Olympiad problems

2017 AIME Problems, 9

Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$.

2010 AIME Problems, 4

Tags: probability , AMC , AIME , number theory , relatively prime

Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $ 100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $ 400$ feet or less to the new gate be a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

2025 AIME, 14

Tags: AIME , AIME I , geometry

Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\angle B=\angle E=60^{\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2007 ITest, 40

Tags: floor function , AMC , AIME , calculus , integration , desargues involution theorem

Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$.

2020 AIME Problems, 4

Tags: AMC , AIME , AIME I , 2020 AIME I , 2020 AMC , Divisors

Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020$, and when the last four digits are removed, the result is a divisor of $N$. For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020$. Find the sum of all the digits of all the numbers in $S$. For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.

2006 AMC 12/AHSME, 1

Tags: AMC , AIME

Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? $ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$

2015 AIME Problems, 5

Tags: AMC , AIME , AIME I

In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the remaining 8 socks at random and on Wednesday two of the reaining 6 socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2001 AIME Problems, 10

Tags: floor function , modular arithmetic , AMC , AIME , function , number theory

How many positive integer multiples of 1001 can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?

2023 AIME, 3

Tags: AMC , AIME , AIME I

A plane contains $40$ lines, no $2$ of which are parallel. Suppose there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.