Olympiad problems

2014-2015 SDML (High School), 4

Tags: geometry , 3D geometry , pyramid , 2014-15 SDML Round 1 , SDML High School , SDML Middle School

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

2014-2015 SDML (Middle School), 4

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

If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?

2014-2015 SDML (High School), 3

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

At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes? $\text{(A) }10\qquad\text{(B) }11\qquad\text{(C) }12\qquad\text{(D) }13\qquad\text{(E) }14$

2014-2015 SDML (High School), 2

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

A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?

2014-2015 SDML (High School), 1

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

If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?

2014-2015 SDML (High School), 14

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

What is the greatest integer $n$ such that $$n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}?$$ $\text{(A) }31\qquad\text{(B) }59\qquad\text{(C) }74\qquad\text{(D) }88\qquad\text{(E) }112$

2014-2015 SDML (Middle School), 12

Tags: function , real analysis , 2014-15 SDML Round 1 , SDML High School , SDML Middle School

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

2014-2015 SDML (High School), 12

Tags: algebra , polynomial , trigonometry , SDML High School , 2014-15 SDML Round 1

Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root? $\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$

2014-2015 SDML (Middle School), 8

Tags: geometry , 3D geometry , pyramid , 2014-15 SDML Round 1 , SDML High School , SDML Middle School

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

2014-2015 SDML (High School), 11

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

Kyle found the sum of the digits of $2014^{2014}$. Then, Shannon found the sum of the digits of Kyle's result. Finally, James found the sum of the digits of Shannon's result. What number did James find? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }11\qquad\text{(D) }16\qquad\text{(E) }18$

2014-2015 SDML (High School), 3

Tags: algebra , functional equation , function , 2014-15 SDML Round 1 , SDML High School

Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$

2014-2015 SDML (Middle School), 15

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

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), 8

Tags: geometry , conics , ellipse , 2014-15 SDML Round 1 , SDML High School

What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$? $\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$

2014-2015 SDML (Middle School), 2

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

Suppose $a=1332$ and $b=-222$. Find $c$ such that $\left(\frac{a}{c}\right)^3=\sqrt{b^6}$.

2014-2015 SDML (High School), 4

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

What is the maximum number of points that can be placed in the interior of an equilateral triangle of side length $2$ such that the distance between any two points is greater than one? $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$

2014-2015 SDML (High School), 6

Tags: function , real analysis , 2014-15 SDML Round 1 , SDML High School , SDML Middle School

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

2014-2015 SDML (Middle School), 3

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

Layna wants to paint a rectangular wall green, but she only has blue and yellow paint. She finds that a $2:1$ mix of blue paint to yellow paint produces the color green she wants, and she knows that one gallon of paint will cover $80$ square feet of wall. If the wall is $8$ feet tall and $21$ feet long, how many gallons of blue paint does Layna need? Express your answer as a fraction in simplest form.

2014-2015 SDML (Middle School), 5

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

A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?

2014-2015 SDML (High School), 7

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

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 $ac$? $\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$

2014-2015 SDML (High School), 13

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

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), 15

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

How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by $120$? $\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }12\qquad\text{(D) }24\qquad\text{(E) }32$

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? $\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$