Olympiad problems

1959 AMC 12/AHSME, 46

A student on vacation for $d$ days observed that $(1)$ it rained $7$ times, morning or afternoon $(2)$ when it rained in the afternoon, it was clear in the morning $(3)$ there were five clear afternoons $(4)$ there were six clear mornings. Then $d$ equals: $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $

1993 AMC 8, 14

Tags: AMC

The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \[\begin{tabular}{|c|c|c|} \hline 1 & & \\ \hline & 2 & A \\ \hline & & B \\ \hline \end{tabular}\] $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

2013 AMC 8, 2

Tags: AMC

A sign at the fish market says, '50\% off, today only: half-pound packages for just \$3 per package.' What is the regular price for a full pound of fish, in dollars? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$

2009 AIME Problems, 11

Tags: geometry , analytic geometry , vector , calculus , integration , absolute value , AMC

Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer.

2010 AIME Problems, 1

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

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

2021 AMC 10 Fall, 24

Tags: geometry , 3D geometry , AMC , AMC 10 , AMC 10 B , AMC 12 , AMC 12 B

A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

2011 AMC 12/AHSME, 4

Tags: AMC

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His errorneous product was $161$. What is the correct value of the product of $a$ and $b$? $ \textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224 $

1959 AMC 12/AHSME, 24

Tags: AMC

A chemist has $m$ ounces of salt that is $m\%$ salt. How many ounces of salt must he add to make a solution that is $2m\%$ salt? $ \textbf{(A)}\ \frac{m}{100+m} \qquad\textbf{(B)}\ \frac{2m}{100-2m}\qquad\textbf{(C)}\ \frac{m^2}{100-2m}\qquad\textbf{(D)}\ \frac{m^2}{100+2m}\qquad\textbf{(E)}\ \frac{2m}{100+2m} $

1994 AMC 12/AHSME, 3

Tags: modular arithmetic , AMC , AMC 12

How many of the following are equal to $x^x+x^x$ for all $x>0$? $\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textbf{IV:}\ (2x)^{2x}$ $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

2008 AMC 10, 11

Tags: AMC

Suppose that $ \left(u_n\right)$ is a sequence of real numbers satisfying $ u_{n \plus{} 2} \equal{} 2u_{n \plus{} 1} \plus{} u_{n}$, and that $ u_3 \equal{} 9$ and $ u_6 \equal{} 128$. What is $ u_5$? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 53 \qquad \textbf{(C)}\ 68 \qquad \textbf{(D)}\ 88 \qquad \textbf{(E)}\ 104$

1963 AMC 12/AHSME, 3

Tags: AMC

If the reciprocal of $x+1$ is $x-1$, then $x$ equals: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ -1 \qquad \textbf{(D)}\ \pm 1 \qquad \textbf{(E)}\ \text{none of these}$

2022 AMC 12/AHSME, 6

Tags: 2022 AMC , 2022 AMC 10a , 2022 AMC 12A , AMC 10 , AMC 12 , AMC , arithmetic mean

A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$? $\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$

1968 AMC 12/AHSME, 4

Tags: function , AMC

Define an operation $*$ for positve real numbers as $a*b=\dfrac{ab}{a+b}$. Then $4*(4*4)$ equals: $\textbf{(A)}\ \frac{3}{4} \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \dfrac{4}{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \dfrac{16}{3} $

1991 AIME Problems, 9

Tags: trigonometry , AMC , AIME

Suppose that $\sec x+\tan x=\frac{22}7$ and that $\csc x+\cot x=\frac mn,$ where $\frac mn$ is in lowest terms. Find $m+n.$

2014 AMC 12/AHSME, 7

Tags: ratio , geometric sequence , AMC

The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term? $\textbf{(A) }1\qquad \textbf{(B) }\sqrt[7]3\qquad \textbf{(C) }\sqrt[8]3\qquad \textbf{(D) }\sqrt[9]3\qquad \textbf{(E) }\sqrt[10]3\qquad$

1968 AMC 12/AHSME, 24

Tags: geometry , ratio , AMC

A painting $18''\ \text{X}\ 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: $\textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:2 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 3:4 \qquad\textbf{(E)}\ 1:1$

1998 AMC 8, 10

Tags: AMC

Each of the letters $W$, $X$, $Y$,and $Z$ represents a different integer in the set $ \{ 1,2,3,4\} $, but not necessarily in that order. If , $ \frac{\text{W}}{\text{X}}-\frac{\text{Y}}{\text{Z}}=1 $ then the sum of $W$ and $Y$ is $ \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7 $

2021 AMC 12/AHSME Spring, 2

Tags: AMC , AMC 12 , AMC 12 A

Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers? $\textbf{(A) }$ It is never true. $\textbf{(B) }$ It is true if and only if $ab=0$. $\textbf{(C) }$ It is true if and only if $a+b\ge 0$. $\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\ge 0$. $\textbf{(E) }$ It is always true.

2012 AMC 10, 18

Tags: probability , conditional probability , expected value , AMC

Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate; in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to $p$? $ \textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$

2018 AIME Problems, 13

Tags: AMC , AIME , AIME I , geometry , similar triangles , Spiral Similarity , trigonometry

Let $\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$, respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $ X$ varies along $\overline{BC}$.

1969 AMC 12/AHSME, 5

Tags: Vieta , AMC

If a number $N$, $N\neq 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is: $\textbf{(A) }\dfrac1R\qquad \textbf{(B) }R\qquad \textbf{(C) }4\qquad \textbf{(D) }\dfrac14\qquad \textbf{(E) }-R$

2016 AMC 10, 17

Tags: probability , AMC , AMC 10 , AMC 10 A , 2016 AMC 10A

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? $\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

2017 AMC 10, 13

Tags: AMC , AMC 10 , AMC 10 B , 2017 AMC 10B

There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes? $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5 $

1964 AMC 12/AHSME, 8

Tags: AMC

The smaller root of the equation $ \left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0$ is: ${{ \textbf{(A)}\ -\frac{3}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{5}{8}\qquad\textbf{(D)}\ \frac{3}{4} }\qquad\textbf{(E)}\ 1 } $

1972 AMC 12/AHSME, 25

Tags: AMC

Inscribed in a circle is a quadrilateral having sides of lengths $25,~39,~52,$ and $60$ taken consecutively. The diameter of this circle has length $\textbf{(A) }62\qquad\textbf{(B) }63\qquad\textbf{(C) }65\qquad\textbf{(D) }66\qquad \textbf{(E) }69$