problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Without using a calculator, find the largest prime factor of \(16^4 + 2 \times 16^2 + 1 - 15^4\).
|
241
| 0.916667 |
Ryan has 4 red lava lamps and 2 blue lava lamps. He arranges all 6 lamps in a row on a shelf randomly, then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, the third from the left is red and off, and the rightmost lamp is red and on?
|
\frac{3}{100}
| 0.333333 |
There are $30$ different complex numbers $z$ such that $z^{30}=1$. For how many of these is $z^5$ a real number?
|
10
| 0.833333 |
Let \(a\), \(b\), and \(c\) be the roots of the polynomial \(x^3 - 5x + 7 = 0\). Find the monic polynomial, in \(x\), whose roots are \(a - 3\), \(b - 3\), and \(c - 3\).
|
x^3 + 9x^2 + 22x + 19
| 0.833333 |
The science club now has 23 members: 13 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?
|
\frac{7930}{8855}
| 0.75 |
For what values of $z$ is $z^2-40z+400 \leq 36$? Express your answer in interval notation.
|
[14, 26]
| 0.916667 |
Calculate $\left[\left(\frac{18^{18}}{18^{17}}\right)^2 \cdot 9^2\right] \div 3^4$.
|
324
| 0.916667 |
A sports league has 16 teams in two divisions of 8 each. How many games are in a complete season if each team plays every other team in its own division thrice and each team in the other division once?
|
232
| 0.916667 |
Compute $26 \times 43 + 57 \times 26$.
|
2600
| 0.916667 |
Jason is trying to remember the five-digit combination to his safe. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Jason need to try?
|
486
| 0.916667 |
Calculate $7 \cdot 9\frac{2}{5}$.
|
65 \frac{4}{5}
| 0.416667 |
How many sequences of 8 digits $x_1, x_2, \ldots, x_8$ can we form such that no two adjacent digits have the same parity? Leading zeroes are allowed.
|
781,250
| 0.916667 |
The force needed to push a car varies inversely with the number of people pushing it. If 4 people can push a car with an effort of 120 newtons each, how much effort would each person need to exert if 6 people are pushing the car?
|
80
| 0.833333 |
The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?
|
31254
| 0.166667 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 120$.
|
\frac{1}{36}
| 0.916667 |
Compute the exact value of the expression $\left|\pi - | \pi - \frac{11}{2} | \right|$. Write your answer using only integers, π, and fractions, without any absolute value signs.
|
2\pi - \frac{11}{2}
| 0.75 |
The operation $\odot$ is redefined as $a \odot b = a + \frac{5a}{2b}$. What is the value of $10 \odot 4$?
|
16.25
| 0.166667 |
Tiffany is constructing a fence around a rectangular garden next to her tennis court. She must use exactly 400 feet of fencing. The fence must enclose all four sides of the garden. Regulations state that the length of the fence enclosure must be at least 100 feet and the width must be at least 50 feet. Tiffany wants the area enclosed by the fence to be as large as possible to include various plants and decorations. What is the optimal area, in square feet?
|
10000
| 0.25 |
If $\mathbf{c}$ and $\mathbf{d}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{c},$ $\mathbf{d} + 2\mathbf{d} \times \mathbf{c},$ and $\mathbf{d}.$
|
1
| 0.916667 |
How many distinct arrangements of the letters in the word "balloon" are there?
|
1260
| 0.666667 |
Solve
\[\frac{1}{x + 5} + \frac{1}{x + 3} = \frac{1}{x + 6} + \frac{1}{x + 2}.\]
|
-4
| 0.916667 |
Find the greatest common factor of \(7!\) and \(8!\).
|
5040
| 0.916667 |
The lattice shown is continued for $15$ rows. What will be the fourth number in the $15$th row if the number of elements per row increases to $7$ and starts from 3? \begin{tabular}{rccccccc}
Row 1: & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
Row 2: & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
Row 3: & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\
Row 4: & 24 & 25 & 26 & 27 & 28 & 29 & 30
\end{tabular}
|
104
| 0.5 |
A ball is thrown downward from a height with its height (in feet) after $t$ seconds given by the expression $-20t^2 - 40t + 50$. Determine the time at which it hits the ground.
|
-1 + \frac{\sqrt{14}}{2}
| 0.333333 |
The area of a region formed by four congruent squares is 144 square centimeters. What is the perimeter of the region, in centimeters?
[asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((10,0)--(20,0)--(20,10)--(10,10)--cycle);
draw((0,10)--(10,10)--(10,20)--(0,20)--cycle);
draw((10,10)--(20,10)--(20,20)--(10,20)--cycle);
[/asy]
|
48
| 0.833333 |
Evaluate the sum
\[ \text{cis } 80^\circ + \text{cis } 90^\circ + \text{cis } 100^\circ + \dots + \text{cis } 130^\circ \]
and express it in the form \( r \, \text{cis } \theta \), where \( r > 0 \) and \( 0^\circ \le \theta < 360^\circ \). Find \( \theta \) in degrees.
|
105^\circ
| 0.416667 |
In triangle $DEF$, $\cos(3D-E) + \sin(D+E) = 2$ and $DE = 6$. What is $EF$?
|
3\sqrt{2 - \sqrt{2}}
| 0.333333 |
How many positive four-digit integers of the form $\_\_25$ are divisible by 25?
|
90
| 0.833333 |
Suppose that $d, e,$ and $f$ are positive integers satisfying $(d+e+f)^3 - d^3 - e^3 - f^3 = 300$. Find $d+e+f$.
|
7
| 0.416667 |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively, where $A(0,8)$, $B(0,0)$, and $C(10,0)$. Determine the area of $\triangle DBC$.
|
20
| 0.916667 |
Determine $\cos B$ in the right triangle shown below where side AC is 7 units and side BC (hypotenuse) is 25 units.
[asy]
pair A,B,C;
A = (0,0);
B = (24,0);
C = (0,7);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,10));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$25$",(B+C)/2,NE);
label("$7$",C/2,W);
[/asy]
|
\frac{24}{25}
| 0.833333 |
Two circles are drawn in a 15-inch by 20-inch rectangle. Each circle has a diameter of 4 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?
|
\sqrt{377}\text{ inches}
| 0.75 |
Let $p(x) = x^2 + px + q$ and $r(x) = x^2 + sx + t$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $p$ is a root of $r,$ and the $x$-coordinate of the vertex of $r$ is a root of $p,$ and both $p$ and $r$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50, -50),$ what is the value of $p + s$?
|
-200
| 0.75 |
Calculate:
\[\cos \left( 6 \arccos \frac{1}{4} \right).\]
|
-\frac{7}{128}
| 0.5 |
John arranges his playing cards in a suit in the order $$K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2, A, K, Q, J, \cdots.$$ What is the $45^{th}$ card?
|
8
| 0.916667 |
Consider a rectangle with dimensions $x + 3$ and $3x - 4$ where the area of the rectangle is $5x + 14$. Find the value of x.
|
\frac{\sqrt{78}}{3}
| 0.75 |
A 4x4x4 cube is formed by assembling 64 unit cubes. Four unit squares are painted on the center of each of the six faces of the cube. How many of the 64 unit cubes have no paint on them?
|
40
| 0.25 |
A right triangle has sides of 8, 15, and 17. Calculate the radius of the circumscribed circle and the distance between the circumcenter and the incenter of the triangle.
|
\frac{\sqrt{85}}{2}
| 0.583333 |
The first term of the given sequence is 2, and each subsequent term is the sum of the squares of all previous terms. What is the value of the first term which exceeds 10000?
|
176820
| 0.916667 |
A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side?
|
14
| 0.916667 |
Two calculations are given: $\left(6^2-3^2\right)^4$ and $\left(7^2-2^2\right)^4$. What is the sum of the results of these two calculations?
|
4632066
| 0.5 |
The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$.
|
300\pi
| 0.833333 |
Determine the function $g(x)$ if $g(1) = 2$ and
\[ g(x + y) = 5^y g(x) + 4^x g(y) \]
for all real numbers $x$ and $y$.
|
2(5^x - 4^x)
| 0.25 |
The first stage of a geometric shape made entirely of matchsticks is a square and each subsequent stage adds an additional square on one side, each requiring 5 new matchsticks to complete without breaking any matches. How many matchsticks are needed to complete the 100th stage?
|
499
| 0.916667 |
Compute without using a calculator: $10! - 9! + 8! - 7!$.
|
3301200
| 0.666667 |
Solve the inequality
\[\frac{x^2 - 49}{x + 7} < 0.\]
|
(-\infty, -7) \cup (-7, 7)
| 0.916667 |
Compute $12^{-1} \pmod{997}$.
|
914
| 0.916667 |
Let $c = \frac{4}{7}$ and let $d=\frac{5}{8}$. Compute $c^{-2}d^{3}\sqrt{d}$.
|
\frac{6125 \sqrt{10}}{32768}
| 0.916667 |
If $\log_{27} (x-3) = \frac{1}{3}$, find $\log_{343} x$.
|
\frac{1}{3} \log_7 6
| 0.5 |
How many degrees are there in the measure of angle $P$?
[asy]
size (6cm);
pair A,B,C,D,E,F;
A=(0,0);
B=(1,-1);
C=(3,-1);
D=(4,0);
E=(3,2);
F=(1,2);
draw (A--B--C--D--E--F--A, linewidth(1));
label("$P$", D, NE);
label("$135^\circ$", shift(0,-0.6) * A);
label("$120^\circ$", B, S);
label("$105^\circ$", C, S);
label("$150^\circ$", E, N);
label("$110^\circ$", F, N);
draw(anglemark(E,D,C), blue);
[/asy]
|
100^\circ
| 0.5 |
What is the product of the digits in the base 8 representation of $8654_{10}$?
|
0
| 0.916667 |
The solution of the equation $6^{x+6} = 9^x$ can be expressed in the form $x = \log_b 6^6$. What is $b$?
|
\frac{3}{2}
| 0.5 |
In right triangle $ABC$ with $\angle C = 90^\circ$, we have $AB = 15$ and $AC = 7$. Find $\sin A$ and $\cos A$.
|
\sin A = \frac{4\sqrt{11}}{15}, \quad \cos A = \frac{7}{15}
| 0.5 |
The points $(2, 9), (14, 18)$ and $(6, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?
|
24
| 0.166667 |
Find $n$ such that $2^8 \cdot 3^4 \cdot 5^1 \cdot n = 10!$.
|
35
| 0.75 |
Calculate the dot product of the vectors $\begin{pmatrix} 4 \\ 5 \\ -6 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} -3 \\ 7 \\ 0 \\ -4 \end{pmatrix}$.
|
15
| 0.916667 |
Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be three mutually orthogonal unit vectors, such that
\[\mathbf{u} = s (\mathbf{u} \times \mathbf{v}) + t (\mathbf{v} \times \mathbf{w}) + u (\mathbf{w} \times \mathbf{u})\]
for some scalars \(s\), \(t\), and \(u\), and \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 1.\) Find \(s + t + u.\)
|
1
| 0.916667 |
Isosceles triangle $ABF$ has an area of 180 square inches and is divided by line $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 135 square inches. If the altitude of triangle $ABF$ from $A$ is 30 inches, determine the length of $\overline{CD}$.
|
6 \text{ inches}
| 0.75 |
Find the integer $m$, $-90 \le m \le 90$, such that $\sin m^\circ = \sin 710^\circ$.
|
-10
| 0.833333 |
Fourteen stones are arranged in a circle. They are counted clockwise from 1 to 14, and then continue to count counter-clockwise starting from stone 14 again as 15, onward until reaching stone 2 again as 27. This pattern continues indefinitely. Determine which original stone is counted as 99.
|
10
| 0.083333 |
Compute the product $94 \times 106$.
|
9964
| 0.916667 |
Express $\sqrt{a} \div \sqrt{b}$ as a common fraction, given:
$$\frac{{\left(\frac{1}{3}\right)}^2 + {\left(\frac{1}{4}\right)}^2}{{\left(\frac{1}{5}\right)}^2 + {\left(\frac{1}{6}\right)}^2} = \frac{25a}{61b}$$
|
\frac{5}{2}
| 0.833333 |
What is the product of the positive odd divisors of $180$?
|
91125
| 0.916667 |
A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side?
|
11
| 0.916667 |
Jenna distributes her stamp collection evenly across pages and then stores these pages into three different stamp books. One book contains 924 stamps, another has 1386 stamps, and the third contains 1848 stamps. Determine the greatest number of stamps Jenna could be putting on each page.
|
462
| 0.916667 |
Two dice each with 8 sides are rolled. What is the probability that the sum rolled is a perfect square?
|
\dfrac{3}{16}
| 0.916667 |
Add $24_8 + 157_8.$ Express your answer in base 8.
|
203_8
| 0.916667 |
Let $T = 2 - 3 + 4 - 5 + \cdots + 2010 - 2011$. What is the residue of $T$, modulo 2011?
|
1006
| 0.916667 |
If $x^{2y} = 4$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.
|
1
| 0.833333 |
Let $x$, $y$, $z$, and $w$ be real numbers with $|x-y|=1$, $|y-z|=2$, and $|z-w|=3$. What is the sum of all possible values of $|x-w|$?
|
12
| 0.666667 |
What is the remainder when $22277 + 22279 + 22281 + 22283 + 22285 + 22287 + 22289 + 22291$ is divided by $8$?
|
0
| 0.833333 |
Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between?
|
10
| 0.083333 |
In triangle $ABC$, we have $\angle B = 60^\circ$ and $\cos A = \frac{3}{5}$. Find $\sin C$.
|
\sin C = \frac{3\sqrt{3} + 4}{10}
| 0.833333 |
If $x + \frac{1}{x} = \sqrt{2}$, then find $x^{12}$.
|
-1
| 0.666667 |
An infinite geometric series has a common ratio of $\frac{1}{4}$ and sum $10$. What is the second term of the sequence?
|
\frac{15}{8}
| 0.833333 |
What is the remainder when $5x^7 - 3x^6 - 8x^5 + 3x^3 + 5x^2 - 20$ is divided by $3x - 9$?
|
6910
| 0.75 |
Find the quadratic polynomial \( q(x) \) such that \( q(-2) = 7 \), \( q(1) = 4 \), and \( q(3) = 10 \).
|
q(x) = \frac{4}{5}x^2 - \frac{1}{5}x + \frac{17}{5}
| 0.916667 |
What is the probability that the square root of a randomly selected two-digit whole number is less than nine? Express your answer as a common fraction.
|
\frac{71}{90}
| 0.833333 |
Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$.
|
064
| 0.166667 |
The sum of 36 consecutive integers is $6^5$. What is their median?
|
216
| 0.916667 |
A polynomial of degree $15$ is divided by $d(x)$ to give a quotient of degree $9$ and a remainder of $2x^4 + 3x^3 - 5x + 7$. What is $\deg d$?
|
6
| 0.916667 |
Let $a,$ $b,$ and $c$ be the roots of
\[x^3 - 4x^2 + 6x - 3 = 0.\]Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$
|
x^3 - 13x^2 + 57x - 84
| 0.583333 |
Let \(P\), \(Q\), and \(R\) be points on a circle of radius \(12\). If \(\angle PRQ = 90^\circ\), what is the circumference of the minor arc \(PQ\)? Express your answer in terms of \(\pi\).
|
12\pi
| 0.833333 |
Let $x = (1 + \sqrt{2})^{500},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[x(1 - f).\]
|
1
| 0.75 |
Find the largest value less than 1000 that is common to the arithmetic progressions $\{4, 9, 14, \ldots \}$ and $\{5, 13, 21, \ldots \}$.
|
989
| 0.666667 |
The quadratic \( x^2 - 20x + 49 \) can be written in the form \( (x+b)^2 + c \), where \( b \) and \( c \) are constants. Find the value of \( b+c \).
|
-61
| 0.916667 |
Robert decides to visit the milk bottling plant daily over a week-long period. Each day, the plant has a 3/4 chance of bottling chocolate milk. What is the probability that the plant bottles chocolate milk exactly 5 out of the 7 days he visits?
|
\frac{5103}{16384}
| 0.833333 |
Two rays with common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=2$. What is the maximum possible length of $\overline{OB}$?
|
2\sqrt{2}
| 0.75 |
How many even four-digit integers have the property that their digits, read left to right, are in strictly increasing order?
|
46
| 0.583333 |
Points $P$, $Q$, $R$, and $S$ lie on a line, in that order. If $PQ=4$ units, $QR=10$ units, and $PS=28$ units, calculate the ratio of $PR$ to $QS$. Express your answer as a common fraction.
|
\frac{7}{12}
| 0.916667 |
Piravena needs to travel from city $X$ to city $Y$, then from $Y$ to city $Z$, and finally from $Z$ back to $X$. Each segment of her journey is completed entirely by bus or by airplane. The cities form a right-angled triangle with city $Z$ as the right angle. The distance from $X$ to $Z$ is 4000 km and from $X$ to $Y$ is 5000 km. Bus fare is $\$0.20$ per kilometer, and airplane travel costs a $\$120$ booking fee plus $\$0.12$ per kilometer. Piravena chooses the least expensive travel option for each segment. Determine the total cost of her trip.
|
\$1800
| 0.333333 |
Consider the lines:
\[
y = 4x + 6, \quad 2y = 6x + 8, \quad 3y = 12x - 3, \quad 2y = 3x - 4, \quad 4y = 2x - 7.
\]
Determine how many pairs of lines are either parallel or perpendicular.
|
1
| 0.833333 |
What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 43\cdot 2x + 43^2$ is a multiple of 53?
|
5
| 0.25 |
Given the number $A485B6$, where $A$ and $B$ are single digits, find the sum of all possible values of $A+B$ such that the number is divisible by 9.
|
17
| 0.5 |
A rhombus has an area of 192 square units. The lengths of its diagonals have a ratio of 4 to 3. Determine the length of the longest diagonal, in units.
|
16\sqrt{2}
| 0.833333 |
Calculate the number of different rectangles with sides parallel to the grid that can be formed by connecting four of the dots in a $5\times 5$ square array of dots. (Two rectangles are different if they do not share all four vertices.)
|
100
| 0.166667 |
Let $f(x) = 4x^2 - 3$ and $g(f(x)) = x^2 - x + 2$. Find the sum of all possible values of $g(47)$.
|
29
| 0.916667 |
In the diagram below, we have $\sin \angle RPQ = \frac{3}{5}$. What is $\sin \angle RPT$?
[asy]
pair R,P,Q,T;
T = (4,0);
P = (0,0);
Q = (2,0);
R = rotate(aSin(3/5))*(1.5,0);
dot("$T$",T,E);
dot("$Q$",Q,S);
dot("$R$",R,N);
dot("$P$",P,S);
draw(Q--T);
draw(P--R);
[/asy]
|
\frac{3}{5}
| 0.666667 |
What is $\left(\frac{8}{9}\right)^2 \cdot \left(\frac{1}{3}\right)^2 \cdot \left(\frac{1}{4}\right)^2$?
|
\frac{4}{729}
| 0.333333 |
Cory needs to buy 18 identical notebooks and has a budget of $160. He must pay a $5 entrance fee at the store. Each notebook costs the same whole-dollar amount and a 5% sales tax is added to the price at checkout. What is the greatest possible price per notebook that allows Cory to stay within his budget?
|
8
| 0.833333 |
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