problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
There are 203 students in the third grade, which is 125 fewer than the fourth grade. How many students are there in total in the third and fourth grades? | 531 | 1 | 331.3125 | 331.3125 | -1 | |
Find the remainder when $x^{44} + x^{33} + x^{22} + x^{11} + 1$ is divided by $x^4 + x^3 + x^2 + x + 1.$ | 0 | 0.875 | 5,187.3125 | 4,758.071429 | 8,192 | |
Given real numbers \( a, b, c \) satisfy
\[
a^{2}+b^{2}-4a \leqslant 1, \quad b^{2}+c^{2}-8b \leqslant -3, \quad c^{2}+a^{2}-12c \leqslant -26,
\]
what is the value of \( (a+b)^{c} \)? | 27 | 0.3125 | 7,723.125 | 6,691.6 | 8,192 | |
Given that the variance of a set of data is $S^{2}=\frac{1}{5}[\left(6-10\right)^{2}+\left(9-10\right)^{2}+\left(a-10\right)^{2}+\left(11-10\right)^{2}+\left(b-10\right)^{2}]=6.8$, then the value of $a^{2}+b^{2}$ is ____. | 296 | 0.8125 | 4,261.125 | 3,667.769231 | 6,832.333333 | |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | 1 | 2,281.625 | 2,281.625 | -1 | |
Assign five volunteers randomly to three different positions A, B, and C for service, with each position having at least one volunteer.
(1) Calculate the probability that exactly two volunteers are assigned to position A.
(2) Let the random variable $\xi$ represent the number of these five volunteers serving at pos... | \frac{2}{5} | 0.3125 | 4,375.9375 | 3,680.2 | 4,692.181818 | |
Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score of the four students is 0, calculate the nu... | 44 | 0.0625 | 7,526.5625 | 6,095 | 7,622 | |
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\frac{1}{3}$ of all the ninth graders are paired with $\frac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth ... | \frac{4}{11} | 1. **Define Variables:**
Let $s$ be the number of sixth graders, and $n$ be the number of ninth graders.
2. **Set Up Equations:**
According to the problem, $\frac{1}{3}$ of all the ninth graders are paired with $\frac{2}{5}$ of all the sixth graders. This implies that the number of ninth graders paired with sixt... | 0.8125 | 2,220.125 | 2,200.384615 | 2,305.666667 |
If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ . | 1/9 | 0.5 | 7,020.25 | 5,848.5 | 8,192 | |
On bookshelf A, there are 4 English books and 2 Chinese books, while on bookshelf B, there are 2 English books and 3 Chinese books.
$(Ⅰ)$ Without replacement, 2 books are taken from bookshelf A, one at a time. Find the probability of getting an English book on the first draw and still getting an English book on the s... | \frac{93}{280} | 0.5 | 4,933 | 4,678.375 | 5,187.625 | |
Given that $x, y,$ and $z$ are real numbers that satisfy: \begin{align*} x &= \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}, \\ y &= \sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}, \\ z &= \sqrt{x^2 - \frac 1{36}}+\sqrt{y^2-\frac 1{36}}, \end{align*} and that $x+y+z = \frac{m}{\sqrt{n}},$ where $m$ and $n$ are posit... | 9 | Note that none of $x,y,z$ can be zero.
Each of the equations is in the form \[a=\sqrt{b^2-d^2}+\sqrt{c^2-d^2}\]
Isolate a radical and square the equation to get \[b^2-d^2=a^2-2a\sqrt{c^2-d^2}+c^2-d^2\]
Now cancel, and again isolate the radical, and square the equation to get \[a^4+b^4+c^4+2a^2c^2-2a^2b^2-2b^2c^2=4a^... | 0.0625 | 8,059.25 | 6,068 | 8,192 |
Let \( x, y, z \) be positive integers such that
\[ (x+y)(y+z) = 2016 \]
\[ (x+y)(z+x) = 1080 \]
Determine the smallest possible value for \( x+y+z \). | 61 | 0.625 | 6,929.875 | 6,383.7 | 7,840.166667 | |
On a given circle, six points $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. | \[
\frac{3}{10}
\] | First we give the circle an orientation (e.g., letting the circle be the unit circle in polar coordinates). Then, for any set of six points chosen on the circle, there are exactly $6!$ ways to label them one through six. Also, this does not affect the probability we wish to calculate. This will, however, make calculat... | 0 | 8,192 | -1 | 8,192 |
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111, R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=R_{24}/R_4$ is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in $Q$ is: | 15 | 1. **Expression for $R_k$:**
The number $R_k$ is defined as an integer consisting of $k$ ones in its decimal representation. We can express $R_k$ as:
\[
R_k = 111\ldots1 \quad (\text{$k$ times})
\]
This can be written as a sum of powers of 10:
\[
R_k = 1 + 10 + 10^2 + \ldots + 10^{k-1}
\]
U... | 0.1875 | 7,982 | 7,972.333333 | 7,984.230769 |
What is the volume, in cubic units, of a cube whose surface area is 600 square units? | 1000 | 1 | 1,003.5625 | 1,003.5625 | -1 | |
How many units are in the sum of the lengths of the two longest altitudes in a right triangle with sides $9$, $40$, and $41$? | 49 | 0.8125 | 2,608.5625 | 2,403.384615 | 3,497.666667 | |
For any positive integers \( m \) and \( n \), define \( r(m, n) \) as the remainder of \( m \div n \) (for example, \( r(8,3) \) represents the remainder of \( 8 \div 3 \), so \( r(8,3)=2 \)). What is the smallest positive integer solution satisfying the equation \( r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4 ... | 120 | 0 | 8,192 | -1 | 8,192 | |
In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$ | 112 | Since a $13-14-15$ triangle is a $5-12-13$ triangle and a $9-12-15$ triangle "glued" together on the $12$ side, $[ABC]=\frac{1}{2}\cdot12\cdot14=84$.
There are six points of intersection between $\Delta ABC$ and $\Delta A'B'C'$. Connect each of these points to $G$.
[asy] size(8cm); pair A,B,C,G,D,E,F,A_1,A_2,B_1,B_2,C... | 0 | 8,192 | -1 | 8,192 |
What is the modulo $13$ residue of $247+5 \cdot 39 + 7 \cdot 143 +4 \cdot 15?$ | 8 | 0.9375 | 3,512.9375 | 3,201 | 8,192 | |
A pipe with inside diameter 10'' is to carry water from a reservoir to a small town in an arid land. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of 2''-inside-diameter pipes of the same length needed to carry the same volume of water to the arid town? | 25 | 1 | 3,068.375 | 3,068.375 | -1 | |
In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is: | 6\dfrac{1}{2} | 1. **Identify Given Angles and Sides**: We are given that $\angle RFS = \angle FDR = \theta$, and the side lengths $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, and $FS = 7\frac{1}{2}$ inches.
2. **Apply the Law of Cosines in $\triangle FDR$**:
\[
FR^2 = FD^2 + DR^2 - 2 \cdot FD \cdot DR \cdot \cos(\theta)
... | 0 | 5,767.6875 | -1 | 5,767.6875 |
When $x = -1$, the value of the algebraic expression $2ax^{3} - 3bx + 8$ is $18$. Express the value of the algebraic expression $9b - 6a + 2$ in terms of the given information. | 32 | 1 | 1,779.875 | 1,779.875 | -1 | |
A polynomial $P$ with integer coefficients is called tricky if it has 4 as a root. A polynomial is called $k$-tiny if it has degree at most 7 and integer coefficients between $-k$ and $k$, inclusive. A polynomial is called nearly tricky if it is the sum of a tricky polynomial and a 1-tiny polynomial. Let $N$ be the num... | 64912347 | A tricky 7-tiny polynomial takes the form $$\left(c_{6} x^{6}+\ldots+c_{1} x+c_{0}\right)(x-4)$$ For each fixed value of $k, c_{k}-4 c_{k+1}$ should lie in $[-7,7]$, so if we fix $c_{k}$, there are around $15 / 4$ ways of choosing $c_{k+1}$. Therefore if we pick $c_{0}, \ldots, c_{6}$ in this order, there should be aro... | 0 | 8,192 | -1 | 8,192 |
How many ordered integer pairs $(x,y)$ ($0 \leq x,y < 31$) are there satisfying $(x^2-18)^2 \equiv y^2 \pmod{31}$? | 60 | 0.5625 | 6,291.9375 | 4,814.111111 | 8,192 | |
Let $S=\{1,2,4,8,16,32,64,128,256\}$. A subset $P$ of $S$ is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set $Q$ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets. | 5 | Clearly we may biject squarely sets with binary representations of perfect squares between 1 and $2^{0}+\cdots+2^{8}=2^{9}-1=511$, so there are 22 squarely sets, corresponding to $n^{2}$ for $n=1,2, \ldots, 22$. For convenience, we say $N$ is (super) squarely if and only if the set corresponding to $N$ is (super) squar... | 0 | 8,192 | -1 | 8,192 |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 0 | 6,602.125 | -1 | 6,602.125 | |
Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of
\[\frac{4z}{2x + y} + \frac{4x}{y + 2z} + \frac{y}{x + z}.\] | 3 | 0.375 | 7,398.0625 | 6,074.833333 | 8,192 | |
Let $M = 36 \cdot 36 \cdot 85 \cdot 128$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | \frac{1}{4094} | 0.375 | 7,174.1875 | 6,021 | 7,866.1 | |
If the solution set of the inequality system about $x$ is $\left\{\begin{array}{l}{x+1≤\frac{2x-5}{3}}\\{a-x＞1}\end{array}\right.$ is $x\leqslant -8$, and the solution of the fractional equation about $y$ is $4+\frac{y}{y-3}=\frac{a-1}{3-y}$ is a non-negative integer, then the sum of all integers $a$ that satisfy the c... | 24 | 0.375 | 6,363.75 | 6,747.166667 | 6,133.7 | |
What three-digit number with units digit 2 and hundreds digit 4 is divisible by 9? | 432 | 1 | 1,462.3125 | 1,462.3125 | -1 | |
Find the total length of the intervals on the number line where the inequalities $x < 1$ and $\sin (\log_{2} x) < 0$ hold. | \frac{2^{\pi}}{1+2^{\pi}} | 0 | 7,344.6875 | -1 | 7,344.6875 | |
Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$.
(1) Determine the range of the real number $a$ and the expression for $S_n$;
(2) Does there exist a real number $a$ such that $S_1$, $S_3$, $S_2$ form an arithmetic sequence? If it ex... | \frac{1}{2} | 0.125 | 7,008.625 | 7,441.5 | 6,946.785714 | |
If the number 79777 has the digit 9 crossed out, the result is the number 7777. How many different five-digit numbers exist from which 7777 can be obtained by crossing out one digit? | 45 | 0 | 7,739 | -1 | 7,739 | |
Adam and Simon start on bicycle trips from the same point at the same time. Adam travels east at 8mph and Simon travels south at 6mph. After how many hours are they 60 miles apart? | 6 | 1 | 1,515.25 | 1,515.25 | -1 | |
The number of real roots of the equation $\frac{x}{100} = \sin x$ is:
(32nd United States of America Mathematical Olympiad, 1981) | 63 | 0 | 8,192 | -1 | 8,192 | |
Rectangle $ABCD$ has $AB=8$ and $BC=6$. Point $M$ is the midpoint of diagonal $\overline{AC}$, and $E$ is on $AB$ with $\overline{ME} \perp \overline{AC}$. What is the area of $\triangle AME$? | \frac{75}{8} | 1. **Calculate the length of diagonal $AC$**:
Since $ABCD$ is a rectangle, by the Pythagorean Theorem, we have:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10.
\]
2. **Determine the length of $AM$ and $MC$**:
Since $M$ is the midpoint of diagonal $\overline{AC}$, we hav... | 1 | 3,973.4375 | 3,973.4375 | -1 |
In a certain company, there are 100 shareholders, and any 66 of them own no less than 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own? | 25 | 0 | 8,192 | -1 | 8,192 | |
The stem and leaf plot represents the heights, in inches, of the players on the Spring Vale Middle School girls' basketball team. What is the mean height of the players on the team? (Note: $5|3$ represents 53 inches.)
Height of the Players on the Basketball Team (inches)
$4|9$
$5|2\;3\;5\;8\;8\;9$
$6|0\;1\;1\;2\;6\... | 60 | 0.3125 | 848.3125 | 1,004.2 | 777.454545 | |
Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is | 80 | 1. **Identify the total cost and the number of people sharing the cost:**
The total cost to send a person to Mars is estimated to be $20$ billion dollars. This cost is shared equally among $250$ million people in the U.S.
2. **Convert the total cost into a consistent unit for easy calculation:**
Since the co... | 1 | 2,827.0625 | 2,827.0625 | -1 |
If the equation with respect to \( x \), \(\frac{x \lg^2 a - 1}{x + \lg a} = x\), has a solution set that contains only one element, then \( a \) equals \(\quad\) . | 10 | 0 | 5,215.625 | -1 | 5,215.625 | |
A toy factory has a total of 450 labor hours and 400 units of raw materials for production. Producing a bear requires 15 labor hours and 20 units of raw materials, with a selling price of 80 yuan; producing a cat requires 10 labor hours and 5 units of raw materials, with a selling price of 45 yuan. Under the constraint... | 2200 | 0.6875 | 3,880.1875 | 3,931.818182 | 3,766.6 | |
Fill in the blanks with numbers $1 \sim 3$, so that each row and each column contains exactly one number appearing twice. The numbers outside the table indicate how many numbers are visible from that direction. A number can block equal or smaller numbers. What is the four-digit number $\overline{\mathrm{ABCD}}$? | 2213 | 0 | 8,186.0625 | -1 | 8,186.0625 | |
Given the function $f(x)= \begin{cases} 2x-10, & x\leqslant 7 \\ \frac {1}{f(x-2)}, & x > 7 \end{cases}$, and the sequence ${a_{n}}={f(n)}$ where $n\in\mathbb{N}^{*}$, find the sum of the first 50 terms of the sequence ${a_{n}}$. | \frac {225}{4} | 0 | 8,184.8125 | -1 | 8,184.8125 | |
Given sets $A=\{x,\frac{y}{x},1\}$ and $B=\{{x}^{2},x+y,0\}$, if $A=B$, then $x^{2023}+y^{2024}=\_\_\_\_\_\_.$ | -1 | 0.25 | 6,450.9375 | 5,161 | 6,880.916667 | |
Two friends are playing tic-tac-toe. If Amy wins $\frac{3}{8}$ of the time, Lily wins $\frac{3}{10}$ of the time, and they tie the rest of the time, then what fraction of the time do they tie? | \frac{13}{40} | 1 | 1,335.875 | 1,335.875 | -1 | |
If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point | (1997,0) | We are given that $m$ and $b$ are real numbers and $mb > 0$. This implies that both $m$ and $b$ are either both positive or both negative.
1. **Plugging in $(0, 1997)$ into $y = mx + b$:**
\[
1997 = 0 \cdot m + b \implies b = 1997
\]
Since $b$ is positive, $m$ must also be positive to satisfy $mb > 0$.
2.... | 0 | 5,939.1875 | -1 | 5,939.1875 |
Let \( x, y, z \) be positive numbers that satisfy the following system of equations:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=12 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=28
\end{array}\right.
$$
Find the value of the expression \( x y + y z + x z \). | 16 | 0.625 | 7,156.6875 | 6,535.5 | 8,192 | |
Find the least positive integer $n$ such that $$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$$ | 1 | 0 | 8,192 | -1 | 8,192 | |
Define $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$? | h | 1. **Define the operation $\otimes$:** Given $x \otimes y = x^3 - y$.
2. **Calculate $h \otimes h$:** Using the definition, substitute $x = h$ and $y = h$:
\[
h \otimes h = h^3 - h.
\]
3. **Calculate $h \otimes (h \otimes h)$:** Now, substitute $x = h$ and $y = h \otimes h$ into the definition:
\[
h \o... | 1 | 2,263.1875 | 2,263.1875 | -1 |
The conference has 12 teams. Each team plays every other team twice and an additional 6 games against non-conference opponents. Calculate the total number of games in a season involving the conference teams. | 204 | 0.3125 | 4,092.4375 | 4,960.2 | 3,698 | |
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{2y} = \frac{1}{7}$, what is the least possible value of $xy$? | 98 | 1 | 4,582.125 | 4,582.125 | -1 | |
Group the set of positive odd numbers $\{1, 3, 5, \cdots\}$ in increasing order such that the $n$-th group has $(2n-1)$ odd numbers:
\[
\{1\}, \quad \{3, 5, 7\}, \quad \{9, 11, 13, 15, 17\}, \cdots
\]
(first group)(second group)(third group)
Determine which group 1991 belongs to. | 32 | 0.375 | 7,408.75 | 6,103.333333 | 8,192 | |
If $\log_5 (x+4)=3$, find $\log_{11} x$. | 2 | 1 | 1,457.1875 | 1,457.1875 | -1 | |
Let the function \( f(x) \) defined on \( (0, +\infty) \) satisfy \( f(x) > -\frac{3}{x} \) for any \( x \in (0, +\infty) \) and \( f\left(f(x) + \frac{3}{x}\right) = 2 \). Find \( f(5) \). | \frac{7}{5} | 0 | 7,086.375 | -1 | 7,086.375 | |
Suppose a point $P$ has coordinates $(m, n)$, where $m$ and $n$ are the points obtained by rolling a dice twice consecutively. The probability that point $P$ lies outside the circle $x^{2}+y^{2}=16$ is _______. | \frac {7}{9} | 0.5 | 7,167.8125 | 6,143.625 | 8,192 | |
An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is 11/5. Find the ratio of shaded region $D$ to the area of shaded region $A.$
[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,... | 408 | Let the distances from the apex to the parallel lines be $x$ and $y$ and the distance between the intersections be $a,b.$ We know the area ratio means $\frac{(x+4a)(y+4b)-(x+3a)(y+3b)}{(x+2a)(y+2b)-(x+a)(y+b)} =\frac{5}{11}$ which simplifying yields $ab = 3ay+3bx.$ The ratio we seek is $\frac{(x+6a)(y+6b)-(x+5a)(y+5b)}... | 0 | 8,192 | -1 | 8,192 |
Let $x_1<x_2<x_3$ be the three real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$. | 2 | Let $x =\frac{y}{\sqrt{2014}}.$ The original equation simplifies to $\frac{y^3}{2014} -\frac{4029y^2}{2014}+2 = 0 \implies y^3 - 4029y^2 + 4028=0.$ Here we clearly see that $y=1$ is a root. Dividing $y-1$ from the sum we find that $(y-1)(y^2-4028y-4028)=0.$ From simple bounding we see that $y=1$ is the middle root. The... | 0 | 7,878.625 | -1 | 7,878.625 |
Let $a_0 = 3,$ $b_0 = 4,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\] for all $n \ge 0.$ Calculate $b_7 = \frac{4^p}{3^q}$ for some integers $p$ and $q.$ | (1094,1093) | 0 | 7,627.125 | -1 | 7,627.125 | |
Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum of 10 at $x = 1$, find the value of $f(2)$. | 18 | 0.9375 | 4,274.6875 | 4,013.533333 | 8,192 | |
A right octagonal pyramid has two cross sections obtained by slicing the pyramid with planes parallel to the octagonal base. The area of the smaller cross section is $256\sqrt{2}$ square feet and the area of the larger cross section is $576\sqrt{2}$ square feet. The distance between the two planes is $12$ feet. Determi... | 36 | 0.75 | 4,834.75 | 4,155.916667 | 6,871.25 | |
Acme T-Shirt Company charges a $\$50$ set-up fee plus $\$9$ for each shirt printed. Beta T-shirt Company has no set up fee, but charges $\$14$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme? | 11 | 1 | 1,615.5 | 1,615.5 | -1 | |
Given the decimal representation of $\frac{1}{30^{30}}$, determine how many zeros immediately follow the decimal point. | 44 | 0.5 | 7,142.5625 | 6,093.125 | 8,192 | |
Let squares of one kind have a side of \(a\) units, another kind have a side of \(b\) units, and the original square have a side of \(c\) units. Then the area of the original square is given by \(c^{2}=n a^{2}+n b^{2}\).
Numbers satisfying this equation can be obtained by multiplying the equality \(5^{2}=4^{2}+3^{2}\... | 15 | 0.5625 | 6,736.5625 | 6,300.333333 | 7,297.428571 | |
Solve for $\log_{3} \sqrt{27} + \lg 25 + \lg 4 + 7^{\log_{7} 2} + (-9.8)^{0} = \_\_\_\_\_\_\_\_\_\_\_$. | \frac{13}{2} | 1 | 3,366.875 | 3,366.875 | -1 | |
The graph shows the birth month of 100 famous Americans. What percent of these people have March as their birth month?
[asy]
draw((0,0)--(24,0)--(24,16)--(0,16)--(0,0)--cycle,linewidth(1));
for(int i = 1; i < 16; ++i)
{
draw((-.5,i)--(24,i),gray);
}
filldraw((.5,0)--(1.5,0)--(1.5,5)--(.5,5)--(.5,0)--cycle,blue,line... | 8 | 0.1875 | 470.25 | 454.666667 | 473.846154 | |
Find the greatest value of $a$ such that $a^2-10a+21 \le 0$. | 7 | 1 | 1,412.8125 | 1,412.8125 | -1 | |
Find the exact value of $1+\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}}$. | \sqrt{2} | Let $x$ be what we are trying to find. $x-1=\frac{1}{1+\frac{2}{1+\frac{1}{1+\frac{2}{1+\ldots}}}} \Rightarrow \frac{1}{x-1}-1=\frac{2}{1+\frac{1}{1+\frac{2}{1+\cdots}}} \Rightarrow \frac{2}{\frac{1}{x-1}-1}=x \Rightarrow x^{2}-2=0$, so $x=\sqrt{2}$ since $x>0$. | 0.4375 | 5,881.375 | 4,730.142857 | 6,776.777778 |
Let $ABCDE$ be a convex pentagon with $AB \parallel CE, BC \parallel AD, AC \parallel DE, \angle ABC=120^\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
| 484 | 0 | 7,308.8125 | -1 | 7,308.8125 | |
Several different positive integers are written on a blackboard. The product of the smallest two of them is 16. The product of the largest two of them is 225. What is the sum of all the integers written on the blackboard? | 44 | 0.6875 | 6,072.0625 | 5,108.454545 | 8,192 | |
Simplify $$\frac{13!}{10! + 3\cdot 9!}$$ | 1320 | 1 | 2,632.125 | 2,632.125 | -1 | |
Find all five-digit numbers \(\overline{abcde}\) that are divisible by 9, and \(\overline{ace} - \overline{bda} = 760\). | 81828 | 0 | 8,192 | -1 | 8,192 | |
Which of the following is equal to $2017 - \frac{1}{2017}$? | $\frac{2018 \times 2016}{2017}$ | 0 | 7,393.4375 | -1 | 7,393.4375 | |
Given $m>0$ and $|x+1|+|2x-1|\geq m$ always holds, $a, b, c \in \mathbb{R}$ satisfy $a^2+2b^2+3c^2=m$. Find the minimum value of $a+2b+3c$. | -3 | 0.875 | 6,314.1875 | 6,045.928571 | 8,192 | |
On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges conne... | 34 | 0 | 8,192 | -1 | 8,192 | |
The difference between two numbers is 7.02. If the decimal point of the smaller number is moved one place to the right, it becomes the larger number. The larger number is \_\_\_\_\_\_, and the smaller number is \_\_\_\_\_\_. | 0.78 | 0.125 | 494.625 | 514 | 491.857143 | |
A teacher drew a rectangle $ABCD$ on the board. A student named Petya divided this rectangle into two rectangles with a line parallel to side $AB$. It turned out that the areas of these parts are in the ratio 1:2, and their perimeters are in the ratio 3:5 (in the same order). Another student named Vasya divided this re... | 20/19 | 0 | 7,368.6875 | -1 | 7,368.6875 | |
How many ways are there to arrange the letters of the word $\text{BA}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3$, in which the three A's and the two N's are considered different? | 720 | 0.8125 | 3,574 | 3,788.461538 | 2,644.666667 | |
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 28 | Let the large tetrahedron be $ABCD$, and the small tetrahedron be $WXYZ$, with $W$ on $ABC$, $X$ on $BCD$, $Y$ on $ACD$, and $Z$ on $ABD$. Clearly, the two regular tetrahedrons are similar, so if we can find the ratio of the sides, we can find the ratio of the volumes. Let $AB=1$, for our convenience. Dropping an altit... | 0.6875 | 7,080.8125 | 6,575.727273 | 8,192 |
A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? | 15 | To find the minimum number of small bottles needed to completely fill a large bottle, we need to divide the capacity of the large bottle by the capacity of the small bottle and then round up to the nearest whole number if necessary.
1. **Calculate the number of small bottles needed:**
\[
\text{Number of small bo... | 1 | 2,321.9375 | 2,321.9375 | -1 |
Given $a=1$, $b=2$, $C=\frac{2π}{3}$ in triangle $\triangle ABC$, calculate the value of $c$. | \sqrt{9} | 0 | 2,358.5 | -1 | 2,358.5 | |
In parallelogram $ABCD$ , $AB = 10$ , and $AB = 2BC$ . Let $M$ be the midpoint of $CD$ , and suppose that $BM = 2AM$ . Compute $AM$ . | 2\sqrt{5} | 0.8125 | 4,720.25 | 4,311.230769 | 6,492.666667 | |
A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted lin... | \frac{4}{5} | 1. **Assumptions and Setup**: Assume the width of the paper is $w$ and the length is $\sqrt{3}w$. The area of the paper, $A$, is given by:
\[
A = w \times \sqrt{3}w = \sqrt{3}w^2.
\]
Without loss of generality, let $w = 1$. Then the area $A = \sqrt{3}$.
2. **Dividing the Paper**: The paper is divided into ... | 0 | 8,192 | -1 | 8,192 |
Find the sum of the ages of everyone who wrote a problem for this year's HMMT November contest. If your answer is $X$ and the actual value is $Y$, your score will be $\max (0,20-|X-Y|)$ | 258 | There was one problem for which I could not determine author information, so I set the author as one of the problem czars at random. Then, I ran the following command on a folder containing TeX solutions files to all four contests: ``` evan@ArchMega ~/Downloads/November $ grep --no-filename "Proposed by: " *.tex | sort... | 0 | 6,806.625 | -1 | 6,806.625 |
Let $ 2^{1110} \equiv n \bmod{1111} $ with $ 0 \leq n < 1111 $ . Compute $ n $ . | 1024 | 0.375 | 7,389.1875 | 6,051.166667 | 8,192 | |
In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\overline{YE}$ and $\overline{XD}$, respectively. Find... | 67 | 0 | 8,192 | -1 | 8,192 | |
How many ordered pairs \((b, g)\) of positive integers with \(4 \leq b \leq g \leq 2007\) are there such that when \(b\) black balls and \(g\) gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is \(\frac{1}{2}\)? | 59 | 0.375 | 7,624.125 | 7,039 | 7,975.2 | |
What is the probability of drawing a number that is a multiple of 15 from a five-digit number formed without repeating digits using 1, 2, 3, 4, and 5? | \frac{1}{5} | 0.8125 | 3,186.4375 | 3,001.538462 | 3,987.666667 | |
Jasmine wishes to purchase some trading cards. She has $\$7.50$ and the cards each cost $\$0.85$, tax included. What is the most number of cards she can buy? | 8 | 1 | 3,707.5 | 3,707.5 | -1 | |
Suppose that $x = 2-t$ and $y = 4t + 7$. If $x = -3$, what is $y$? | 27 | 1 | 1,602.4375 | 1,602.4375 | -1 | |
Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient? | 231 | 0 | 8,192 | -1 | 8,192 | |
How many multiples of 15 are between 25 and 225? | 14 | 0.0625 | 4,257.4375 | 4,048 | 4,271.4 | |
In triangle $ABC$, $AB = 11$, $AC = 13$, and $BC = 20$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let $P$ be the foot of the altitude from $G$ to $BC$. Find $GP$.
[asy]
unitsize(0.3 cm);
pair A, B, C, D, E, F, G, P;
A = (44/5,33/5);
B = (0,0);
C = (20,0);
D = (B + C)/2;
E =... | \frac{11}{5} | 0.875 | 4,923.375 | 4,456.428571 | 8,192 | |
Mila and Zhenya each came up with a number and wrote down all the natural divisors of their numbers on the board. Mila wrote down 10 numbers, Zhenya wrote down 9 numbers, and the number 6 appeared twice. How many distinct numbers are on the board in total? | 18 | 0.1875 | 6,465.6875 | 6,677.666667 | 6,416.769231 | |
(1) In an arithmetic sequence $\{ a_n \}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.
(2) Determine the interval of decrease for the function $y = \log_{\frac{1}{2}}(x^2 + 2x - 3)$.
(3) In triangle $ABC$, given $a = \sqrt{3}$, $b = 1$, and $\angle A = \frac{\pi}{3}$, find the value of $\cos B$.
... | 10\pi | 0.3125 | 7,867.875 | 7,154.8 | 8,192 | |
Within a cube with edge length 6, there is a regular tetrahedron with edge length \( x \) that can rotate freely inside the cube. What is the maximum value of \( x \)? | 2\sqrt{6} | 0 | 8,075.8125 | -1 | 8,075.8125 | |
If $M = 2098 \div 2$, $N = M \times 2$, and $X = M + N$, what is the value of $X$? | 3147 | 1 | 248.1875 | 248.1875 | -1 | |
Let $M=123456789101112\dots5253$ be the $96$-digit number that is formed by writing integers from $1$ to $53$ in order. What is the remainder when $M$ is divided by $55$? | 53 | 0 | 7,500.0625 | -1 | 7,500.0625 | |
Erin walks $\frac{3}{5}$ of the way home in 30 minutes. If she continues to walk at the same rate, how many minutes will it take her to walk the rest of the way home? | 20 | Since Erin walks $\frac{3}{5}$ of the way home in 30 minutes, then she walks $\frac{1}{5}$ of the way at the same rate in 10 minutes. She has $1-\frac{3}{5}=\frac{2}{5}$ of the way left to walk. This is twice as far as $\frac{1}{5}$ of the way. Since she continues to walk at the same rate and $\frac{1}{5}$ of the way t... | 1 | 1,701.5 | 1,701.5 | -1 |
What is the product of all the positive integer divisors of 12? | 1728 | 1 | 2,710.5625 | 2,710.5625 | -1 | |
A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged... | 3 : 4 : 13 | 0.1875 | 4,678.3125 | 3,694.666667 | 4,905.307692 |
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