# Difference between revisions of "User:Rowechen"

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[[2013 AIME I Problems/Problem 15|Solution]] | [[2013 AIME I Problems/Problem 15|Solution]] | ||

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==Problem 14== | ==Problem 14== | ||

For positive integers <math>n</math> and <math>k</math>, let <math>f(n, k)</math> be the remainder when <math>n</math> is divided by <math>k</math>, and for <math>n > 1</math> let <math>F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)</math>. Find the remainder when <math>\sum\limits_{n=20}^{100} F(n)</math> is divided by <math>1000</math>. | For positive integers <math>n</math> and <math>k</math>, let <math>f(n, k)</math> be the remainder when <math>n</math> is divided by <math>k</math>, and for <math>n > 1</math> let <math>F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)</math>. Find the remainder when <math>\sum\limits_{n=20}^{100} F(n)</math> is divided by <math>1000</math>. | ||

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[[2013 AIME II Problems/Problem 15|Solution]] | [[2013 AIME II Problems/Problem 15|Solution]] | ||

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==Problem 13== | ==Problem 13== | ||

On square <math>ABCD</math>, points <math>E,F,G</math>, and <math>H</math> lie on sides <math>\overline{AB},\overline{BC},\overline{CD},</math> and <math>\overline{DA},</math> respectively, so that <math>\overline{EG} \perp \overline{FH}</math> and <math>EG=FH = 34</math>. Segments <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, CGPF,</math> and <math>DHPG</math> are in the ratio <math>269:275:405:411.</math> Find the area of square <math>ABCD</math>. | On square <math>ABCD</math>, points <math>E,F,G</math>, and <math>H</math> lie on sides <math>\overline{AB},\overline{BC},\overline{CD},</math> and <math>\overline{DA},</math> respectively, so that <math>\overline{EG} \perp \overline{FH}</math> and <math>EG=FH = 34</math>. Segments <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, CGPF,</math> and <math>DHPG</math> are in the ratio <math>269:275:405:411.</math> Find the area of square <math>ABCD</math>. | ||

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[[2014 AIME I Problems/Problem 15|Solution]] | [[2014 AIME I Problems/Problem 15|Solution]] | ||

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## Revision as of 16:38, 29 May 2020

Here's the AIME compilation I will be doing:

## Contents

## Problem 3

Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person.

## Problem 4

In equiangular octagon , and . The self-intersecting octagon enclosed six non-overlapping triangular regions. Let be the area enclosed by , that is, the total area of the six triangular regions. Then , where and are relatively prime positive integers. Find .

## Problem 5

Suppose that , , and are complex numbers such that , , and , where . Then there are real numbers and such that . Find .

## Problem 7

Let be the set of all integers such that . For example, is the set . How many of the sets do not contain a perfect square?

## Problem 7

Define an ordered triple of sets to be if and . For example, is a minimally intersecting triple. Let be the number of minimally intersecting ordered triples of sets for which each set is a subset of . Find the remainder when is divided by .

**Note**: represents the number of elements in the set .

## Problem 7

At each of the sixteen circles in the network below stands a student. A total of coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.

## Problem 11

Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, *N*, of play blocks which satisfies the conditions:

(a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and

(b) There are three integers such that when , , or students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of *N* satisfying the above conditions.

## Problem 12

Let be a triangle with and . A regular hexagon with side length 1 is drawn inside so that side lies on , side lies on , and one of the remaining vertices lies on . There are positive integers and such that the area of can be expressed in the form , where and are relatively prime, and c is not divisible by the square of any prime. Find .

## Problem 11

Let , and let be the number of functions from set to set such that is a constant function. Find the remainder when is divided by .

## Problem 11

In , and . . Let be the midpoint of segment . Point lies on side such that . Extend segment through to point such that . Then , where and are relatively prime positive integers, and is a positive integer. Find .

## Problem 15

Let be the number of ordered triples of integers satisfying the conditions (a) , (b) there exist integers , , and , and prime where , (c) divides , , and , and (d) each ordered triple and each ordered triple form arithmetic sequences. Find .

## Problem 14

For positive integers and , let be the remainder when is divided by , and for let . Find the remainder when is divided by .

## Problem 15

Let be angles of an acute triangle with There are positive integers , , , and for which where and are relatively prime and is not divisible by the square of any prime. Find .

## Problem 13

On square , points , and lie on sides and respectively, so that and . Segments and intersect at a point , and the areas of the quadrilaterals and are in the ratio Find the area of square .

## Problem 15

In and . Circle intersects at and at and and at and . Given that and length where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find .