problem
stringlengths 18
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|---|---|---|
Calculate the sum $S = \sum_{k=0}^{49} (-1)^k \binom{100}{2k+1} = \binom{100}{1} - \binom{100}{3} + \binom{100}{5} - \cdots - \binom{100}{99}$.
|
0
| 0.833333 |
Nalia needs to travel from point $X$ to $Y$, then from $Y$ to $Z$, and then from $Z$ to $X$. Each of these segments is either by road or rail. The cities form a right-angled triangle with $Z$ located 4000 km from $X$ and $Y$ located 5000 km from $X$. Compute the total distance Nalia travels on her journey.
|
12000
| 0.166667 |
The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?
|
44
| 0.916667 |
Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/3 chance that she will toss it to Bob and a 2/3 chance that she will keep it. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after three turns?
|
\frac{203}{432}
| 0.25 |
A right triangle with sides of 8, 15, and 17 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
|
\frac{\sqrt{85}}{2}
| 0.75 |
The real number $x$ satisfies $x^2 - 4x + 3 < 0.$ Find all possible values of $x^2 + 4x + 3.$
|
(8, 24)
| 0.916667 |
Given a function \( f(x) \) defined for all real numbers \( x \) such that for all non-zero values \( x \),
\[ 2f(x) + f\left(\frac{1}{x}\right) = 6x + 3 \]
Let \( T \) denote the sum of all values of \( x \) for which \( f(x) = 2023 \). Compute the integer nearest to \( T \).
|
506
| 0.833333 |
How many factors of \(5400\) are perfect squares?
|
8
| 0.916667 |
When a polynomial is divided by $3x^3 - 5x^2 + 3x - 8,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
|
0, 1, 2
| 0.833333 |
Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2 &=& c^2+d^2 &=& 2016, \\
ac &=& bd &=& 1024.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$.
|
127
| 0.083333 |
When the base-12 integer $2625_{12}$ is divided by $10$, what is the remainder?
|
9
| 0.666667 |
Let \( g(x) = 3x^5 - 4x^4 + 2x^3 - 28x^2 + 15x - 90 \). Find \( g(6) \) and evaluate \( g'(6) \) where \( g'(x) \) is the derivative of \( g(x) \).
|
15879
| 0.166667 |
I have 6 books, three of which are identical copies of a science book, and the other three books are different. In how many ways can I arrange them on a shelf?
|
120
| 0.833333 |
What is the sum of all integers in the set $\{1,2,3,4,5,6,7,8,9,10\}$ that are primitive roots $\pmod {11}$?
|
23
| 0.916667 |
Given that $\binom{24}{5}=42504$, and $\binom{24}{6}=134596$, find $\binom{26}{6}$.
|
230230
| 0.833333 |
Find the sum of the squares of the solutions to the equation
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\]
|
\frac{2008}{1005}
| 0.083333 |
Find the projection of the vector $\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$ onto the plane given by the equation $2x + 3y - z = 0$.
|
\begin{pmatrix} \frac{17}{7} \\ \frac{-5}{14} \\ \frac{53}{14} \end{pmatrix}
| 0.083333 |
In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(-12, 5)$?
|
13
| 0.416667 |
Determine the radius of the circle with equation \(x^2 + 8x + y^2 - 4y + 20 = 0\).
|
0
| 0.416667 |
Let \( m = \underbrace{33333333}_{8 \text{ digits}} \) and \( n = \underbrace{666666666}_{9 \text{ digits}} \).
What is \(\gcd(m, n)\)?
|
3
| 0.833333 |
If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 72$ and $\gcd(a,c) = 240$, what is the smallest possible value of $\gcd(b,c)$?
|
24
| 0.833333 |
At the conclusion of a match, each of the six members of a basketball team shakes hands with each of the six members of the opposite team, and all of the players shake hands with three referees. How many handshakes occur?
|
72
| 0.916667 |
What is the greatest positive integer that must divide the sum of the first twelve terms of any arithmetic sequence whose terms are positive integers?
|
6
| 0.916667 |
Find the largest constant $m$, so that for any positive real numbers $a, b, c, d,$ and $e$,
\[\sqrt{\frac{a}{b+c+d+e}} + \sqrt{\frac{b}{a+c+d+e}} + \sqrt{\frac{c}{a+b+d+e}} + \sqrt{\frac{d}{a+b+c+e}} > m.\]
|
2
| 0.916667 |
Simplify $\left((9 \times 10^8) \times 2^2\right) \div \left(3 \times 2^3 \times 10^3\right)$.
|
150,000
| 0.833333 |
How many numbers less than 30 are abundant numbers?
|
4
| 0.916667 |
I have four distinct mystery novels, four distinct fantasy novels, and four distinct biographies. I want to take exactly two books from different genres with me on a vacation. How many possible combinations of books can I choose?
|
48
| 0.916667 |
A circle has a radius of five inches. The distance from the center of the circle to chord $PQ$ is four inches. How many inches long is chord $PQ$, and what is the area of the circle? Express the length of $PQ$ in simplest radical form and the area in terms of $\pi$.
|
25\pi
| 0.333333 |
Let $f(x) = 3x - 2$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x) = f(x^{-1})$.
|
-8
| 0.916667 |
Determine the sum of the rational roots of the polynomial $h(x) = x^3 - 7x^2 + 8x + 6$.
|
0
| 0.5 |
A binary operation $\diamondsuit$ is defined such that $a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b) \cdot c$ and $a \diamondsuit a = 1$ for all nonzero $a, b, c$. Solve the equation $504 \diamondsuit (3 \diamondsuit x) = 50$.
|
x = \frac{25}{84}
| 0.583333 |
Find an ordered pair $(x, y)$ that solves the system: \begin{align*}
7x &= -10 - 3y, \\
4x &= 5y - 35.
\end{align*}
|
\left(-\frac{155}{47}, \frac{205}{47}\right)
| 0.25 |
In the diagram, $ABCD$ is a parallelogram with an area of 27. $CD$ is thrice the length of $AB$. What is the area of $\triangle ABC$?
[asy]
draw((0,0)--(2,3)--(10,3)--(8,0)--cycle);
draw((2,3)--(0,0));
label("$A$",(0,0),W);
label("$B$",(2,3),NW);
label("$C$",(10,3),NE);
label("$D$",(8,0),E);
[/asy]
|
13.5
| 0.583333 |
Let $A$ be the number of four-digit even numbers, and let $B$ be the number of four-digit numbers that are multiples of both 5 and 3. Calculate $A + B$.
|
5100
| 0.666667 |
A parallelogram has adjacent sides of lengths \( s \) units and \( 3s \) units, forming a 60-degree angle. What is the area of the parallelogram?
|
\frac{3s^2\sqrt{3}}{2}
| 0.916667 |
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
| 0.333333 |
For a real number \(x,\) find the maximum value of
\[
\frac{x^4}{x^8 + 2x^6 + 4x^4 + 8x^2 + 16}.
\]
|
\frac{1}{20}
| 0.166667 |
Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend?
|
12
| 0.75 |
Find the remainder when the polynomial \( x^{100} \) is divided by the polynomial \( (x^2 + 1)(x - 1) \).
|
1
| 0.75 |
A street has parallel curbs 50 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet and each stripe is 60 feet long. Find the distance, in feet, between the stripes.
|
\frac{50}{3}
| 0.333333 |
Two standard 8-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is either a prime number or a perfect square? Express your answer as a common fraction.
|
\frac{35}{64}
| 0.75 |
In a certain sequence, the first term is \(a_1 = 1010\) and the second term is \(a_2 = 1011\). The values of the remaining terms are chosen so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \(n \geq 1\). Determine \(a_{1000}\).
|
1676
| 0.166667 |
Suppose a drawer contains 3 shirts, 6 pairs of shorts, and 7 pairs of socks. If I randomly select four articles of clothing, what is the probability that I get one shirt, two pairs of shorts, and one pair of socks?
|
\frac{63}{364}
| 0.333333 |
Rationalize the denominator of $\frac{7}{\sqrt{175} - \sqrt{75}}$.
|
\frac{7(\sqrt{7} + \sqrt{3})}{20}
| 0.916667 |
Find the sum of the first seven terms in the geometric sequence $\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots$.
|
\frac{16383}{49152}
| 0.583333 |
How many integers between $200$ and $250$ have three different digits, all in increasing order?
|
11
| 0.166667 |
For what real value of $t$ is $\frac{-15-\sqrt{145}}{6}$ a root of $3x^2 + 15x + t$?
|
\frac{20}{3}
| 0.916667 |
The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 30, $x$ is three times $y$. Determine the value of $y$ when $x=-12$.
|
-14.0625
| 0.666667 |
For what value of $n$ is the four-digit number $315n$, with units digit $n$, divisible by 12?
|
6
| 0.916667 |
How many 3-letter words can we make from the letters A, B, C, D, and E, if we are allowed to repeat letters, and we must use the letters A and B at least once each in every word? (Here, a word is an arbitrary sequence of letters.)
|
24
| 0.666667 |
Determine the number of ways to arrange the letters of the word BANANAS.
|
420
| 0.916667 |
Find the sum of the series $2_6 + 4_6 + 6_6 + \cdots + 100_6$ in base $6$.
|
1330_6
| 0.666667 |
Let \(p\) and \(q\) be the solutions to the equation \(3x^2 - 7x - 6 = 0\). Compute the value of \((5p^3 - 5q^3)(p - q)^{-1}\).
|
\frac{335}{9}
| 0.916667 |
The pond in Jenny's garden grows algae such that the surface area covered by algae doubles each day. By day $24$, the pond is completely covered. Determine on which day the pond was only $12.5\%$ covered with algae.
|
21
| 0.75 |
Let $p(x)$ be a monic polynomial of degree 7 such that $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ and $p(6) = 6$,$p(0) = 0.$ Find $p(7).$
|
5047
| 0.583333 |
What is the median number of moons per celestial body? (Include Pluto, Eris, and Ceres in the list.) \begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0\\
Venus & 0\\
Earth & 1\\
Mars & 2\\
Jupiter & 20\\
Saturn & 25\\
Uranus & 15\\
Neptune & 2\\
Pluto & 5\\
Eris & 1\\
Ceres & 0\\
\end{tabular}
|
2
| 0.916667 |
How many ways are there to distribute 6 distinguishable balls into 4 indistinguishable boxes?
|
187
| 0.333333 |
Calculate the greatest integer less than or equal to the expression \[\frac{4^{150} + 3^{150}}{4^{145} + 3^{145}}.\]
|
1023
| 0.166667 |
A sphere intersects the xy-plane in a circle centered at $(1,3,0)$ with a radius of 2. The sphere also intersects the yz-plane in a circle centered at $(0,3,-8),$ with radius $r.$ Find $r$ and verify the radius of the sphere.
|
2\sqrt{17}
| 0.083333 |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one?
|
\frac{3}{4}
| 0.416667 |
A hurricane in Melbourne, Australia, caused $\$$45 million in damage. Initially, the exchange rate was 1.8 Australian dollars for 1 British pound. Due to economic changes during the storm, the exchange rate shifted to 1.7 Australian dollars for 1 British pound. Calculate the damage in British pounds considering an average exchange rate for the period.
|
25,714,286
| 0.666667 |
Billy and Bobbi each selected a positive integer less than 250. Billy's number is a multiple of 20, and Bobbi's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction.
|
\frac{1}{24}
| 0.833333 |
What is the value of $103^{4} - 4 \cdot 103^{3} + 6 \cdot 103^2 - 4 \cdot 103 + 1$?
|
108243216
| 0.333333 |
Let $c$ and $d$ be positive real numbers such that each of the equations $x^2 + cx + 3d = 0$ and $x^2 + 3dx + c = 0$ has real roots. Find the smallest possible value of $c + 3d$.
|
8
| 0.916667 |
In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find $GH$ to the nearest tenth. You may use a calculator for this problem.
|
14.3
| 0.333333 |
Let $p$, $q$, and $r$ be the roots of $x^3 - x - 6 = 0$. Find $\frac{1}{p+2} + \frac{1}{q+2} + \frac{1}{r+2}$.
|
\frac{11}{12}
| 0.833333 |
What is the simplified value of $$(10^{0.25})(10^{0.25})(10^{0.5})(10^{0.5})(10^{0.75})(10^{0.75})?$$
|
1000
| 0.833333 |
For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $15n$?
|
7
| 0.916667 |
The polynomial $g(x) = x^4 + px^3 + qx^2 + rx + s$ has real coefficients, and $g(3i) = g(1+2i) = 0$. What is $p + q + r + s$?
|
39
| 0.916667 |
Find all integer values of $n$ such that $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 5$.
|
11
| 0.75 |
Expanding $(1+0.1)^{2000}$ by the binomial theorem yields
\[ \sum_{k=0}^{2000} \binom{2000}{k} (0.1)^k = A_0 + A_1 + \dots + A_{2000}, \]
where $A_k = \binom{2000}{k} (0.1)^k$ for $k = 0, 1, 2, \ldots, 2000$. Determine the value of $k$ for which $A_k$ is the largest.
|
181
| 0.75 |
Solve for $x$: $|x - 4| = x^2 - 5x + 6$.
|
2 - \sqrt{2}
| 0.25 |
Construct a quadratic function that has zeroes at $x=1$ and $x=5$, and that takes the value $10$ when $x=3$.
|
-\frac{5}{2}x^2 + 15x - \frac{25}{2}
| 0.833333 |
There are two positive integers \( c \) for which the equation \[ 3x^2 + 7x + c = 0 \] has rational solutions. What is the product of those two values of \( c \)?
|
8
| 0.916667 |
Let $G$ and $H$ denote the centroid and orthocenter of triangle $ABC$, respectively. If $AG = AH$, find all possible values of angle $A$ (in degrees).
|
60^\circ
| 0.666667 |
A curve is described parametrically by:
\[
(x,y) = (3 \cos t + \sin t, 3 \sin t).
\]
The graph of the curve can be expressed in the form:
\[
ax^2 + bxy + cy^2 = 1.
\]
Determine the ordered triple $(a, b, c)$.
|
\left( \frac{1}{9}, -\frac{2}{27}, \frac{10}{81} \right)
| 0.666667 |
Let $\mathbf{u}_0$ be a vector. The vector $\mathbf{u}_0$ is projected onto $\begin{pmatrix} 4 \\ 2 \end{pmatrix},$ resulting in the vector $\mathbf{u}_1.$ The vector $\mathbf{u}_1$ is then projected onto $\begin{pmatrix} 2 \\ 2 \end{pmatrix},$ resulting in the vector $\mathbf{u}_2.$ Find the matrix that takes $\mathbf{u}_0$ to $\mathbf{u}_2.$
|
\begin{pmatrix} \frac{3}{5} & \frac{3}{10} \\ \frac{3}{5} & \frac{3}{10} \end{pmatrix}
| 0.75 |
Given that $x+\cos y=3000$ and $x+3000 \sin y=2999$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$.
|
2999
| 0.75 |
In triangle $ABC$, $AB = 25$ and $BC = 20$. Find the largest possible value of $\tan A$ and $\sin A$.
|
\frac{4}{5}
| 0.083333 |
A regular octahedron is formed by joining the centers of adjoining faces of a cube with a side length of 2 units. Calculate the ratio of the volume of the octahedron to the volume of the cube.
|
\frac{1}{6}
| 0.333333 |
How much greater, in square inches, is the area of a circle with a radius of 12 inches and a surrounding ring of 2 inches thickness than a circle of radius 10 inches? Express your answer in terms of $\pi$.
|
96\pi
| 0.083333 |
What is the largest prime factor of $1739$?
|
47
| 0.333333 |
Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $n$ be the number of ordered pairs of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.
|
250
| 0.416667 |
Points $A$, $B$, $C$, and $P$ are in space such that each of $\overline{PA}$, $\overline{PB}$, and $\overline{PC}$ is perpendicular to the other two. If $PA = PB = 12$ and $PC = 7$, what is the volume of pyramid $PABC$?
|
168 \text{ cubic units}
| 0.833333 |
Evaluate the expression $\sqrt{80} - 4\sqrt{5} + 3\sqrt{\frac{180}{3}}$ and express it as $\sqrt{M}$, where $M$ is an integer.
|
540
| 0.25 |
A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing?
|
\frac{3}{20}
| 0.5 |
Let $F_n$ be the Fibonacci sequence, defined by $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute:
\[
\sum_{n=0}^{\infty} \frac{F_n}{5^n}.
\]
|
\frac{5}{19}
| 0.416667 |
Find the minimum value of
\[(15 - x)(13 - x)(15 + x)(13 + x).\]
|
-784
| 0.083333 |
If
\[\tan x = \frac{3ab}{a^2 - b^2},\]where $a > b > 0$ and $0^\circ < x < 90^\circ,$ then find $\sin x$ in terms of $a$ and $b.$
|
\frac{3ab}{\sqrt{a^4 + 7a^2b^2 + b^4}}
| 0.833333 |
Commander Krel from Planet Verx has received 30 resumes for four specialized roles on his starship: Chief Science Officer, Head Engineer, Navigation Expert, and Communications Officer. Unfortunately, Commander Krel finds only 10 of these candidates appropriate for any role. Additionally, for the position of Chief Science Officer, only 4 of the suitable candidates have the specific scientific expertise required. In how many ways can Commander Krel staff his starship, considering these constraints?
|
2016
| 0.916667 |
Both roots of the quadratic equation $x^2 - 107x + k = 0$ are prime numbers. Find the number of possible values of $k.$
|
0
| 0.75 |
Bob's password now consists of a non-negative single-digit number followed by any letter (uppercase or lowercase) and another non-negative single-digit number. What is the probability that Bob's password consists of an odd single-digit number followed by an uppercase letter and a positive single-digit number?
|
\frac{9}{40}
| 0.083333 |
Solve $\log_8 x + \log_4 x^3 = 9$.
|
2^{\frac{54}{11}}
| 0.916667 |
Simplify the expression $(3x-2)\cdot(5x^{12} + 3x^{11} + 7x^9 + 3x^8)$ and express your answer as a polynomial in descending order of degrees.
|
15x^{13} - x^{12} - 6x^{11} + 21x^{10} - 5x^9 - 6x^8
| 0.666667 |
The average of five different positive whole numbers is $5.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the sum of the two smallest numbers after setting aside the smallest and largest numbers?
|
5
| 0.666667 |
In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 13$, and $BC = \sqrt{75}$. What is $\tan A$?
|
\frac{5\sqrt{3}}{\sqrt{94}}
| 0.416667 |
Suppose a real number $x$ satisfies \[\sqrt{64-x^2}-\sqrt{36-x^2}=4.\] What is the value of $\sqrt{64-x^2}+\sqrt{36-x^2}$?
|
7
| 0.833333 |
What are the last two digits in the sum of the factorials of the first 15 positive integers?
|
13
| 0.75 |
Alex has 12 different kinds of lunch meat and 11 different kinds of cheese. He can make a sandwich with one kind of meat and up to two kinds of cheese (it does not matter in which order he chooses the cheese). Additionally, there are 3 different kinds of bread he can choose from. How many different sandwiches could Alex make?
|
2412
| 0.083333 |
Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x-15$, the remainder is $8$, and when $Q(x)$ is divided by $x-19$, the remainder is $10$. What is the remainder when $Q(x)$ is divided by $(x-15)(x-19)$?
|
\frac{1}{2}x + \frac{1}{2}
| 0.833333 |
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