Olympiad problems

2019 Purple Comet Problems, 12

Tags: geometry , HS

The following diagram shows four adjacent $2\times 2$ squares labeled $1, 2, 3$, and $4$. A line passing through the lower left vertex of square $1$ divides the combined areas of squares $1, 3$, and $4$ in half so that the shaded region has area $6$. The difference between the areas of the shaded region within square $4$ and the shaded region within square $1$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [img]https://cdn.artofproblemsolving.com/attachments/7/4/b9554ccd782af15c680824a1fbef278a4f736b.png[/img]

2019 Purple Comet Problems, 6

Tags: geometry , HS

A pentagon has four interior angles each equal to $110^o$. Find the degree measure of the fifth interior angle.

2018 Purple Comet Problems, 29

Tags: number theory , HS

Find the three-digit positive integer $n$ for which $\binom n3 \binom n4 \binom n5 \binom n6 $ is a perfect square.

2022 Purple Comet Problems, 4

Tags: Purple Comet , HS

Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus.

2022 Purple Comet Problems, 1

Tags: Purple Comet , HS

Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.

2022 Purple Comet Problems, 6

Tags: Purple Comet , HS

Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$.

2022 Purple Comet Problems, 8

Tags: Purple Comet , HS

The product $$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$ can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.

2022 Purple Comet Problems, 2

Tags: Purple Comet , HS

Call a date mm/dd/yy $\textit{multiplicative}$ if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, $01/21/21$, $03/07/21$, and $07/03/21$ are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.

2022 Purple Comet Problems, 7

Tags: Purple Comet , HS

In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$.

2019 Purple Comet Problems, 18

Tags: combinatorics , probability , HS

A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all five balls are colored blue is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2022 Purple Comet Problems, 10

Tags: Purple Comet , HS

Let $a$ be a positive real number such that $$4a^2+\frac{1}{a^2}=117.$$ Find $$8a^3+\frac{1}{a^3}.$$

2019 Purple Comet Problems, 17

Tags: geometry , HS

The following diagram shows equilateral triangle $\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\vartriangle ABC$ a distance $\frac18 AB$ along a side of $\vartriangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has area $300$. Find the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png[/img]

2022 Purple Comet Problems, 3

Tags: Purple Comet , HS

An isosceles triangle has a base with length $12$ and the altitude to the base has length $18$. Find the area of the region of points inside the triangle that are a distance of at most 3 from that altitude.

2019 Purple Comet Problems, 30

Tags: combinatorics , HS

A [i]derangement [/i] of the letters $ABCDEF$ is a permutation of these letters so that no letter ends up in the position it began such as $BDECFA$. An [i]inversion [/i] in a permutation is a pair of letters $xy$ where $x$ appears before $y$ in the original order of the letters, but $y$ appears before $x$ in the permutation. For example, the derangement $BDECFA$ has seven inversions: $AB, AC, AD, AE, AF, CD$, and $CE$. Find the total number of inversions that appear in all the derangements of $ABCDEF$.

2022 Purple Comet Problems, 9

Tags: Purple Comet , HS

For positive integer $n$ let $z_n=\sqrt{\frac{3}{n}}+i$, where $i=\sqrt{-1}$. Find $|z_1 \cdot z_2 \cdot z_3 \cdots z_{47}|$.

2022 Purple Comet Problems, 5

Tags: Purple Comet , HS

Below is a diagram showing a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids.

2019 Purple Comet Problems, 26

Tags: 3D geometry , geometry , HS

Let $D$ be a regular dodecahedron, which is a polyhedron with $20$ vertices, $30$ edges, and $12$ regular pentagon faces. A tetrahedron is a polyhedron with $4$ vertices, $6$ edges, and $4$ triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of $D$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png[/img]