problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
$P, Q, R,$ and $S$ are distinct positive integers such that the product $PQ = 120$, the product $RS = 120$, and $P - Q = R + S$. What is the value of $P$?
|
30
| 0.166667 |
If \(a+b=\frac{9}{17}\) and \(a-b=\frac{1}{51}\), what is the value of \(a^2-b^2\)? Express your answer as a common fraction.
|
\frac{3}{289}
| 0.833333 |
A biologist wants to estimate the population of a fish species in a large pond. On April 1, she captures and tags 120 fish, then releases them back into the pond. On August 1, she captures another sample of 150 fish, of which 5 are found to be tagged. She estimates that 30% of the fish present on April 1 have left the pond and that 50% of the fish in the August sample were not in the pond in April. Using this information, how many fish were in the pond on April 1?
|
1800
| 0.083333 |
The polynomial $3x^3 + bx + 15$ has a factor of the form $x^2 + px + 1$. Find $b$.
|
-72
| 0.916667 |
Suppose $a,b,$ and $c$ are integers such that $4b = 10 - 3a + c$. Determine how many of the numbers from the set $\{1, 2, 3, 4, 5, 6\}$ must be divisors of $3b + 15 - c$.
|
1
| 0.666667 |
Simplify and write the result as a common fraction: $$\sqrt{\sqrt[4]{\left(\frac{1}{65536}\right)^2}}$$
|
\frac{1}{16}
| 0.833333 |
Find the number of integers \( n \) that satisfy
\[15 < n^2 < 121.\]
|
14
| 0.833333 |
How many integers $n$ satisfy $(n-3)(n+5)<0$?
|
7
| 0.916667 |
What is the area of the circle defined by $x^2 - 8x + y^2 - 16y + 68 = 0$ that lies above the line $y = 4$?
|
12\pi
| 0.416667 |
It is now 3:00:00 PM, as read on a 12-hour digital clock. In 316 hours, 59 minutes, and 59 seconds, the time will be $A:B:C$. What is the value of $A + B + C$?
|
125
| 0.083333 |
Let
\[\mathbf{A} = \begin{pmatrix} 5 & 2 \\ -12 & -5 \end{pmatrix}.\]
Compute $\mathbf{A}^{50}.$
|
\mathbf{A}^{50} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
| 0.916667 |
There are 25 gremlins, 20 imps, and 10 sprites at the Horizon Fantasy Meetup. The imps continue to avoid shaking hands among themselves. However, the sprites only shake hands with each other and with the gremlins, but not with the imps. Each pair of creatures shakes hands only once. How many handshakes occurred at this event?
|
1095
| 0.166667 |
A square is divided into four congruent strips, as shown. If the perimeter of each of these four strips is 40 inches, what is the perimeter of the square, in inches?
[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((0,3)--(4,3));
[/asy]
|
64
| 0.5 |
Solve for $n$: $(n-5)^4 = \left(\frac{1}{16}\right)^{-1}$.
|
7
| 0.083333 |
Find the remainder when \(x^4\) is divided by \(x^2 + 4x + 1\).
|
-56x - 15
| 0.916667 |
The sides of a right triangle measure 8, 15, and 17. Determine the distance between the centers of the triangle's inscribed circle and the circumscribed circle.
|
\frac{\sqrt{85}}{2}
| 0.583333 |
Choose two different prime numbers between $20$ and $30$. Calculate their product, the sum, and the sum of their squares. What number can be obtained by the following expression: $$xy - (x + y) - (x^2 + y^2)?$$
|
-755
| 0.916667 |
A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 616, and the sum of the lengths of its 12 edges is 160. What is $s$?
|
\sqrt{246}
| 0.833333 |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\spadesuit$?
|
\dfrac{1}{52}
| 0.333333 |
What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $76$?
|
459
| 0.916667 |
What is the area, in square units, of a triangle with vertices at $A(2, 3), B(7, 3), C(4, 9)$?
|
15 \text{ square units}
| 0.833333 |
Let \[f(x) = \left\{
\begin{array}{cl}
-x + 3 & \text{if } x \le 0, \\
2x - 5 & \text{if } x > 0.
\end{array}
\right.\]How many solutions does the equation $f(f(x)) = 6$ have?
|
3
| 0.916667 |
Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x-10$, the remainder is $20$, and when $Q(x)$ is divided by $x-7$, the remainder is $10$. What is the remainder when $Q(x)$ is divided by $(x-10)(x-7)$?
|
\frac{10}{3}x - \frac{40}{3}
| 0.666667 |
A frustum of a right circular cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate both the lateral surface area and the volume of the frustum.
|
\frac{560\pi}{3}
| 0.583333 |
If an integer is divisible by 6 and the sum of its last two digits is 15, then what is the product of its last two digits?
|
54
| 0.916667 |
The sum of the first 1010 terms of a geometric sequence is 300. The sum of the first 2020 terms is 540. Find the sum of the first 3030 terms.
|
732
| 0.833333 |
Determine the remainder when $1 + 3 + 3^2 + \cdots + 3^{1000}$ is divided by $500$.
|
1
| 0.583333 |
A whole number is considered "11-heavy" if the remainder when the number is divided by 11 is greater than 7. What is the least four-digit "11-heavy" whole number?
|
1000
| 0.666667 |
Simplify $\frac{5^6 + 5^3}{5^5 - 5^2}$. Express your answer as a common fraction.
|
\frac{315}{62}
| 0.916667 |
Alice and Bob each think of a polynomial. Each of their polynomials is monic, has degree 5, and has the same positive constant term and the same coefficient of $x$. The product of their polynomials is \[x^{10} + 4x^9 + 6x^8 + 4x^7 + 5x^6 + 10x^5 + 5x^4 + 6x^3 + 4x^2 + 4x + 9.\] What is the constant term of Bob's polynomial?
|
3
| 0.916667 |
John scored 84, 88, and 95 on his first three biology exams. If he scores 92 on his fourth exam, how much will his average score change?
|
0.75
| 0.916667 |
What is the greatest integer less than 200 for which the greatest common divisor of that integer and 18 is 6?
|
192
| 0.916667 |
If the odds for pulling a prize out of the box are $5:6$, what is the probability of not pulling out the prize? Express your answer as a common fraction.
|
\frac{6}{11}
| 0.916667 |
Thirteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Dana, starts with the ball, skips the next four girls, and throws to the sixth girl, who then skips the next four girls and throws the ball following the same pattern. If the throwing pattern continues, including Dana's initial throw, how many total throws are necessary for the ball to return to Dana?
|
13
| 0.666667 |
Find the number of real solutions of the equation
\[
\frac{x}{50} = \sin x.
\]
|
31
| 0.25 |
The equation $x^2 - 2x = i$ has two complex solutions. Determine the product of their real parts.
|
\frac{1 - \sqrt{2}}{2}
| 0.166667 |
A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Calculate the length of the longer side of the rectangle, in centimeters. Express your answer in terms of $\pi$.
|
9\pi \text{ cm}
| 0.916667 |
Find the greatest common divisor of 108 and 450.
|
18
| 0.916667 |
How many digits are located to the right of the decimal point when $\frac{5^7}{10^5 \cdot 15625}$ is expressed as a decimal?
|
5
| 0.916667 |
What is the base $2$ representation of $96_{10}$?
|
1100000_2
| 0.666667 |
Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{v} = \begin{pmatrix} 7 \\ 2 \\ -8 \end{pmatrix} \times \mathbf{v}\] for all vectors $\mathbf{v}.$
|
\begin{pmatrix} 0 & 8 & 2 \\ -8 & 0 & -7 \\ -2 & 7 & 0 \end{pmatrix}
| 0.833333 |
Let \(x\) and \(y\) be two distinct positive real numbers, and \(n\) a positive integer. We define sequences \((P_n), (G_n)\) as follows. First, \(P_1\) is \( \sqrt[n]{\frac{x^n + y^n}{2}} \) and \( G_1 \) is \( \sqrt{x \cdot y} \). Then for \( n \ge 2 \), \( P_n \) and \( G_n \) are:
\[ P_{n+1} = \sqrt[n]{\frac{(\sin P_n)^n + (\sin G_n)^n}{2}} \text{ and } G_{n+1} = \sqrt{\sin P_n \cdot \sin G_n} \] respectively.
Determine if the sequences \( (P_n) \) and \( (G_n) \) converge and identify their limits if they exist.
|
0
| 0.25 |
Find the sum of the $1004$ roots of $(x-1)^{1004} + 2(x-2)^{1003} + 3(x-3)^{1002} + ... + 1003(x-1003)^2 + 1004(x-1004).$
|
1002
| 0.333333 |
I have six apples and eight oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)
|
62
| 0.75 |
Convert $\rm{BAC}_{16}$ to a base 10 integer.
|
2988
| 0.75 |
What is the sum of all the two-digit primes that are greater than 20 but less than 90 and are still prime when their two digits are interchanged?
|
291
| 0.416667 |
Compute $\sqrt{75} \cdot \sqrt{45} \cdot \sqrt{20}$.
|
150\sqrt{3}
| 0.666667 |
Find the greatest possible value of $k$ if the roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{85}$.
|
\sqrt{117}
| 0.583333 |
Find the smallest solution to the equation \[\frac{1}{x-1} + \frac{1}{x-5} = \frac{4}{x-4}.\]
|
\frac{5 - \sqrt{33}}{2}
| 0.916667 |
In a right triangle $ABC$ with $\angle A = 90^\circ$, the lengths of $AB = 8$ and $BC = 10$. Find $\sin C$.
|
\frac{4}{5}
| 0.75 |
If the three lines $4y-3x=2$, $x+3y=3$ and $8x-12y=9$ are drawn in the plane, how many points will lie at the intersection of at least two of the three lines?
|
3
| 0.833333 |
What is the base-10 integer 729 when expressed in base 7?
|
2061_7
| 0.666667 |
The arithmetic mean of 15 scores is 75. When the highest and lowest scores are removed, the new mean becomes 77. If the highest of the 15 scores is 98, what is the lowest score?
|
26
| 0.833333 |
In triangle $ABC$, $AB = 6$, $BC = 8$, and $CA = 10$.
Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to $B$ than it is to either $A$ or $C$?
|
\frac{1}{2}
| 0.166667 |
How many positive integers less than 500 are congruent to 5 (mod 13)?
|
39
| 0.916667 |
What is the least positive multiple of 36 for which the product of its digits is also a positive multiple of 9?
|
36
| 0.583333 |
Solve for $y$: $(y-3)^4 = \left(\frac{1}{16}\right)^{-1}$
|
5
| 0.333333 |
Consider that there exists a real number $m$ such that the equation
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 6 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + s \begin{pmatrix} -3 \\ m \end{pmatrix}\]
does not have any solutions in $t$ and $s$. Find $m$.
|
1
| 0.833333 |
My school's math club has 7 boys and 10 girls. We need to select a team of 6 people to send to the state math competition, ensuring that the team includes at least one previously competing boy and girl. Out of these, 1 boy and 1 girl have competition experience. In how many ways can I select the team to have 3 boys and 3 girls including the experienced members?
|
540
| 0.833333 |
A "super ball" is dropped from a window 25 meters above the ground. On each bounce, it rises to 80% of the height of the preceding bounce. How far does the ball travel until it reaches the high point after hitting the ground for the fourth time?
|
132.84 \text{ meters}
| 0.916667 |
Suppose $(v_n)$ is a sequence of real numbers satisfying
\[ v_{n+2} = 2v_{n+1} + v_{n} \]
with $v_4 = 15$ and $v_7 = 255$. Find $v_5$.
|
45
| 0.916667 |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 24, what is the smallest value of $n$ such that $f(n) > 24$?
|
696
| 0.166667 |
Tom’s graduating class has 300 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, what is the sum of all possible values of $x$ where $x$ is the number of students in each row?
|
45
| 0.666667 |
If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)?
|
\frac{2197}{499800}
| 0.083333 |
A portion of the graph of $y = H(x)$ is shown below. The distance between grid lines is $1$ unit. Compute $H(H(H(H(H(2)))))$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
real h(real x) {return ((x-2)*(x-2)/4 - 2);}
draw(graph(h,2-sqrt(2*8),2+sqrt(2*8),operator ..), red);
[/asy]
|
-2
| 0.833333 |
Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_3b_4 = 72$, what is $b_2b_5$?
|
70
| 0.833333 |
Joann consumed a total of 154 lollipops over seven days. Each day after the first, she ate four more lollipops than she had the previous day. How many lollipops did she eat on the fifth day?
|
26
| 0.833333 |
Suppose that $p$ is prime and $1014_p + 307_p + 114_p + 126_p + 7_p = 143_p + 272_p + 361_p$. How many possible values of $p$ are there?
|
0
| 0.25 |
Solve for $t$: $5 \cdot 5^t + \sqrt{25 \cdot 25^t} = 50$.
|
1
| 0.916667 |
What is half of the absolute value of the difference of the squares of 15 and 12?
|
40.5
| 0.833333 |
Calculate \(7 \cdot 9\frac{2}{5}\).
|
65\frac{4}{5}
| 0.75 |
I have four distinct mystery novels, four distinct fantasy novels, four distinct biographies, and four distinct science fiction novels. I'm planning a long trip and want to take two books from different genres. How many possible pairs can I choose?
|
96
| 0.75 |
An integer greater than 9 and less than 100, which is divisible by 7, is randomly chosen. What is the probability that its digits are different?
|
\frac{12}{13}
| 0.5 |
Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3$.
|
12
| 0.416667 |
For how many digits $C$ is the positive four-digit number $1C35$ a multiple of 5?
|
10
| 0.916667 |
Compute $\frac{2468_{10}}{121_{3}} + 3456_{7} - 9876_{9}$. Express your answer in base 10.
|
-5857.75
| 0.166667 |
A round cake has a diameter of $12$ cm and is cut into six equal-sized sector-shaped pieces. Calculate the square of the length ($l^2$) of the longest line segment that may be drawn in one of these pieces.
|
36 \text{ cm}^2
| 0.666667 |
When you simplify $\sqrt[3]{40a^6b^7c^{14}}$, what is the sum of the exponents of the variables that are outside the radical?
|
8
| 0.75 |
Evaluate $\log_4 128\sqrt2$. Express your answer as an improper fraction.
|
\frac{15}{4}
| 0.916667 |
What is $\log_{5}{3120}$ rounded to the nearest integer?
|
5
| 0.916667 |
A collection of five positive integers has a mean of 4.6, unique mode 5, and median 5. If a 10 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth.
|
5.0
| 0.916667 |
How many 3-digit numbers have the property that the units digit is at least three times the tens digit?
|
198
| 0.833333 |
Points $P$ and $Q$ are midpoints of two adjacent sides of a rectangle with length $l$ and width $w$. What fraction of the interior of the rectangle is shaded? Consider the triangle formed by the vertex at the intersection of uncut sides and points $P$ and $Q$ to be unshaded.
[asy]
filldraw((0,0)--(4,0)--(4,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(2,2)--(4,2)--(0,1)--cycle,white,linewidth(1));
label("P",(0,1),W);
label("Q",(2,2),N);
[/asy]
|
\frac{7}{8}
| 0.666667 |
Let $n$ be the number of ordered quadruples $(x_1, x_2, x_3, x_4)$ of positive even integers that satisfy $\sum_{i=1}^4 x_i = 104$. Find $\frac{n}{100}$.
|
208.25
| 0.916667 |
I have fifteen books, of which six are written by one author, and nine by another. How many different pairs can I choose such that each pair contains books by different authors?
|
54
| 0.833333 |
Jenny considers the quadratic equation $x^2 - sx + p$ with roots $r_1$ and $r_2$. She finds out that all power sums of the roots up to 2023 equate to $s$, i.e., $r_1 + r_2 = r_1^2 + r_2^2 = \cdots = r_1^{2023} + r_2^{2023} = s$. Jenny is curious to find out the maximum possible value of $\frac1{r_1^{2024}}+\frac1{r_2^{2024}}$.
|
2
| 0.833333 |
Line \( l_1 \) has equation \( 4x - 3y = 2 \) and passes through \( A = (-2, -3) \). Line \( l_2 \) has equation \( y = 2 \) and intersects line \( l_1 \) at point \( B \). Line \( l_3 \) has a positive slope, goes through point \( A \), and intersects \( l_2 \) at point \( C \). The area of \( \triangle ABC \) is \( 6 \). Find the slope of \( l_3 \).
|
\frac{25}{32}
| 0.083333 |
Jessica has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Jessica's 30 coins represents states that joined the union during the decade 1800 through 1809? Express your answer as a common fraction.
[asy]size(200);
label("1780",(6,0),S);
label("1800",(12,0),S);
label("1820",(18,0),S);
label("1840",(24,0),S);
label("1860",(30,0),S);
label("1880",(36,0),S);
label("1900",(42,0),S);
label("1950",(48,0),S);
label("to",(6,-4),S);
label("to",(12,-4),S);
label("to",(18,-4),S);
label("to",(24,-4),S);
label("to",(30,-4),S);
label("to",(36,-4),S);
label("to",(42,-4),S);
label("to",(48,-4),S);
label("1789",(6,-8),S);
label("1809",(12,-8),S);
label("1829",(18,-8),S);
label("1849",(24,-8),S);
label("1869",(30,-8),S);
label("1889",(36,-8),S);
label("1909",(42,-8),S);
label("1959",(48,-8),S);
draw((0,0)--(50,0));
draw((0,2)--(50,2));
draw((0,4)--(50,4));
draw((0,6)--(50,6));
draw((0,8)--(50,8));
draw((0,10)--(50,10));
draw((0,12)--(50,12));
draw((0,14)--(50,14));
draw((0,16)--(50,16));
draw((0,18)--(50,18));
fill((4,0)--(8,0)--(8,12)--(4,12)--cycle,gray(0.8));
fill((10,0)--(14,0)--(14,5)--(10,5)--cycle,gray(0.8));
fill((16,0)--(20,0)--(20,7)--(16,7)--cycle,gray(0.8));
fill((22,0)--(26,0)--(26,6)--(22,6)--cycle,gray(0.8));
fill((28,0)--(32,0)--(32,7)--(28,7)--cycle,gray(0.8));
fill((34,0)--(38,0)--(38,5)--(34,5)--cycle,gray(0.8));
fill((40,0)--(44,0)--(44,4)--(40,4)--cycle,gray(0.8));
[/asy]
|
\frac{1}{6}
| 0.666667 |
Given that $x$ is a multiple of $32515$, what is the greatest common divisor of $g(x) = (3x+5)(5x+3)(11x+7)(x+17)$ and $x$?
|
35
| 0.083333 |
During her summer break, Julie works for 48 hours per week for 10 weeks, earning a total of $\$5000$. For the school year, which lasts 40 weeks, she hopes to earn a total of $6000$. At the same rate of pay, how many hours per week must she work during the school year?
|
14.4
| 0.833333 |
What is the smallest integer value of $x$ for which $7 + 3x < 26$?
|
6
| 0.833333 |
How many ways are there to put 5 distinguishable balls into 3 distinguishable boxes, provided that one of the boxes, Box C, must contain at least 2 balls?
|
131
| 0.916667 |
A square and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$? Express your answer as a common fraction.
|
\frac{130}{1}
| 0.25 |
When four standard dice are tossed, the numbers $a, b, c, d$ are obtained. Find the probability that $abcd = 2$.
|
\frac{1}{324}
| 0.916667 |
If the product $(4x^2 - 3x + 6)(9 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $8a + 4b + 2c + d$ evaluated at $x=3$.
|
48
| 0.916667 |
In triangle $ABC$, $AB = 8$, $AC = 17$, and $BC = 15$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. The centroid divides the median into the ratio $2:1$ with the longer segment being from the vertex to the centroid. Let $P$ be the foot of the altitude from $G$ to side $BC$. Find the length of segment $GP$.
|
GP = \frac{8}{3}
| 0.333333 |
What is the 52nd number in the row of Pascal's triangle that contains 55 numbers?
|
24804
| 0.583333 |
How many three-digit whole numbers have no 5's, 7's, 8's, or 9's as digits?
|
180
| 0.25 |
Determine how many positive integral values of $b$ ensure that $x = 3$ is the only positive integer solution of the system of inequalities:
$$
\begin{cases}
3x > 4x - 4 \\
4x - b > -8
\end{cases}
$$
|
4
| 0.75 |
In rectangle $ABCD,$ $AB=15$ and $AC=17.$ What is the area of rectangle $ABCD?$ Additionally, find the length of the diagonal $BD.$
|
17
| 0.416667 |
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