Found problems: 3632
2020 AMC 10, 5
What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$
$\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$
1970 AMC 12/AHSME, 19
The sum of an infinite geometric series with common ratio $r$ such that $|r|<1$, is $15$, and the sum of the squares of the terms of this series is $45$. The first term of the series is
$\textbf{(A) }12\qquad\textbf{(B) }10\qquad\textbf{(C) }5\qquad\textbf{(D) }3\qquad \textbf{(E) }2$
2022 AMC 12/AHSME, 15
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
2013 AIME Problems, 6
Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.
2023 AMC 12/AHSME, 24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
2023 AMC 12/AHSME, 20
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
(I forgot answer choices)
2009 AIME Problems, 3
A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
1994 AMC 12/AHSME, 18
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$
[asy]
draw(Circle((0,0), 5));
draw((0,5)--(3,-4)--(-3,-4)--cycle);
label("A", (0,5), N);
label("B", (-3,-4), SW);
label("C", (3,-4), SE);
dot((0,5));
dot((3,-4));
dot((-3,-4));
[/asy]
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18 $
1983 AMC 12/AHSME, 29
A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A$, $B$, $C$ and $D$. Also, let the distances from $P$ to $A$, $B$ and $C$, respectively, be $u$, $v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$?
$ \textbf{(A)}\ 1 + \sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{2}\qquad\textbf{(C)}\ 2 + \sqrt{2}\qquad\textbf{(D)}\ 3\sqrt{2}\qquad\textbf{(E)}\ 3 + \sqrt{2}$
2016 AIME Problems, 11
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
1994 AIME Problems, 8
The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
2016 AMC 12/AHSME, 17
In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?
[asy]draw((0,0)--(7,0));
draw((0,0)--(33/7,7.66651));
draw((33/7,7.66651)--(7,0));
draw((11/5,7*7.66651/15)--(7,0));
draw((63/17,0)--(33/7,7.66651));
draw((0,0)--(45/7,7.66651/4));
dot((0,0));
label("A",(0,0),SW);
dot((7,0));
label("B",(7,0),SE);
dot((33/7,7.66651));
label("C",(33/7,7.66651),N);
dot((11/5,7*7.66651/15));
label("D",(11/5+.2,7*7.66651/15-.25),S);
dot((63/17,0));
label("E",(63/17,0),NE);
dot((45/7,7.66651/4));
label("H",(44/7,7.66651/4),NW);
dot((27/7,3*7.66651/20));
label("P",(27/7,3*7.66651/20),NW);
dot((5,7*7.66651/36));
label("Q",(5,7*7.66651/36),N);
label("9",(33/14,7.66651/2),NW);
label("8",(41/7,7.66651/2),NE);
label("7",(3.5,0),S);[/asy]
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \frac{5}{8}\sqrt{3} \qquad
\textbf{(C)}\ \frac{4}{5}\sqrt{2} \qquad
\textbf{(D)}\ \frac{8}{15}\sqrt{5} \qquad
\textbf{(E)}\ \frac{6}{5}$
2015 AMC 12/AHSME, 1
What is the value of $2-(-2)^{-2}$?
$ \textbf{(A) } -2
\qquad\textbf{(B) } \dfrac{1}{16}
\qquad\textbf{(C) } \dfrac{7}{4}
\qquad\textbf{(D) } \dfrac{9}{4}
\qquad\textbf{(E) } 6
$
1998 AMC 8, 8
A child's wading pool contains $200$ gallons of water. If water evaporates at the rate of $0.5$ gallons per day and no other water is added or removed, how many gallons of water will be in the pool after $30$ days?
$ \text{(A)}\ 140\qquad\text{(B)}\ 170\qquad\text{(C)}\ 185\qquad\text{(D)}\ 198.5\qquad\text{(E)}\ 199.85 $
2011 AMC 10, 11
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $
2019 AIME Problems, 8
The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f(\tfrac{1+\sqrt{3}i}{2})=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.
2019 AIME Problems, 6
In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.
2014 Contests, 2
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?
2010 Contests, 3
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
2014 AMC 10, 20
For how many integers is the number $x^4-51x^2+50$ negative?
$ \textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }14\qquad\textbf{(E) }16\qquad $
2002 AMC 12/AHSME, 25
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
[asy]//Choice A
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125;
}
real g(real x) {
return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(A)}$", (-5,4.5));
[/asy]
[asy]//Choice B
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2;
}
real g(real x) {
return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(B)}$", (-5,4.5));
[/asy]
[asy]//Choice C
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.21875 x^2+0.28125 x+0.5;
}
real g(real x) {
return -0.375 x^2-0.75 x+0.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(C)}$", (-5,4.5));
[/asy]
[asy]//Choice D
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875;
}
real g(real x) {
return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625;
}
real z=3.14;
draw(graph(f,-z, z), heavygray);
draw(graph(g,-z, z), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(D)}$", (-5,4.5));
[/asy]
[asy]//Choice E
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598;
}
real g(real x) {
return -0.166667 x^3+0.125 x^2+0.479167 x-0.375;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(E)}$", (-5,4.5));
[/asy]
2006 AMC 10, 9
How many sets of two or more consecutive positive integers have a sum of 15?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2019 AMC 10, 24
Define a sequence recursively by $x_0=5$ and
\[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\]
for all nonnegative integers $n.$ Let $m$ be the least positive integer such that
\[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie?
$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$
2019 AMC 10, 2
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
2013 AIME Problems, 2
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5$,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5$.