SDML Round 2

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

2012-2013 SDML (High School), 15

Tags: geometry , SDML High School , 2012-13 SDML Round 2

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? $\text{(A) }\frac{25}{3}\qquad\text{(B) }\frac{26}{3}\qquad\text{(C) }9\qquad\text{(D) }\frac{28}{3}\qquad\text{(E) }\frac{29}{3}$

2012-2013 SDML (Middle School), 14

Tags: SDML High School , 2012-13 SDML Round 2

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? $\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$

2012-2013 SDML (High School), 12

Tags: SDML High School , 2012-13 SDML Round 2

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? $\text{(A) }24\qquad\text{(B) }33\qquad\text{(C) }36\qquad\text{(D) }45\qquad\text{(E) }126$

2012-2013 SDML (High School), 6

Tags: SDML High School , 2012-13 SDML Round 2

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

Tags: SDML High School , 2012-13 SDML Round 2

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

Tags: SDML High School , 2012-13 SDML Round 2

What is the largest two-digit integer for which the product of its digits is $17$ more than their sum?