Olympiad problems

2017 Purple Comet Problems, 18

Tags: Purple Comet

Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2016 Purple Comet Problems, 19

Tags: Purple Comet

Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$.

2016 Purple Comet Problems, 11

Tags: Purple Comet

Find the number of three-digit positive integers which have three distinct digits where the sum of the digits is an even number such as 925 and 824.

2016 Purple Comet Problems, 8

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The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles. [center][img]https://i.snag.gy/Ik094i.jpg[/img][/center]

2021 Purple Comet Problems, 17

Tags: Purple Comet , logarithms

For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, 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 $$8a^3+\frac{1}{a^3}.$$

2016 Purple Comet Problems, 4

Tags: Purple Comet

One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the perimeter of the rectangle.

2015 Purple Comet Problems, 18

Tags: Purple Comet

Deﬁne the determinant $D_1$ = $|1|$, the determinant $D_2$ = $|1 1|$ $|1 3|$ , and the determinant $D_3=$ |1 1 1| |1 3 3| |1 3 5| . In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its ﬁrst row and ﬁrst column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015.

2017 Purple Comet Problems, 2

Tags: Purple Comet , geometry

The figure below was made by gluing together 5 non-overlapping congruent squares. The figure has area 45. Find the perimeter of the figure. [center][img]https://snag.gy/ZeKf4q.jpg[/center][/img]

2015 Purple Comet Problems, 1

Tags: Purple Comet

The ﬁve numbers $17$, $98$, $39$, $54$, and $n$ have a mean equal to $n$. Find $n$.

2022 Purple Comet Problems, 15

Tags: Purple Comet

Find the number of rearrangements of the nine letters $\text{AAABBBCCC}$ where no three consecutive letters are the same. For example, count $\text{AABBCCABC}$ and $\text{ACABBCCAB}$ but not $\text{ABABCCCBA}.$

2016 Purple Comet Problems, 6

Tags: Purple Comet

Find the number of three-digit positive integers where the digits are three different prime numbers. For example, count 235 but not 553.

2016 Purple Comet Problems, 14

Tags: Purple Comet , number theory , prime numbers

Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.

2016 Purple Comet Problems, 15

Tags: Purple Comet

The real numbers $x$, $y$, and $z$ satisfy the system of equations $$x^2 + 27 = -8y + 10z$$ $$y^2 + 196 = 18z + 13x$$ $$z^2 + 119 = -3x + 30y$$ Find $x + 3y + 5z$.

2022 Purple Comet Problems, 20

Tags: geometry , 3D geometry , Purple Comet

Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$

2015 Purple Comet Problems, 26

Tags: probability , Purple Comet

Seven people of seven diﬀerent ages are attending a meeting. The seven people leave the meeting one at a time in random order. Given that the youngest person leaves the meeting sometime before the oldest person leaves the meeting, the probability that the third, fourth, and ﬁfth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

2021 Purple Comet Problems, 26

Tags: Purple Comet

The product $$\left(\frac{1}{2^3-1}+\frac12\right)\left(\frac{1}{3^3-1}+\frac12\right)\left(\frac{1}{4^3-1}+\frac12\right)\cdots\left(\frac{1}{100^3-1}+\frac12\right)$$ can be written as $\frac{r}{s2^t}$ where $r$, $s$, and $t$ are positive integers and $r$ and $s$ are odd and relatively prime. Find $r+s+t$.

2016 Purple Comet Problems, 20

Tags: Purple Comet

Positive integers a, b, c, d, and e satisfy the equations $$(a + 1)(3bc + 1) = d + 3e + 1$$ $$(b + 1)(3ca + 1) = 3d + e + 13$$ $$(c + 1)(3ab + 1) = 4(26-d- e) - 1$$ Find $d^2+e^2$.

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.

2015 Purple Comet Problems, 6

Tags: Purple Comet , number theory , divisibility tests

Find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple of 27. For example, 87999921 is one such number.

2015 Purple Comet Problems, 15

Tags: Purple Comet

How many positive integers less than 2015 have exactly 9 positive integer divisors?

2016 Purple Comet Problems, 13

Tags: Purple Comet

In $\triangle$ABC shown below, AB = AC, AF = EF, and EH = CH = DH = GH = DG = BG. Also, ∠CHE = ∠F GH. Find the degree measure of ∠BAC. [center][img]https://i.snag.gy/ZyxQVX.jpg[/img][/center]

2016 Purple Comet Problems, 5

Tags: Purple Comet

Julius has a set of ﬁve positive integers whose mean is 100. If Julius removes the median of the set of ﬁve numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum possible value of the largest of the ﬁve numbers Julius has.

2021 Purple Comet Problems, 20

Tags: Purple Comet

Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$. Find the maximum possible value of $AB$.

2016 Purple Comet Problems, 27

Tags: Purple Comet

A container the shape of a pyramid has a 12 × 12 square base, and the other four edges each have length 11. The container is partially filled with liquid so that when one of its triangular faces is lying on a flat surface, the level of the liquid is half the distance from the surface to the top edge of the container. Find the volume of the liquid in the container. [center][img]https://snag.gy/CdvpUq.jpg[/img][/center]