Found problems: 3632
1990 AMC 12/AHSME, 24
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?
\[ \begin{tabular}{c c c c}
{} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams and Baker} \\
\textbf{Boys:} & 71 & 81 & 79 \\
\textbf{Girls:} & 76 & 90 & ? \\
\textbf{Boys and Girls:} & 74 & 84 & \\
\end{tabular}
\]
$ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 84 \quad\textbf{(E)}\ 85 $
1972 AMC 12/AHSME, 8
If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then
$\textbf{(A) }x=0\qquad\textbf{(B) }y=1\qquad\textbf{(C) }x=0\text{ and }y=1\qquad$
$\textbf{(D) }x(y-1)=0\qquad \textbf{(E) }\text{None of these}$
2019 AMC 10, 19
What is the least possible value of
$$(x+1)(x+2)(x+3)(x+4)+2019$$where $x$ is a real number?
$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$
2020 AMC 12/AHSME, 6
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$
[asy]
import olympiad;
unitsize(25);
filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7));
filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7));
filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7));
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 6; ++j) {
pair A = (j,i);
}
}
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 6; ++j) {
if (j != 5) {
draw((j,i)--(j+1,i));
}
if (i != 4) {
draw((j,i)--(j,i+1));
}
}
}
[/asy]
$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2012 AMC 10, 9
Two integers have a sum of $26$. When two more integers are added to the first two integers the sum is $41$. Finally when two more integers are added to the sum of the previous four integers the sum is $57$. What is the minimum number of even integers among the $6$ integers?
${{ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4}\qquad\textbf{(E)}\ 5} $
2017 AMC 10, 21
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2012 AMC 8, 23
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon?
$\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}4\sqrt3 \qquad \textbf{(E)}\hspace{.05in}6\sqrt3 $
2000 USAMO, 5
Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$
1961 AMC 12/AHSME, 22
If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by:
${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $
2014 AMC 12/AHSME, 5
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$
2021 AMC 10 Spring, 20
In how many ways can the sequence $1,2,3,4,5$ be arranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
$\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 24 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 44$
2023 AIME, 14
The following analog clock has two hands that can move independently of each other.
[asy]
unitsize(2cm);
draw(unitcircle,black+linewidth(2));
for (int i = 0; i < 12; ++i) {
draw(0.9*dir(30*i)--dir(30*i));
}
for (int i = 0; i < 4; ++i) {
draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2));
}
for (int i = 0; i < 12; ++i) {
label("\small" + (string) i, dir(90 - i * 30) * 0.75);
}
draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp));
draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp));
[/asy]
Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.
Let $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.
1969 AMC 12/AHSME, 34
The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as:
$\textbf{(A) }2^{100}-1\qquad
\textbf{(B) }2^{100}(x-1)-(x-2)\qquad
\textbf{(C) }2^{100}(x-3)\qquad$
$\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad
\textbf{(E) }2^{100}(x+1)-(x+2)$
1992 AIME Problems, 7
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
1959 AMC 12/AHSME, 15
In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is:
$ \textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 75^{\circ} $
2013 AMC 12/AHSME, 24
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle ACB$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\bigtriangleup BXN$ is equilateral and $AC=2$. What is $BN^2$?
$\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad\textbf{(B)}\ \frac{2}{9} \qquad\textbf{(C)}\ \frac{5\sqrt{2} - 3\sqrt{3}}{8} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(E)}\ \frac{3\sqrt{3} - 4}{5}$.
2018 AMC 10, 3
In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained?
$
\textbf{(A) }2 \qquad
\textbf{(B) }3\qquad
\textbf{(C) }4 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }24 \qquad
$
1993 AMC 8, 3
Which of the following numbers has the largest prime factor?
$\text{(A)}\ 39 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 77 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 121$
2010 AIME Problems, 1
Let $ N$ be the greatest integer multiple of $ 36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $ N$ is divided by $ 1000$.
2022 AMC 10, 14
Suppose that $S$ is a subset of $\{1, 2, 3,...,25\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain?
$\textbf{(A) }12 \qquad \textbf{(B) }13 \qquad \textbf{(C) }14 \qquad \textbf{(D) }15 \qquad \textbf{(E) }16$
2023 AMC 10, 22
How many distinct values of $x$ satisfy $\lfloor x \rfloor ^2 – 3x + 2 = 0$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$?
$\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$
1960 AMC 12/AHSME, 27
Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then
$ \textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad$
$\textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad$
$\textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad$
$\textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad$
$\textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular} $
1978 AMC 12/AHSME, 19
A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is
$\textbf{(A) }.05\qquad\textbf{(B) }.065\qquad\textbf{(C) }.08\qquad\textbf{(D) }.09\qquad \textbf{(E) }.1$
1963 AMC 12/AHSME, 27
Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:
$\textbf{(A)}\ 16 \qquad
\textbf{(B)}\ 20\qquad
\textbf{(C)}\ 22 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 26$
1972 AMC 12/AHSME, 19
The sum of the first $n$ terms of the sequence \[1,~(1+2),~(1+2+2^2),~\dots ~(1+2+2^2+\dots +2^{n-1})\] in terms of $n$ is
$\textbf{(A) }2^n\qquad\textbf{(B) }2^n-n\qquad\textbf{(C) }2^{n+1}-n\qquad\textbf{(D) }2^{n+1}-n-2\qquad \textbf{(E) }n\cdot 2^n$