Olympiad problems

2018-2019 SDML (High School), 15

Pentagon $ABCDE$ is such that all five diagonals $AC, BD, CE, DA,$ and $EB$ lie entirely within the pentagon. If the area of each of the triangles $ABC, BCD, CDE,$ and $DEA$ is equal to $1$ and the area of triangle $EAB$ is equal to $2$, the area of the pentagon $ABCDE$ is closest to $ \mathrm{(A) \ } 4.42 \qquad \mathrm{(B) \ } 4.44 \qquad \mathrm {(C) \ } 4.46 \qquad \mathrm{(D) \ } 4.48 \qquad \mathrm{(E) \ } 4.5$

2018-2019 SDML (High School), 4

Tags: SDML High School , 2018-2019 Round 2

How many $3$-element subsets of $\left\{1, 2, 3, \dots, 11\right\}$ are there, such that the sum of the three elements is a multiple of $3$?

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 $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

2011-2012 SDML (High School), 8

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

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

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

Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate? $\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$

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? $\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$

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 $n$? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }6\qquad\text{(E) }10$

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? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }11\qquad\text{(D) }16\qquad\text{(E) }18$

2018-2019 SDML (High School), 1

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

Find the remainder when $1! + 2! + 3! + \dots + 1000!$ is divided by $9$.

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 $n$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2011-2012 SDML (High School), 4

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

What is the imaginary part of the complex number $\frac{-4+7i}{1+2i}$? $\text{(A) }-\frac{1}{2}\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }\frac{7}{2}\qquad\text{(E) }-\frac{18}{5}$

2018-2019 SDML (High School), 5

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

Let $f(x) = x^2 + ax + b$, where $a$ and $b$ are real numbers. If $f(f(1)) = f(f(2)) = 0$, then find $f(0)$.

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 $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

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 $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$

2011-2012 SDML (High School), 4

Tags: geometry , NEEDS DIAGRAM , 2011-12 SDML Round 2 , SDML High School

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

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

What is the greatest number of regions into which four planes can divide three-dimensional space?

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), 15

Tags: quadratics , 2014-15 SDML Round 2 , SDML High School , floor function

Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$. $\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$

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?

2014-2015 SDML (High School), 3

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

A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds? $\text{(A) }148\qquad\text{(B) }154\qquad\text{(C) }167\qquad\text{(D) }217\qquad\text{(E) }223$

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? $\text{(A) }30\qquad\text{(B) }31\qquad\text{(C) }60\qquad\text{(D) }61\qquad\text{(E) }62$

2011-2012 SDML (High School), 11

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

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 $a+b$? $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$