Olympiad problems

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 $k$? $\text{(A) }\frac{25}{4}\qquad\text{(B) }\frac{19}{3}\qquad\text{(C) }25\qquad\text{(D) }\frac{25}{3}\qquad\text{(E) }-\frac{7}{3}$

2014-2015 SDML (High School), 2

Tags: SDML High School , 2014-15 SDML Round 2 , Divisibility , SDML Middle School

Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?

2018-2019 SDML (High School), 8

Tags: SDML High School , 2018-2019 Round 2

Five consecutive positive integers have the property that the sum of the second, third, and fourth is a perfect square, while the sum of all five is a perfect cube. If $m$ is the first of these five integers, then the minimum possible value of $m$ satisfies $ \mathrm{(A) \ } m \leq 200 \qquad \mathrm{(B) \ } 200 < m \leq 400 \qquad \mathrm {(C) \ } 400 < m \leq 600 \qquad \mathrm{(D) \ } 600 < m \leq 800 \qquad \mathrm{(E) \ } 800 < m$

2012-2013 SDML (Middle School), 12

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-$6$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$

2014-2015 SDML (High School), 6

Tags: SDML High School , 2014-15 SDML Round 2

Let $a$ and $b$ be positive reals such that $$a=1+\frac{a}{b}$$$$b=3+\frac{4+a}{b-2}$$ What is $a$? $\text{(A) }\sqrt{2}\qquad\text{(B) }2+\sqrt{2}\qquad\text{(C) }2+\sqrt{2}+\sqrt[3]{2}\qquad\text{(D) }\sqrt{2}+\sqrt[3]{2}\qquad\text{(E) }\sqrt[3]{2}$

2018-2019 SDML (High School), 9

Tags: SDML High School , 2018-2019 Round 2 , geometry

Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\overline{BC}$ such that the perpendicular distances from $P$ to $\overline{AB}$ and $\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\triangle ABC$, in cm$^2$, is $ \mathrm{(A) \ } 1254 \qquad \mathrm{(B) \ } 1640 \qquad \mathrm {(C) \ } 1950 \qquad \mathrm{(D) \ } 2535 \qquad \mathrm{(E) \ } 2942$

2018-2019 SDML (High School), 8

Tags: NEEDS DIAGRAM , SDML High School , 2018-2019 Round 2 , geometry , rhombus

The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square. [DIAGRAM NEEDED]

2012-2013 SDML (High School), 5

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? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

2011-2012 SDML (High School), 2

Tags: SDML High School , 2011-12 SDML Round 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

Tags: geometry , 3D geometry , 2011-12 SDML Round 2 , SDML High School

How many numbers among $1,2,\ldots,2012$ have a positive divisor that is a cube other than $1$? $\text{(A) }346\qquad\text{(B) }336\qquad\text{(C) }347\qquad\text{(D) }251\qquad\text{(E) }393$

2014-2015 SDML (High School), 5

Tags: SDML High School , 2014-15 SDML Round 1

Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers? $\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$

2014-2015 SDML (High School), 6

Tags: algebra , polynomial , sum of roots , 2014-15 SDML Round 2 , SDML High School

Find the largest integer $k$ such that $$k\leq\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\cdots+\sqrt[2015]{\frac{2015}{2014}}.$$

2014-2015 SDML (High School), 12

Tags: SDML High School , 2014-15 SDML Round 2

An ant starts at the bottom left corner of a $5\times5$ grid of dots and walks to the top right corner. It can walk from one dot to any dot that is horizontally or vertically adjacent to it. If it never walks between the same pair of dots twice, what is the length of the longest path the ant can take? $\text{(A) }30\qquad\text{(B) }31\qquad\text{(C) }32\qquad\text{(D) }33\qquad\text{(E) }34$

2018-2019 SDML (High School), 13

Tags: NEEDS DIAGRAM , SDML High School , 2018-2019 Round 2 , geometry , 3D geometry

A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube? [NEEDS DIAGRAM] $ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$

2014-2015 SDML (High School), 1

Tags: 2014-15 SDML Round 2 , SDML High School , colorings

How many ways are there to color the vertices of a square green, red, or blue so that no two adjacent vertices have the same color? (Two colorings are considered different even if one coloring can be rotated to product the other coloring.)

2011-2012 SDML (High School), 1

Tags: SDML High School , 2011-12 SDML Round 2

If $\left(0.67\right)^x=0.5$, then find the value of $16\cdot\left(0.67\right)^{3x}$. $\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }16\qquad\text{(D) }64\qquad\text{(E) }128$

2018-2019 SDML (High School), 6

Tags: SDML High School , 2018-2019 Round 2

Find the largest prime $p$ less than $210$ such that the number $210 - p$ is composite.

2012-2013 SDML (Middle School), 9

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? $\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$

2014-2015 SDML (High School), 8

Tags: analytic geometry , 2014-15 SDML Round 2 , SDML High School

A penny is placed in the coordinate plane $\left(0,0\right)$. The penny can be moved $1$ unit to the right, $1$ unit up, or diagonally $1$ unit to the right and $1$ unit up. How many different ways are there for the penny to get to the point $\left(5,5\right)$? $\text{(A) }8\qquad\text{(B) }25\qquad\text{(C) }99\qquad\text{(D) }260\qquad\text{(E) }351$

2012-2013 SDML (Middle School), 6

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?