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 |
|---|---|---|---|---|---|---|
How many positive integers less than $200$ are multiples of $5$, but not multiples of either $10$ or $6$? | 20 | 0.375 | 7,675.5625 | 7,080.833333 | 8,032.4 | |
The volume of a regular triangular prism is $8$, the base edge length that minimizes the surface area of the prism is __________. | 2\sqrt[3]{4} | 0.625 | 5,351 | 5,018 | 5,906 | |
Calculate \( \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \frac{8}{7} \). Express the answer in decimal form, accurate to two decimal places. | 1.67 | 0 | 5,855.375 | -1 | 5,855.375 | |
Simplify: \\( \dfrac {\sin 7 ^{\circ} + \cos 15 ^{\circ} \sin 8 ^{\circ} }{\cos 7 ^{\circ} - \sin 15 ^{\circ} \sin 8 ^{\circ} }= \) \_\_\_\_\_\_ . | 2- \sqrt {3} | 0 | 6,276.75 | -1 | 6,276.75 | |
The Fibonacci sequence is defined as follows: $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \geq 2$. Find the smallest positive integer $m$ such that $F_{m} \equiv 0(\bmod 127)$ and $F_{m+1} \equiv 1(\bmod 127)$. | 256 | First, note that 5 is not a quadratic residue modulo 127. We are looking for the period of the Fibonacci numbers $\bmod 127$. Let $p=127$. We work in $\mathbb{F}_{p^{2}}$ for the remainder of this proof. Let $\alpha$ and $\beta$ be the roots of $x^{2}-x-1$. Then we know that $F_{n}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\b... | 0.0625 | 7,534.625 | 3,921 | 7,775.533333 |
A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them? | 144 | 0.125 | 7,904.375 | 6,059.5 | 8,167.928571 | |
A random point \(N\) on a line has coordinates \((t, -2-t)\), where \(t \in \mathbb{R}\). A random point \(M\) on a parabola has coordinates \(\left( x, x^2 - 4x + 5 \right)\), where \(x \in \mathbb{R}\). The square of the distance between points \(M\) and \(N\) is given by \(\rho^2(x, t) = (x - t)^2 + \left( x^2 - 4x... | \frac{19\sqrt{2}}{8} | 0 | 5,887 | -1 | 5,887 | |
Let $S=\{1,2,\ldots ,98\}$ . Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$ -element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other ... | 50 | 0.0625 | 7,924.8125 | 8,111 | 7,912.4 | |
Given that in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\sin A+a\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $b=2$, find the maximum area of $\triangle ABC$. | \sqrt{2}-1 | 0.9375 | 6,037.3125 | 5,893.666667 | 8,192 | |
Given the polynomial expression \( x^4 - 61x^2 + 60 \), for how many integers \( x \) is the expression negative. | 12 | 0.8125 | 4,276.5625 | 4,448.846154 | 3,530 | |
In the sequence \(\left\{a_{n}\right\}\), it is known that \(a_{1}=1\) and \(a_{n+1}>a_{n}\), and that \(a_{n+1}^{2}+a_{n}^{2}+1=2\left(a_{n+1}+a_{n}+2 a_{n+1} a_{n}\right)\). Find \(\lim \limits_{n \rightarrow \infty} \frac{S_{n}}{n a_{n}}\). | 1/3 | 0 | 8,192 | -1 | 8,192 | |
What is the value of $\sqrt[3]{3^5 + 3^5 + 3^5}$? | 9 | 1 | 1,553.375 | 1,553.375 | -1 | |
If $a>0$ and $b>0,$ a new operation $\Delta$ is defined as follows: $$a \Delta b = \frac{a^2 + b^2}{1 + ab}.$$ Calculate $(2 \Delta 3) \Delta 4$. | \frac{6661}{2891} | 0 | 6,420.1875 | -1 | 6,420.1875 | |
Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room. | 360 | 0.25 | 8,058.375 | 7,657.5 | 8,192 | |
Simplify and find the value of:<br/>$(1)$ If $a=2$ and $b=-1$, find the value of $(3{a^2}b+\frac{1}{4}a{b^2})-(\frac{3}{4}a{b^2}-{a^2}b)$.<br/>$(2)$ If the value of the algebraic expression $(2x^{2}+ax-y+6)-(2bx^{2}-3x+5y-1)$ is independent of the variable $x$, find the value of the algebraic expression $5ab^{2}-[a^{2}... | -60 | 0.875 | 4,211.1875 | 3,642.5 | 8,192 | |
Given that the random variable $\xi$ follows the normal distribution $N(1, 4)$, if $P(\xi > 4) = 0.1$, then $P(-2 \leq \xi \leq 4)$ equals _______. | 0.8 | 0.375 | 6,789.75 | 4,957 | 7,889.4 | |
Given that the ratio of bananas to yogurt to honey is 3:2:1, and that Linda has 10 bananas, 9 cups of yogurt, and 4 tablespoons of honey, determine the maximum number of servings of smoothies Linda can make. | 13 | 0 | 2,565.8125 | -1 | 2,565.8125 | |
What is the sum of the positive integer divisors of 23? | 24 | 1 | 1,262.25 | 1,262.25 | -1 | |
The diagram shows a triangle \(ABC\) and two lines \(AD\) and \(BE\), where \(D\) is the midpoint of \(BC\) and \(E\) lies on \(CA\). The lines \(AD\) and \(BE\) meet at \(Z\), the midpoint of \(AD\). What is the ratio of the length \(CE\) to the length \(EA\)? | 2:1 | 0.125 | 6,662.4375 | 4,798.5 | 6,928.714286 | |
What is the ratio of the sides of a triangle in which the sum of the lengths of the altitudes taken two at a time corresponds to the ratio 5:7:8? | 10:15:6 | 0.75 | 5,004.8125 | 4,506.416667 | 6,500 | |
What is the largest factor of $130000$ that does not contain the digit $0$ or $5$ ? | 26 | 0 | 8,100.625 | -1 | 8,100.625 | |
Given a rectangular coordinate system with origin point $O$, vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfy $|\overrightarrow {a}|=|\overrightarrow {b}|=1$ and $\overrightarrow {a}\cdot \overrightarrow {b}=\frac {1}{2}$. Let $\overrightarrow {c} = (m, 1-m)$ and $\overrightarrow {d} = (n, 1-n)$. For any... | \frac{\sqrt{6} - \sqrt{2}}{4} | 0 | 8,050.8125 | -1 | 8,050.8125 | |
Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________. | 111111_{(2)} | 0.6875 | 2,317.75 | 2,048.545455 | 2,910 | |
Given a cube $ABCD$-$A\_1B\_1C\_1D\_1$ with edge length $1$, point $M$ is the midpoint of $BC\_1$, and $P$ is a moving point on edge $BB\_1$. Determine the minimum value of $AP + MP$. | \frac{\sqrt{10}}{2} | 0 | 7,156.8125 | -1 | 7,156.8125 | |
Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$ | \frac{\sqrt{5} + 1}{2} | 0 | 7,472.8125 | -1 | 7,472.8125 | |
Box $A$ contains 1 red ball and 5 white balls, and box $B$ contains 3 white balls. Three balls are randomly taken from box $A$ and placed into box $B$. After mixing thoroughly, three balls are then randomly taken from box $B$ and placed back into box $A$. What is the probability that the red ball moves from box $A$ to ... | 1/4 | 0.0625 | 7,446.4375 | 8,192 | 7,396.733333 | |
Given that $x$, $y$, and $z$ are all non-negative numbers and $x + y + z = 2$, find the minimum value of $\frac{1}{3}x^{3} + y^{2} + z$. | \frac{13}{12} | 0.6875 | 6,122.875 | 5,182.363636 | 8,192 | |
Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$ Find the maximum value of
\[7x + 2y + 8z.\] | \frac{9}{2} | 0.875 | 5,102.375 | 4,661 | 8,192 | |
In $\triangle ABC$, it is given that $\cos A= \frac{5}{13}$, $\tan \frac{B}{2}+\cot \frac{B}{2}= \frac{10}{3}$, and $c=21$.
1. Find the value of $\cos (A-B)$;
2. Find the area of $\triangle ABC$. | 126 | 1 | 3,884.0625 | 3,884.0625 | -1 | |
Compute $\sin 315^\circ$. | -\frac{\sqrt{2}}{2} | 0 | 2,141.875 | -1 | 2,141.875 | |
In the figure, the visible gray area within the larger circle is equal to three times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.
[asy]size(101);
filldraw(Circle((0,0),2)^^Circle((.8,-.3),1),gray... | \frac{1}{2} | 0.5 | 7,194.1875 | 6,196.375 | 8,192 | |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, ... | 20\sqrt{5} | 0 | 8,192 | -1 | 8,192 | |
In the diagram below, we have $AB = 24$ and $\angle ADB =90^\circ$. If $\sin A = \frac23$ and $\sin C = \frac13$, then what is $DC$?
[asy]
pair A,B,C,D;
A = (0,0);
B = (8*sqrt(5),16);
D = (8*sqrt(5),0);
C = (8*sqrt(5) + 32*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label... | 32\sqrt{2} | 0.625 | 5,575.4375 | 4,279.3 | 7,735.666667 | |
Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to both the radius lines $OA$ and $OB$, and the arc $AB$. Determine the radius of the inscribed circle in centimeters. Express your answer in simplest radical form. | 6\sqrt{2} - 6 | 0.4375 | 4,298.5625 | 4,366.714286 | 4,245.555556 | |
There are 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positive integers). If the total number of candie... | 105 | 0 | 7,830.25 | -1 | 7,830.25 | |
Given that $x^2 + y^2 = 14x + 6y + 6,$ find the largest possible value of $3x + 4y.$ | 73 | 1 | 4,146.4375 | 4,146.4375 | -1 | |
Given that EF = 40 units, FG = 30 units, and one diagonal EH = 50 units, calculate the perimeter of parallelogram EFGH. | 140 | 0.875 | 2,128.875 | 2,373.285714 | 418 | |
You are given the digits $0$, $1$, $2$, $3$, $4$, $5$. Form a four-digit number with no repeating digits.
(I) How many different four-digit numbers can be formed?
(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit? | 100 | 0 | 7,745.5625 | -1 | 7,745.5625 | |
If $x = 3$ and $y = 5$, what is the value of $\frac{3x^4 + 2y^2 + 10}{8}$? | 37 | 0 | 2,265 | -1 | 2,265 | |
Compute the remainder when 10002000400080016003200640128025605121024204840968192 is divided by 100020004000800160032. | 40968192 | Let $X_{k}$ denote $2^{k}$ except with leading zeroes added to make it four digits long. Let $\overline{a b c \cdots}$ denote the number obtained upon concatenating $a, b, c, \ldots$ We have $$2^{6} \cdot \overline{X_{0} X_{1} \ldots X_{5}}=\overline{X_{6} X_{7} \ldots X_{11}}$$ Therefore, $\overline{X_{0} X_{1} \ldots... | 0.125 | 8,122.3125 | 7,634.5 | 8,192 |
Given that x > 0, y > 0, z > 0, and x + $\sqrt{3}$y + z = 6, find the minimum value of x³ + y² + 3z. | \frac{37}{4} | 0.9375 | 4,789.375 | 4,562.533333 | 8,192 | |
A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to... | 36\pi | 1. **Understanding the Problem:**
The paper is wrapped around a cardboard tube, forming concentric circles. Each wrap increases the diameter of the roll by the thickness of the paper, which is $5$ cm. The total number of wraps is $600$, and the final diameter of the roll is $10$ cm, starting from a diameter of $2$ c... | 0 | 7,794.125 | -1 | 7,794.125 |
In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are located on sides $YZ$, $XZ$, and $XY$, respectively. The cevians $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$. Given that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the value of $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$. | 98 | 0 | 8,023.9375 | -1 | 8,023.9375 | |
Let \\(f(x)=3\sin (\omega x+ \frac {\pi}{6})\\), where \\(\omega > 0\\) and \\(x\in(-\infty,+\infty)\\), and the function has a minimum period of \\(\frac {\pi}{2}\\).
\\((1)\\) Find \\(f(0)\\).
\\((2)\\) Find the expression for \\(f(x)\\).
\\((3)\\) Given that \\(f( \frac {\alpha}{4}+ \frac {\pi}{12})= \frac {9}{5}... | \frac {4}{5} | 0.375 | 6,981.875 | 5,332.833333 | 7,971.3 | |
Let \( M(x, y, z) \) represent the minimum of the three numbers \( x, y, z \). If the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c > 0 \)) has a zero, determine the maximum value of \( M \left( \frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c} \right) \). | \frac{5}{4} | 0.0625 | 7,894.8125 | 6,432 | 7,992.333333 | |
Find $\frac{7}{17} - \frac{4}{51}$. Reduce your answer to simplest form. | \frac{1}{3} | 1 | 1,840 | 1,840 | -1 | |
Arrange the positive integers whose digits sum to 4 in ascending order. Which position does the number 2020 occupy in this sequence? | 28 | 0.125 | 8,061.125 | 7,375 | 8,159.142857 | |
For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together? | $87.50 | Let $B$ represent the amount Ben spent and $D$ represent the amount David spent at the bagel store. According to the problem, for every dollar Ben spent, David spent $25$ cents less. This can be expressed as:
\[ D = B - 0.25B = 0.75B \]
This simplifies to:
\[ D = \frac{3}{4}B \]
We are also given that Ben paid $12.50 ... | 0 | 1,699.5625 | -1 | 1,699.5625 |
Given that $a$ is a positive integer and $a = b - 2005$, if the equation $x^2 - ax + b = 0$ has a positive integer solution, what is the minimum value of $a$?
(Hint: First, assume the two roots of the equation are $x_1$ and $x_2$, then…) | 95 | 0.9375 | 4,336.75 | 4,079.733333 | 8,192 | |
Let $x, y$ and $z$ be consecutive integers such that
\[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\]
Find the maximum value of $x + y + z$ . | 402 | 0 | 7,404.0625 | -1 | 7,404.0625 | |
The 5 on the tenths place is \_\_\_\_\_ more than the 5 on the hundredths place. | 0.45 | 0.1875 | 278.5 | 318.666667 | 269.230769 | |
In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$,
$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. Find the area of $\triangle BEC$. | 4.5 | 1. **Identify the Shape and Properties**: Given that $ABCD$ is a trapezoid with $\overline{AD} \perpendicular \overline{DC}$, and $AD = AB = 3$, $DC = 6$. Since $\overline{BE} \parallel \overline{AD}$ and both $AD$ and $BE$ are perpendicular to $DC$, $ABED$ forms a square.
2. **Determine the Length of $BE$ and $DE$**:... | 0 | 4,287.125 | -1 | 4,287.125 |
Find the smallest six-digit number that is divisible by 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits, and equal to the sum of the third and sixth digits. | 100122 | 0 | 8,192 | -1 | 8,192 | |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? | 42 | 0.8125 | 4,700.125 | 3,894.307692 | 8,192 | |
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be w... | 574 | Let $BC=a$, $AC=b$, and $AB=c$. Let $\cos\angle A=k$. Then $AP=bk$ and $AQ=ck$.
By Power of a Point theorem, \begin{align} AP\cdot BP=XP\cdot YP \quad &\Longrightarrow \quad b^2k^2-bck+400=0\\ AQ\cdot CQ=YQ\cdot XQ \quad &\Longrightarrow \quad c^2k^2-bck+525=0 \end{align} Thus $bck = (bk)^2+400=(ck)^2+525 = u$. Then $... | 0 | 8,192 | -1 | 8,192 |
Find the principal (smallest positive) period of the function
$$
y=(\arcsin (\sin (\arccos (\cos 3 x))))^{-5}
$$ | \frac{\pi}{3} | 0.0625 | 8,158.1875 | 8,192 | 8,155.933333 | |
Eight chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least four adjacent chairs. | 288 | 0 | 8,192 | -1 | 8,192 | |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=1+\dfrac{\sqrt{2}}{2}t \\ y=2+\dfrac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, an... | 2\sqrt{7} | 0.875 | 5,020.125 | 4,567 | 8,192 | |
Ms. Linda teaches mathematics to 22 students. Before she graded Eric's test, the average score for the class was 84. After grading Eric's test, the class average rose to 85. Determine Eric's score on the test. | 106 | 0.125 | 3,353.125 | 2,046 | 3,539.857143 | |
Two concentric circles have the same center, labeled $C$. The larger circle has a radius of $12$ units while the smaller circle has a radius of $7$ units. Determine the area of the ring formed between these two circles and also calculate the circumference of the larger circle. | 24\pi | 1 | 1,157.625 | 1,157.625 | -1 | |
A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches. At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer $n$, the king stirs his drink vigorously and takes a sip such that the he... | \frac{216 \pi^{3}-2187 \sqrt{3}}{8 \pi^{2}} | First, we find the total amount of juice consumed. We can simply subtract the amount of juice remaining at infinity from the initial amount of juice in the cup, which of course is simply the volume of the cup; we'll denote this value by $V$. Since volume in the cup varies as the cube of height, the amount of juice rema... | 0 | 7,605.875 | -1 | 7,605.875 |
If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0.8125 | 4,679.1875 | 3,868.538462 | 8,192 | |
How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
| 750 | 0.9375 | 5,547.875 | 5,371.6 | 8,192 | |
Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $d.$
[asy]import TrigMacros;
size(400);
real f(real x)
{
return 2*sin(3*x + pi) + 1;
}
draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-4,4,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);... | 1 | 1 | 2,327.625 | 2,327.625 | -1 | |
It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4? | 32 |
To determine how many numbers \( 2^k \), for \( 1 \leq k \leq 332 \), have the first digit as 4, we can approach the problem using logarithms to examine the leading digits.
### Step 1: Understanding the Leading Digit
For a number \( 2^k \) to have a first digit of 4, it must satisfy:
\[
4 \times 10^m \leq 2^k < 5 \t... | 0.375 | 7,854.625 | 7,292.333333 | 8,192 |
A factory produces a certain type of component that undergoes two processes. The probability of producing a defective component in the first and second processes is 0.01 and 0.03, respectively. The production of defective components in each process is independent. What is the probability that the component is of accept... | 0.960 | 0.875 | 2,750.375 | 2,494.5 | 4,541.5 | |
If $\sec x + \tan x = \frac{5}{2},$ then find $\sec x - \tan x.$ | \frac{2}{5} | 1 | 2,251.4375 | 2,251.4375 | -1 | |
Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$. | - \frac {3\pi}{4} | 0.3125 | 6,821.25 | 6,316.6 | 7,050.636364 | |
A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid). | 34 | 0.1875 | 6,419.5 | 4,764.333333 | 6,801.461538 | |
With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.
There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of ... | 1881 | 0.75 | 4,810.8125 | 4,035.416667 | 7,137 | |
Find the number of positive integers less than 1000000 which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\max \left(0,20-80\left|1-\frac{X}{Y}\right|\right)$ rounded to the nearest integer. | 247548 | $\mathrm{N}=1000000$ $\mathrm{s}=[0] * \mathrm{~N}$ ans $=0$ for i in range(1, N): if i <= s[i]: ans $+=1$ for $j$ in range(i + i, N, i): $s[j]+=$ i print(ans) | 0 | 7,874 | -1 | 7,874 |
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{7+3 \operatorname{tg} x}{(\sin x+2 \cos x)^{2}} d x
$$ | 3 \ln \left(\frac{3}{2}\right) + \frac{1}{6} | 0 | 7,303.25 | -1 | 7,303.25 | |
If $a = \log 9$ and $b = \log 16,$ compute
\[4^{a/b} + 3^{b/a}.\] | 7 | 1 | 3,794.125 | 3,794.125 | -1 | |
Let \(a\), \(b\), \(c\), and \(d\) be positive integers with \(a < 3b\), \(b < 4c\), and \(c < 5d\). If \(d < 80\), find the largest possible value for \(a\). | 4724 | 0.9375 | 5,846.5 | 5,690.133333 | 8,192 | |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$ | -48 | 1 | 1,915.1875 | 1,915.1875 | -1 | |
Our club has 20 members, 10 boys and 10 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender? Assume no one can hold both offices. | 180 | 1 | 2,606.625 | 2,606.625 | -1 | |
Given the Fibonacci sequence $\{a_n\}$, where each number from the third one is equal to the sum of the two preceding numbers, find the term of the Fibonacci sequence that corresponds to $\frac{{a_1}^2 + {a_2}^2 + {a_3}^2 + … + {a_{2017}}^2}{a_{2017}}$. | 2018 | 0 | 3,178.3125 | -1 | 3,178.3125 | |
Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$
(1) Find the minimum of $f_{2020}$.
(2) Find the minimum of $f_{2020} \cdot f_{2021}$. | 2 |
Let \(\{ z_n \}_{n \ge 1}\) be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer \(k\), \(|z_k z_{k+1}|=2^k\). Denote \(f_n=|z_1+z_2+\cdots+z_n|,\) for \(n=1,2,\cdots\).
1. To find the minimum of \(f_{2020}\):
Write \(a_k=z_k\) for \(k\) odd and ... | 0 | 8,192 | -1 | 8,192 |
How many natural numbers with up to six digits contain the digit 1? | 468559 | 0.3125 | 6,682.875 | 4,688.4 | 7,589.454545 | |
Given $|\vec{a}| = |\vec{b}| = 2$, and $(\vec{a} + 2\vec{b}) \cdot (\vec{a} - \vec{b}) = -2$, find the angle between $\vec{a}$ and $\vec{b}$. | \frac{\pi}{3} | 0.1875 | 2,479.75 | 2,221.666667 | 2,539.307692 | |
Find constants $A,$ $B,$ and $C$ so that
\[\frac{x^2 - 7}{(x - 2)(x - 3)(x - 5)} = \frac{A}{x - 2} + \frac{B}{x - 3} + \frac{C}{x - 5}.\]Enter the ordered triple $(A,B,C).$ | (-1,-1,3) | 0.9375 | 3,612.8125 | 3,307.533333 | 8,192 | |
Define a operation "\*" between sets A and B, where A\*B = {p | p = x + y, x ∈ A, y ∈ B}. If A = {1, 2, 3} and B = {1, 2}, then the sum of all elements in the set A\*B is ____. | 14 | 0.875 | 4,111.75 | 3,886.928571 | 5,685.5 | |
Given that the function $f(x)=2\cos x-3\sin x$ reaches its minimum value when $x=\theta$, calculate the value of $\tan \theta$. | \frac{3}{2} | 0 | 4,664.3125 | -1 | 4,664.3125 | |
A 6x6x6 cube is formed by assembling 216 unit cubes. Ten unit squares are painted on each of the six faces of the cube, leaving some rows and columns unpainted. Specifically, two non-adjacent columns and two non-adjacent rows on each face are left unpainted. How many of the 216 unit cubes have no paint on them? | 168 | 0 | 8,173.8125 | -1 | 8,173.8125 | |
Which triplet of numbers has a sum NOT equal to 1? | 1.1 + (-2.1) + 1.0 | To find which triplet of numbers has a sum NOT equal to 1, we will add the numbers in each triplet and check the result.
1. **Triplet (A):** $(1/2, 1/3, 1/6)$
\[
\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1
\]
2. **Triplet (B):** $(2, -2, 1)$
\[
2 ... | 0 | 1,037.1875 | -1 | 1,037.1875 |
A three-digit number has different digits in each position. By writing a 2 to the left of this three-digit number, we get a four-digit number; and by writing a 2 to the right of this three-digit number, we get another four-digit number. The difference between these two four-digit numbers is 945. What is this three-digi... | 327 | 0.6875 | 5,373.0625 | 4,652.545455 | 6,958.2 | |
How many decreasing sequences $a_1, a_2, \ldots, a_{2019}$ of positive integers are there such that $a_1\le 2019^2$ and $a_n + n$ is even for each $1 \le n \le 2019$ ? | \binom{2037171}{2019} | 0 | 8,192 | -1 | 8,192 | |
Let $x_1$ and $x_2$ be such that $x_1 \not= x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals | -\frac{h}{3} | 1. **Identify the quadratic equation**: Given the equations $3x_1^2 - hx_1 = b$ and $3x_2^2 - hx_2 = b$, we can rewrite these as $3x_1^2 - hx_1 - b = 0$ and $3x_2^2 - hx_2 - b = 0$. Since $x_1 \neq x_2$, both $x_1$ and $x_2$ are distinct roots of the quadratic equation:
\[
3x^2 - hx - b = 0.
\]
2. **Apply Vie... | 0 | 2,367.375 | -1 | 2,367.375 |
Print 90,000 five-digit numbers
$$
10000, 10001, \cdots, 99999
$$
on cards, with each card displaying one five-digit number. Some numbers printed on the cards (e.g., 19806 when reversed reads 90861) can be read in two different ways and may cause confusion. How many cards will display numbers that do not cause confusi... | 89100 | 0.25 | 3,630.875 | 2,026.75 | 4,165.583333 | |
How many four-digit numbers whose digits add up to $9$ are divisible by $11$? | 0 | 0.5625 | 6,820.875 | 5,754.444444 | 8,192 | |
In the cells of a $100 \times 100$ square, the numbers $1, 2, \ldots, 10000$ were placed, each exactly once, such that numbers differing by 1 are recorded in adjacent cells along the side. After that, the distances between the centers of each two cells, where the numbers in those cells differ exactly by 5000, were calc... | 50\sqrt{2} | 0.0625 | 7,921.6875 | 7,598 | 7,943.266667 | |
Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120^\circ$ and $\stackrel \frown {BC} = 72^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$ is:
$\textbf{(A)}\ \frac... | \frac{1}{3} | 0 | 7,476.4375 | -1 | 7,476.4375 | |
Without using any measuring tools or other auxiliary means, how can you cut exactly half a meter from a piece of cloth that is $\frac{8}{15}$ meters long? | 1/2 | 0.125 | 8,009.6875 | 6,733.5 | 8,192 | |
A store received a cistern of milk. The seller has a balance scale without weights (you can place cans on the pans of the scale) and three identical cans, two of which are empty and the third contains 1 liter of milk. How can you measure exactly 85 liters of milk into one can using no more than eight weighings? | 85 | 0 | 8,192 | -1 | 8,192 | |
Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right and then subtracting $n$ from the resulting number. For example, $b_4 = 1234 - 4 = 1230$ and $b_{12} = 123456789101112 - 12 = 123456789101100$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9? | 22 | 0 | 8,192 | -1 | 8,192 | |
Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n$. Compute the number of positive integers $n$ at most $10^{4}$ that satisfy $$s(11 n)=2 s(n)$$ | 2530 | Note $2 s(n)=s(10 n)+s(n)=s(11 n)$, so there cannot be any carries when adding $n$ and $10 n$. This is equivalent to saying no two consecutive digits of $n$ sum to greater than 9 . We change the problem to nonnegative integers less than $10^{4}$ (as both 0 and $10^{4}$ satisfy the condition) so that we simply consider ... | 0 | 8,192 | -1 | 8,192 |
Given two perpendicular lines, $2x + my - 1 = 0$ and $3x - 2y + n = 0$, with the foot of the perpendicular from the point $(2, p)$, find the value of $m + n + p$. | -6 | 1 | 4,422.5 | 4,422.5 | -1 | |
Find the smallest composite number that has no prime factors less than 15. | 323 | 0 | 7,454.25 | -1 | 7,454.25 | |
A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths? | 60 | 0.5625 | 2,721.5 | 1,937.222222 | 3,729.857143 | |
Suppose that $x$ is an integer that satisfies the following congruences: \begin{align*}
3+x &\equiv 2^2 \pmod{3^3} \\
5+x &\equiv 3^2 \pmod{5^3} \\
7+x &\equiv 5^2 \pmod{7^3}
\end{align*}What is the remainder when $x$ is divided by $105$? | 4 | 0.25 | 7,713.3125 | 6,277.25 | 8,192 |
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