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 |
|---|---|---|---|---|---|---|
Given any positive integer, we can write the integer in base 12 and add together the digits of its base 12 representation. We perform this operation on the number $7^{6^{5^{3^{2^{1}}}}}$ repeatedly until a single base 12 digit remains. Find this digit. | 4 | For a positive integer $n$, let $s(n)$ be the sum of digits when $n$ is expressed in base 12. We claim that $s(n) \equiv n(\bmod 11)$ for all positive integers $n$. Indeed, if $n=d_{k} 12^{k}+d_{k-1} 12^{k-1}+\cdots+d_{0}$ with each $d_{i}$ an integer between 0 and 11, inclusive, because $12 \equiv 1(\bmod 11)$, reduci... | 0.5 | 4,411.5625 | 3,168 | 5,655.125 |
Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\triangle DEF$ is $6$. What is the slope of $l_3$? | \frac{25}{32} | 0.0625 | 8,186.0625 | 8,192 | 8,185.666667 | |
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ | 50 | 1. **Identify the region and lattice points**:
The region is bounded by the lines $y = \pi x$, $y = -0.1$, and $x = 5.1$. We consider only the lattice points (points with integer coordinates) that lie within this region. The relevant lattice points along the x-axis from $x = 0$ to $x = 5$ are:
- $(0,0)$
- $(1,... | 0 | 7,762.0625 | -1 | 7,762.0625 |
If $9s+5t=108$ and $s$ is two less than $t$, what is $t$? | 9 | 1 | 1,466.9375 | 1,466.9375 | -1 | |
Given the functions $f(x)=\log_{a}x$ and $g(x)=\log_{a}(2x+t-2)$, where $a > 0$ and $a\neq 1$, $t\in R$.
(1) If $0 < a < 1$, and $x\in[\frac{1}{4},2]$ such that $2f(x)\geqslant g(x)$ always holds, find the range of values for the real number $t$;
(2) If $t=4$, and $x\in[\frac{1}{4},2]$ such that the minimum value of ... | a=\frac{1}{5} | 0.625 | 5,789.25 | 5,424.3 | 6,397.5 | |
Given that $x+y = 10$ and $2x+y = 13$, evaluate $x^2-y^2$. | -40 | 0.9375 | 2,886.3125 | 2,532.6 | 8,192 | |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? | 1 | 1. **Calculate the percentage of students scoring 95 points**:
Given that $10\%$ scored 70 points, $25\%$ scored 80 points, $20\%$ scored 85 points, and $15\%$ scored 90 points, the remaining percentage of students who scored 95 points is:
\[
100\% - (10\% + 25\% + 20\% + 15\%) = 100\% - 70\% = 30\%
\]
2.... | 0.8125 | 4,914.875 | 4,472.230769 | 6,833 |
A dormitory of a certain high school senior class has 8 people. In a health check, the weights of 7 people were measured to be 60, 55, 60, 55, 65, 50, 50 (in kilograms), respectively. One person was not measured due to some reasons, and it is known that the weight of this student is between 50 and 60 kilograms. The pro... | \frac{1}{2} | 0.5 | 7,431.8125 | 7,314.75 | 7,548.875 | |
A board game spinner is divided into three parts labeled $A$, $B$ and $C$. The probability of the spinner landing on $A$ is $\frac{1}{3}$ and the probability of the spinner landing on $B$ is $\frac{5}{12}$. What is the probability of the spinner landing on $C$? Express your answer as a common fraction. | \frac{1}{4} | 1 | 1,172.5625 | 1,172.5625 | -1 | |
The solid \( T \) consists of all points \((x,y,z)\) such that \( |x| + |y| \le 2 \), \( |x| + |z| \le 1 \), and \( |z| + |y| \le 1 \). Find the volume of \( T \). | \frac{4}{3} | 0.125 | 8,013.25 | 6,762 | 8,192 | |
An oreo shop now sells $5$ different flavors of oreos, $3$ different flavors of milk, and $2$ different flavors of cookies. Alpha and Gamma decide to purchase some items. Since Alpha is picky, he will order no more than two different items in total, avoiding replicas. To be equally strange, Gamma will only order oreos ... | 2100 | 0.0625 | 7,039.8125 | 5,903 | 7,115.6 | |
Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$ , in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$ . What is the largest possible size of $A$ ? | 10 | 0.0625 | 8,123.75 | 8,192 | 8,119.2 | |
Find the smallest value of \(a\) for which the sum of the squares of the roots of the equation \(x^{2}-3ax+a^{2}=0\) is equal to \(0.28\). | -0.2 | 1 | 2,779.3125 | 2,779.3125 | -1 | |
Given that 70% of the light bulbs are produced by Factory A with a pass rate of 95%, and 30% are produced by Factory B with a pass rate of 80%, calculate the probability of buying a qualified light bulb produced by Factory A from the market. | 0.665 | 0.4375 | 3,640.75 | 2,315.428571 | 4,671.555556 | |
Let $k$ be a real number such that the product of real roots of the equation $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$ is $-2013$ . Find the sum of the squares of these real roots. | 4027 | 0.3125 | 7,139.125 | 5,467.4 | 7,899 | |
If for any real number \( x \), the function
\[ f(x)=x^{2}-2x-|x-1-a|-|x-2|+4 \]
always yields a non-negative real number, then the minimum value of the real number \( a \) is . | -2 | 0.1875 | 8,098.25 | 7,787 | 8,170.076923 | |
Jane places six ounces of tea into a ten-ounce cup and six ounces of milk into a second cup of the same size. She then pours two ounces of tea from the first cup to the second and, after stirring thoroughly, pours two ounces of the liquid in the second cup back to the first. What fraction of the liquid in the first cup... | \frac{1}{4} | 0 | 5,039.5625 | -1 | 5,039.5625 | |
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smalle... | 2031 | Let $A$ represent the portion of $N$ to the right of the deleted zero, and $B$ represent the rest of $N$. For example, if the unique zero in $N=12034$ is removed, then $A=34$ and $B=12000$. Then, $\frac{M}{N}=\frac{A+B / 10}{A+B}=1-\frac{9}{10} \frac{B}{N}$. The maximum value for $B / N$ is 1 , which is achieved when $... | 0 | 8,192 | -1 | 8,192 |
How many two-digit numbers are there in which the tens digit is greater than the ones digit? | 45 | 0.6875 | 6,129.125 | 5,191.454545 | 8,192 | |
Given the parabola $y = -2x^2 + 4x + m$.
1. For what value of $m$ does the parabola intersect the x-axis at exactly one point?
2. If two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the parabola have $x$-coordinates satisfying $x_1 > x_2 > 2$, compare the values of $y_1$ and $y_2$. | -2 | 0.3125 | 2,868.3125 | 2,980.8 | 2,817.181818 | |
Solve the equations:<br/>$(1)x^{2}-5x+1=0$;<br/>$(2) 2\left(x-5\right)^{2}+x\left(x-5\right)=0$. | \frac{10}{3} | 0.5 | 1,786.1875 | 2,197.625 | 1,374.75 | |
There are six rock specimens with weights of 8.5 kg, 6 kg, 4 kg, 4 kg, 3 kg, and 2 kg. They need to be distributed into three backpacks such that the heaviest backpack is as light as possible. What is the weight of the rock specimens in the heaviest backpack? | 10 | 0.0625 | 8,144.9375 | 7,919 | 8,160 | |
Given an arithmetic sequence $\{a_n\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \ldots + a_{99}$. | -66 | 0 | 7,088.25 | -1 | 7,088.25 | |
How many six-digit numbers are there in which each subsequent digit is smaller than the previous one? | 210 | 0.6875 | 3,450.875 | 3,633.181818 | 3,049.8 | |
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|.$ | 27 | Playing around with a couple numbers, we see that we can generate the sequence $0, 3, -6, 3, -6, \cdots$, and we can also generate the sequence $3, 6, 9, 12, \cdots$ after each $-6$ value. Thus, we will apply this to try and find some bounds. We can test if the first $1000$ pairs of numbers each sum up to $-3$, and the... | 0 | 8,015.0625 | -1 | 8,015.0625 |
Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\mathcal{R}$ into two regions of equal area. Line $l$'s equation can be expressed in the form $ax=by+c,$ where $a, b,$ and ... | 65 | 0 | 8,099.3125 | -1 | 8,099.3125 | |
Find the sum of all solutions to the equation $(x-6)^2=25$. | 12 | 1 | 1,319.0625 | 1,319.0625 | -1 | |
Given the parametric equation of curve $C\_1$ is $\begin{cases} x=3\cos \alpha \ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C\_2$ is $\rho \cos \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$.
(I) Find the rectangular coordinate equation of curve $C\_2$ an... | \frac{6 \sqrt{3}}{5} | 0 | 7,066 | -1 | 7,066 | |
Find the smallest positive integer $M$ such that both $M$ and $M^2$ end in the same sequence of three digits $xyz$ when written in base $10$, where $x$ is not zero. | 376 | 0.625 | 6,493.5 | 5,474.4 | 8,192 | |
Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? | 2 - \frac{6}{\pi} | The probability is $2 - \frac{6}{\pi}$.
Set coordinates so that the original tiling includes the (filled) square
$S = \{(x,y): 0 \leq x,y \leq 1 \}$. It is then equivalent to choose the second square by first choosing a point uniformly at random in $S$ to be the center of the square, then choosing an angle of rotatio... | 0 | 8,192 | -1 | 8,192 |
Calculate the following infinite product: $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$ | 3^{\frac{3}{4}} | 0 | 4,940.125 | -1 | 4,940.125 | |
If point \( P \) is the circumcenter of \(\triangle ABC\) and \(\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = \mathbf{0}\), where \(\angle C = 120^\circ\), then find the value of the real number \(\lambda\). | -1 | 0.25 | 7,729.25 | 6,600.25 | 8,105.583333 | |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent piece. What is the probability that at least 40 cents worth of coins come up heads? | \frac{9}{16} | 0.4375 | 7,161.0625 | 6,251 | 7,868.888889 | |
What is the total distance on a Cartesian coordinate plane starting at $(2, -3)$, going to $(8, 9)$, and then moving to $(3, 2)?$ | 6\sqrt{5} + \sqrt{74} | 1 | 2,049.6875 | 2,049.6875 | -1 | |
Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism. | 8 | Note that since there are 8 faces to a regular hexagonal prism and a cross-section may only intersect a face once, the upper bound for our answer is 8. Indeed, we can construct a cross-section of the prism with 8 sides. Let $ABCDEF$ and $A'B'C'D'E'F'$ be the two bases of the prism, with $A$ being directly over $A'$. Ch... | 0 | 7,474.5 | -1 | 7,474.5 |
A barn with a roof is rectangular in shape, $12$ yd. wide, $15$ yd. long, and $6$ yd. high. Calculate the total area to be painted. | 828 | 0 | 4,410.5 | -1 | 4,410.5 | |
Find $b^2$ if the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide. | \frac{908}{49} | 0.875 | 4,817.6875 | 4,553.071429 | 6,670 | |
Three players are playing table tennis. The player who loses a game gives up their spot to the player who did not participate in that game. In the end, it turns out that the first player played 10 games, and the second player played 21 games. How many games did the third player play? | 11 | 0 | 8,192 | -1 | 8,192 | |
Container I holds 8 red balls and 4 green balls; containers II and III each hold 2 red balls and 4 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer as a common fraction. | \frac{5}{9} | 0.875 | 4,584.8125 | 4,069.5 | 8,192 | |
If $N$ is a positive integer between 1000000 and 10000000, inclusive, what is the maximum possible value for the sum of the digits of $25 \times N$? | 67 | Since $N$ is between 1000000 and 10000000, inclusive, then $25 \times N$ is between 25000000 and 250000000, inclusive, and so $25 \times N$ has 8 digits or it has 9 digits. We consider the value of $25 \times N$ as having 9 digits, with the possibility that the first digit could be 0. Since $25 \times N$ is a multiple ... | 0 | 8,192 | -1 | 8,192 |
Consider a sequence of positive real numbers where \( a_1, a_2, \dots \) satisfy
\[ a_n = 9a_{n-1} - n \]
for all \( n > 1 \). Find the smallest possible value of \( a_1 \). | \frac{17}{64} | 0.625 | 6,822.125 | 6,000.2 | 8,192 | |
What is the ordered pair of integers $(x,y)$ for which $12x + 21y = 15$ and $21x + 12y = 51$? | (3,-1) | 1 | 2,161 | 2,161 | -1 | |
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it also holds that \( x y z \mid (x+y+z)^{n} \). | 13 | 0 | 8,192 | -1 | 8,192 | |
Given that the two trisection points on the minor axis of an ellipse and its two foci form a square, find the eccentricity of the ellipse. | \frac{\sqrt{10}}{10} | 0 | 6,067.375 | -1 | 6,067.375 | |
Six equilateral triangles, each with side $4$, are arranged in a line such that the midpoint of the base of one triangle is the vertex of the next triangle. Calculate the area of the region of the plane that is covered by the union of the six triangular regions.
A) $16\sqrt{3}$
B) $19\sqrt{3}$
C) $24\sqrt{3}$
D) $18\sq... | 19\sqrt{3} | 0 | 8,121.625 | -1 | 8,121.625 | |
Given the parametric equation of line $l$ as $\begin{cases}x=-\frac{1}{2}+\frac{\sqrt{2}}{2}t \\ y=\frac{1}{2}+\frac{\sqrt{2}}{2}t\end{cases}$ and the parametric equation of ellipse $C$ as $\begin{cases}x=2\cos\theta \\ y=\sqrt{3}\sin\theta\end{cases}$, in a polar coordinate system with the origin as the pole and the p... | \frac{41}{14} | 0.5 | 7,023.25 | 5,946.25 | 8,100.25 | |
Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0,$ and let $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\theta$ counter-clockwise. If
\[\mathbf{R} \mathbf{D} = \begin{pmatrix} 8 & -4 \\ 4 & 8 \end{pmatrix},\]then find $\tan \theta.$ | \frac{1}{2} | 1 | 2,612.375 | 2,612.375 | -1 | |
Maria ordered a certain number of televisions at $R$ \$ 1994.00 each. She noticed that in the total amount to be paid, there are no digits 0, 7, 8, or 9. What was the smallest number of televisions she ordered? | 56 | 0.125 | 7,859.625 | 5,533 | 8,192 | |
If $2 \times 2 \times 3 \times 3 \times 5 \times 6=5 \times 6 \times n \times n$, what is a possible value of $n$? | 6 | We rewrite the left side of the given equation as $5 \times 6 \times(2 \times 3) \times(2 \times 3)$. Since $5 \times 6 \times(2 \times 3) \times(2 \times 3)=5 \times 6 \times n \times n$, then a possible value of $n$ is $2 \times 3$ or 6. | 0.9375 | 1,900.9375 | 1,991.6 | 541 |
A class prepared 5 programs to participate in the Xiamen No.1 Middle School Music Square event (this event only has 5 programs), and the order of the programs has the following requirements: Program A must be in the first two positions, Program B cannot be in the first position, and Program C must be in the last positi... | 10 | 0.5625 | 6,886 | 6,461.222222 | 7,432.142857 | |
Given $x = \frac{2}{3}$ and $y = \frac{5}{2}$, find the value of $\frac{1}{3}x^8y^9$. | \frac{5^9}{2 \cdot 3^9} | 0 | 3,587.4375 | -1 | 3,587.4375 | |
A spiral staircase turns $270^\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. | 17.3 | 0.875 | 6,191.6875 | 5,905.928571 | 8,192 | |
Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations $$\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \text { and } \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4$$ | (-\frac{13}{96}, \frac{13}{40}) | Solution 1: Consider vectors $$\binom{x / \sqrt{x^{2}+y^{2}}}{y / \sqrt{x^{2}+y^{2}}} \text { and }\binom{-1 / x}{1 / y}$$ They are orthogonal and add up to $\binom{7}{4}$, which have length $\sqrt{7^{2}+4^{2}}=\sqrt{65}$. The first vector has length 1, so by Pythagorean's theorem, the second vector has length $\sqrt{6... | 0.375 | 7,223.9375 | 5,610.5 | 8,192 |
A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\pi$. | \frac{\pi}{6} | 1 | 1,449 | 1,449 | -1 | |
The sum of six consecutive positive integers is 2013. What is the largest of these six integers? | 338 | Let the six consecutive integers be $n, n+1, n+2, n+3, n+4, n+5$. The sum of these integers can be expressed as:
\[
n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) = 6n + 15
\]
We are given that this sum equals 2013:
\[
6n + 15 = 2013
\]
To solve for $n$, we first subtract 15 from both sides:
\[
6n = 2013 - 15 = 1998
\]
Next... | 1 | 2,303.9375 | 2,303.9375 | -1 |
Add 3 digits after 325 to make a six-digit number such that it is divisible by 3, 4, and 5, and make this number as small as possible. What is the new six-digit number? | 325020 | 0.25 | 7,771.25 | 6,509 | 8,192 | |
What is the value of $\frac{(2200 - 2096)^2}{121}$? | 89 | 0 | 6,489.5 | -1 | 6,489.5 | |
Alice and Bob live on the same road. At time $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$ , and Bob arrived at Alice's house at $3:29\text{pm}$ . Cha... | 179 | 0.125 | 7,339.8125 | 5,790 | 7,561.214286 | |
Given the equation in terms of \( x \)
$$
x^{4}-16 x^{3}+(81-2a) x^{2}+(16a-142) x + a^{2} - 21a + 68 = 0
$$
where all roots are integers, find the value of \( a \) and solve the equation. | -4 | 0 | 8,192 | -1 | 8,192 | |
Four friends make cookies from the same amount of dough with the same thickness. Art's cookies are circles with a radius of 2 inches, and Trisha's cookies are squares with a side length of 4 inches. If Art can make 18 cookies in his batch, determine the number of cookies Trisha will make in one batch. | 14 | 0.625 | 6,394 | 5,315.2 | 8,192 | |
A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is | 5 | To find the smallest possible range of a set of five observations with an arithmetic mean of $10$ and a median of $12$, we need to consider the properties of the mean and median, and how they affect the arrangement of the numbers in the set.
1. **Understanding the Mean and Median**:
- The arithmetic mean of the obs... | 0.6875 | 6,013.125 | 5,438.909091 | 7,276.4 |
A circle inscribed in triangle \( ABC \) divides median \( BM \) into three equal parts. Find the ratio \( BC: CA: AB \). | 5:10:13 | 0 | 8,192 | -1 | 8,192 | |
How many numbers are in the list $-58, -51, -44, \ldots, 71, 78$? | 20 | 0.125 | 8,056.5625 | 7,108.5 | 8,192 | |
A pentagon is formed by placing an equilateral triangle on top of a square. Calculate the percentage of the pentagon's total area that is made up by the equilateral triangle. | 25.4551\% | 0 | 5,328.25 | -1 | 5,328.25 | |
Given that $2$ boys and $4$ girls are lined up, calculate the probability that the boys are neither adjacent nor at the ends. | \frac{1}{5} | 0.4375 | 7,466.3125 | 6,533.285714 | 8,192 | |
Find all positive integers $k<202$ for which there exist a positive integers $n$ such that
$$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$ | 1, 100, 101, 201 |
To solve the given problem, we need to find all positive integers \( k < 202 \) such that there exists a positive integer \( n \) satisfying the condition:
\[
\left\{\frac{n}{202}\right\} + \left\{\frac{2n}{202}\right\} + \cdots + \left\{\frac{kn}{202}\right\} = \frac{k}{2}
\]
Here, \(\left\{x\right\}\) denotes the ... | 0 | 7,957.9375 | -1 | 7,957.9375 |
In the polar coordinate system, the polar equation of curve $C$ is $\rho =6\sin \theta$, and the polar coordinates of point $P$ are $(\sqrt{2},\frac{\pi }{4})$. Taking the pole as the origin of coordinates and the positive half-axis of the $x$-axis as the polar axis, a plane rectangular coordinate system is established... | 3 \sqrt{2} | 0.5 | 6,937 | 5,952.125 | 7,921.875 | |
An educational center received an object with a volume of around 150 monoliths (a container designed for 150 monoliths that was almost full). Each monolith is identified either as sandy loam or loam and classified by its genesis as either marine or lake-glacial deposits. The relative frequency (statistical probability)... | 80 | 0 | 7,508 | -1 | 7,508 | |
The graph of the function $y=\log_a(x+3)-1$ ($a > 0$ and $a \neq 1$) always passes through a fixed point $A$. If the point $A$ lies on the line $mx+ny+1=0$, where $mn > 0$, then the minimum value of $\frac{1}{m}+\frac{1}{n}$ is | 3+2\sqrt{2} | 0.6875 | 6,488.125 | 5,713.636364 | 8,192 | |
The area of a quadrilateral is 3 cm², and the lengths of its diagonals are 6 cm and 2 cm. Find the angle between the diagonals. | 30 | 0.9375 | 3,394.9375 | 3,075.133333 | 8,192 | |
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ${p}_{a,b}$ be the sum of the zeros of $P(x)$. Find the sum of the ${p}_{a,b}$'s for all possible combinations of $a$ and $b$... | 80 | Since $r+si$ is a root, by the Complex Conjugate Root Theorem, $r-si$ must be the other imaginary root. Using $q$ to represent the real root, we have
$(x-q)(x-r-si)(x-r+si) = x^3 -ax^2 + bx -65$
Applying difference of squares, and regrouping, we have
$(x-q)(x^2 - 2rx + (r^2 + s^2)) = x^3 -ax^2 + bx -65$
So matching c... | 0 | 7,686.6875 | -1 | 7,686.6875 |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existi... | \frac{1}{8} | 0 | 8,007.8125 | -1 | 8,007.8125 | |
Four unit circles are centered at the vertices of a unit square, one circle at each vertex. What is the area of the region common to all four circles? | \frac{\pi}{3}+1-\sqrt{3} | The desired region consists of a small square and four "circle segments," i.e. regions of a circle bounded by a chord and an arc. The side of this small square is just the chord of a unit circle that cuts off an angle of $30^{\circ}$, and the circle segments are bounded by that chord and the circle. Using the law of co... | 0 | 8,192 | -1 | 8,192 |
Juan wants to calculate the area of a large circular garden, whose actual diameter is 50 meters. Unfortunately, his measurement device has an accuracy error of up to 30%. Compute the largest possible percent error, in percent, in Juan’s computed area of the circle in square meters. | 69\% | 0.8125 | 4,182.875 | 3,315.384615 | 7,942 | |
There is a championship where 16 football teams participate, each playing with every other team exactly once. What is the minimum number of games that must be played so that among any three teams, there are at least two that have already played against each other? | 56 | 0.8125 | 5,093.1875 | 4,378.076923 | 8,192 | |
Compute
\[\frac{1}{2^{1990}} \sum_{n = 0}^{995} (-3)^n \binom{1990}{2n}.\] | -\frac{1}{2} | 0.5625 | 6,446.875 | 5,527.555556 | 7,628.857143 | |
Petya can draw only 4 things: a sun, a ball, a tomato, and a banana. Today he drew several things, including exactly 15 yellow items, 18 round items, and 13 edible items. What is the maximum number of balls he could have drawn?
Petya believes that all tomatoes are round and red, all balls are round and can be of any c... | 18 | 0.0625 | 6,964.9375 | 3,381 | 7,203.866667 | |
What is the value of $x$ if $x=\frac{2023^2 - 2023 + 1}{2023}$? | 2022 + \frac{1}{2023} | 0.3125 | 3,873.4375 | 4,443.6 | 3,614.272727 | |
A box contains 5 white balls and 3 black balls. What is the probability that when drawing the balls one at a time, all draws alternate in color starting with a white ball? | \frac{1}{56} | 0.4375 | 5,395.5 | 4,726 | 5,916.222222 | |
The smallest of nine consecutive integers is 2012. These nine integers are placed in the circles to the right. The sum of the three integers along each of the four lines is the same. If this sum is as small as possible, what is the value of $u$? | 2015 | If we have a configuration of the numbers that has the required property, then we can add or subtract the same number from each of the numbers in the circles and maintain the property. (This is because there are the same number of circles in each line.) Therefore, we can subtract 2012 from all of the numbers and try to... | 0 | 8,113.3125 | -1 | 8,113.3125 |
A food factory has made 4 different exquisite cards. Each bag of food produced by the factory randomly contains one card. If all 4 different cards are collected, a prize can be won. Xiaoming buys 6 bags of this food at once. What is the probability that Xiaoming will win the prize? | 195/512 | 0.8125 | 4,858.5 | 4,089.230769 | 8,192 | |
A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? | 24 |
#### Step 1: Understand the properties of $x$ and $x+32$
Given that $x$ is a three-digit palindrome and $x+32$ is a four-digit palindrome, we need to find the possible values of $x$ and $x+32$ that satisfy these conditions.
#### Step 2: Determine the range of $x$
Since $x$ is a three-digit number, $100 \leq x \leq 99... | 0.9375 | 4,026.0625 | 3,748.333333 | 8,192 |
The sequence is $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{2}{4}$, $\frac{3}{4}$, ... , $\frac{1}{m+1}$, $\frac{2}{m+1}$, ... , $\frac{m}{m+1}$, ... Find the $20^{th}$ term. | \frac{6}{7} | 0 | 3,857.4375 | -1 | 3,857.4375 | |
For any positive integers \( n \) and \( k \) (where \( k \leqslant n \)), let \( f(n, k) \) denote the number of positive integers that do not exceed \( \left\lfloor \frac{n}{k} \right\rfloor \) and are coprime with \( n \). Determine the value of \( f(100, 3) \). | 14 | 0.875 | 4,563.9375 | 4,045.642857 | 8,192 | |
Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many pias... | 15 | 0 | 8,192 | -1 | 8,192 | |
If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals | -1 | 1. **Identify the given equation**: We are given that one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$. This can be written as:
\[
1 - \frac{1}{1-x} = \frac{1}{1-x}
\]
2. **Simplify the equation**: To simplify, we can add $\frac{1}{1-x}$ to both sides of the equation:
\[
1 = 2 \cdot ... | 1 | 1,627.3125 | 1,627.3125 | -1 |
When the base-16 number $B1234_{16}$ is written in base 2, how many base-2 digits (bits) does it have? | 20 | 0.8125 | 4,189.625 | 3,266 | 8,192 | |
Let \( u, v, w \) be positive real numbers, all different from 1. If
\[
\log_{u}(vw) + \log_{v}(w) = 5 \quad \text{and} \quad \log_{v}(u) + \log_{w}(v) = 3,
\]
find the value of \( \log_{w}(u) \). | \frac{4}{5} | 0.5625 | 5,484.6875 | 4,472.555556 | 6,786 | |
How many times does the digit 9 appear in the list of all integers from 1 to 700? | 140 | 0.6875 | 6,209 | 5,307.636364 | 8,192 | |
Inside the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is the center $O$ of a sphere with radius 10. The sphere intersects the face $A A_{1} D_{1} D$ along a circle with radius 1, the face $A_{1} B_{1} C_{1} D_{1}$ along a circle with radius 1, and the face $C D D_{1} C_{1}$ along a circle with radius 3. Find the length of ... | 17 | 0.25 | 7,167.875 | 4,760 | 7,970.5 | |
The time on a cell phone is $3:52$. How many minutes will pass before the phone next shows a time using each of the digits 2, 3, and 5 exactly once? | 91 | There are six times that can be made using each of the digits 2, 3, and 5 exactly once: $2:35$, $2:53$, $3:25$, $3:52$, $5:23$, and $5:32$. The first of these that occurs after 3:52 is 5:23. From 3:52 to $4:00$, 8 minutes pass. From 4:00 to 5:00, 60 minutes pass. From 5:00 to 5:23, 23 minutes pass. Therefore, from $3:5... | 0.4375 | 7,533.1875 | 6,781.571429 | 8,117.777778 |
Given $f(x)$ be a differentiable function, satisfying $\lim_{x \to 0} \frac{f(1)-f(1-x)}{2x} = -1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$. | -2 | 0.8125 | 5,690.125 | 5,133.230769 | 8,103.333333 | |
In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$ . A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$ . Points $C$ , $A$ , and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the c... | 9\sqrt{19} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given the function $f(x)=2 \sqrt {3}\tan ( \frac {x}{2}+ \frac {π}{4})\cos ^{2}( \frac {x}{2}+ \frac {π}{4})-\sin (x+π)$.
(I) Find the domain and the minimum positive period of $f(x)$;
(II) If the graph of $f(x)$ is shifted to the right by $\frac {π}{6}$ units to obtain the graph of function $g(x)$, find the maximum an... | -1 | 0.6875 | 5,230.875 | 4,724.909091 | 6,344 | |
Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribu... | 1011 | By stars and bars, the total number of distributions is $\binom{12}{2}^{2}=66^{2}$. If no locations are won, either both distributions are identical or the difference between the two is $(1,0,-1)$, in some order. The first case has 66 possibilities. If the difference is $(1,0,-1)$, we can construct all such possibiliti... | 0.0625 | 7,852.1875 | 8,192 | 7,829.533333 |
Compute \[\lfloor 1 \rfloor + \lfloor 1.6 \rfloor + \lfloor 2.2 \rfloor + \lfloor 2.8 \rfloor + \dots + \lfloor 99.4 \rfloor + \lfloor 100 \rfloor,\]where the arguments of the floor functions are in arithmetic progression. | 8317 | 0 | 8,192 | -1 | 8,192 | |
Xiao Gang goes to buy milk and finds that it's on special offer that day: each bag costs 2.5 yuan, and there is a "buy two, get one free" promotion. Xiao Gang has 30 yuan. What is the maximum number of bags of milk he can buy? | 18 | 0.75 | 1,310.75 | 1,604.166667 | 430.5 | |
In triangle ABC, let the lengths of the sides opposite to angles A, B, and C be a, b, and c respectively, and b = 3, c = 1, A = 2B. Find the value of a. | \sqrt{19} | 0 | 3,604.0625 | -1 | 3,604.0625 | |
How many ways can a 5-person executive board be chosen from a club of 12 people? | 792 | 0.9375 | 1,965 | 2,055.333333 | 610 | |
$(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:<br/>① When $n$ is a positive integer, $f(f(n))=4n+9$;<br/>② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.<br/>$(2)$ The function $f$ is defined on the set of ordered pairs of positive integers, and s... | 364 | 0.25 | 7,323.1875 | 5,897.25 | 7,798.5 |
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