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 a parabola $y=x^2+bx+c$ intersects the y-axis at point Q(0, -3), and the sum of the squares of the x-coordinates of the two intersection points with the x-axis is 15, find the equation of the function and its axis of symmetry. | \frac{3}{2} | 0.9375 | 5,010.3125 | 5,157 | 2,810 | |
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.
| 2500 | 0 | 7,422.75 | -1 | 7,422.75 | |
Given the system of equations in terms of $x$ and $y$: $\left\{\begin{array}{l}{3x+5y=m+2}\\{2x+3y=m}\end{array}\right.$.<br/>$(1)$ Find the relationship between $x$ and $y$ (express $y$ as an algebraic expression only containing $x$);<br/>$(2)$ If the solution to this system satisfies $x+y=-10$, find the value of $m$. | -8 | 1 | 1,957.6875 | 1,957.6875 | -1 | |
Natalie and Harpreet are the same height. Jiayin's height is 161 cm. The average (mean) of the heights of Natalie, Harpreet and Jiayin is 171 cm. What is Natalie's height? | 176 \text{ cm} | Since the average of three heights is 171 cm, then the sum of these three heights is 3 \times 171 \mathrm{~cm} or 513 cm. Since Jiayin's height is 161 cm, then the sum of Natalie's and Harpreet's heights must equal 513 \mathrm{~cm} - 161 \mathrm{~cm} = 352 \mathrm{~cm}. Since Harpreet and Natalie are the same height, t... | 0.875 | 559.5625 | 558.571429 | 566.5 |
Find the value of $$\sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \sum_{c=1}^{\infty} \frac{a b(3 a+c)}{4^{a+b+c}(a+b)(b+c)(c+a)}$$ | \frac{1}{54} | Let $S$ denote the given sum. By summing over all six permutations of the variables $a, b, c$ we obtain $$\begin{aligned} 6 S & =\sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \sum_{c=1}^{\infty} \frac{3\left(a^{2} b+a^{2} c+b^{2} a+b^{2} c+c^{2} a+c^{2} b\right)+6 a b c}{4^{a+b+c}(a+b)(b+c)(c+a)} \\ & =\sum_{a=1}^{\infty} \s... | 0 | 8,168.3125 | -1 | 8,168.3125 |
If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? | 0.15625 | 0 | 7,115.75 | -1 | 7,115.75 | |
Three young married couples were captured by cannibals. Before eating the tourists, the cannibals decided to weigh them. The total weight of all six people was not an integer, but the combined weight of all the wives was exactly 171 kg. Leon weighed the same as his wife, Victor weighed one and a half times more than hi... | 85.5 | 0 | 7,540.4375 | -1 | 7,540.4375 | |
For a natural number $n \geq 1$, it satisfies: $2002 \times n$ is a perfect cube, and $n \div 2002$ is a perfect square. The smallest such $n$ is | 2002^5 | 0.1875 | 7,882.5625 | 6,541.666667 | 8,192 | |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 0.375 | 7,792.9375 | 7,127.833333 | 8,192 | |
Compute the number of dates in the year 2023 such that when put in MM/DD/YY form, the three numbers are in strictly increasing order. For example, $06 / 18 / 23$ is such a date since $6<18<23$, while today, $11 / 11 / 23$, is not. | 186 | January contains 21 such dates, February contains 20, and so on, until December contains 10. The answer is $$21+20+\cdots+10=186$$ | 0.4375 | 7,401.6875 | 6,385.571429 | 8,192 |
In the diagram, the rectangle has a width $w$, a length of $8$, and a perimeter of $24$. What is the ratio of its width to its length? [asy]
pair a = (0, 0); pair b = (8, 0); pair c = (8, 4); pair d = (0, 4);
draw(a--b--c--d--cycle);
label("$w$", midpoint(a--d), W); label("$8$", midpoint(c--d), N);
[/asy] Write your an... | 1 : 2 | 1 | 977.5 | 977.5 | -1 | |
If we express $3x^2 - 6x - 2$ in the form $a(x - h)^2 + k$, then what is $a + h + k$? | -1 | 1 | 1,523.25 | 1,523.25 | -1 | |
Elmer's new car gives $50\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a l... | 20\% | 1. **Define the fuel efficiency and cost per liter:**
- Let the fuel efficiency of Elmer's old car be $x$ kilometers per liter.
- The new car has $50\%$ better fuel efficiency, so it runs at $\frac{3}{2}x$ kilometers per liter.
2. **Convert the new car's efficiency to the old car's metric:**
- The new car's e... | 1 | 2,129.375 | 2,129.375 | -1 |
In a blackboard, it's written the following expression $ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$ We put parenthesis by different ways and then we calculate the result. For example: $ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$ and $ 1-\left(2-2^2 \left(-2^3-2^4 \right)-... | 1024 | 0 | 8,192 | -1 | 8,192 | |
A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squa... | 90 | 0.25 | 7,659.4375 | 6,061.75 | 8,192 | |
How many different positive values of $x$ will make this statement true: there are exactly $2$ positive two-digit multiples of $x$. | 16 | 0.1875 | 7,982.5625 | 7,075 | 8,192 | |
What is the 125th digit beyond the decimal point in the decimal representation of $\frac47$? | 2 | 1 | 2,430 | 2,430 | -1 | |
Given a sequence $\{a_n\}$ that satisfies $a_na_{n+1}a_{n+2}a_{n+3}=24$, and $a_1=1$, $a_2=2$, $a_3=3$, find the sum $a_1+a_2+a_3+\ldots+a_{2013}$. | 5031 | 1 | 3,037.875 | 3,037.875 | -1 | |
A square is cut along a diagonal and reassembled to form a parallelogram \( PQRS \). If \( PR=90 \mathrm{~mm} \), what is the area of the original square, in \( \mathrm{mm}^{2} \)? | 1620 \mathrm{~mm}^{2} | Suppose that the original square had side length \( x \mathrm{~mm} \). We extend \( PQ \) and draw a line through \( R \) perpendicular to \( PQ \), meeting \( PQ \) extended at \( T \). \( SRTQ \) is a square, since it has three right angles at \( S, Q, T \) (which makes it a rectangle) and since \( SR=SQ \) (which ma... | 0 | 3,404.375 | -1 | 3,404.375 |
A $1 \times n$ rectangle ( $n \geq 1 $ ) is divided into $n$ unit ( $1 \times 1$ ) squares. Each square of this rectangle is colored red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$ ? (The... | 37 | 1 | 2,664.25 | 2,664.25 | -1 | |
Simplify completely: $$\sqrt[3]{80^3 + 100^3 + 120^3}.$$ | 20\sqrt[3]{405} | 0 | 5,591.5625 | -1 | 5,591.5625 | |
If \( x=2 \) and \( v=3x \), what is the value of \((2v-5)-(2x-5)\)? | 8 | Since \( v=3x \) and \( x=2 \), then \( v=3 \cdot 2=6 \). Therefore, \((2v-5)-(2x-5)=(2 \cdot 6-5)-(2 \cdot 2-5)=7-(-1)=8\). | 1 | 2,109.75 | 2,109.75 | -1 |
How many cubic feet are in one cubic yard? One yard is equal to three feet.
[asy]import three;
currentprojection=orthographic(1/2,1/2,1);
draw((0,0,0)--(10,0,0)--(10,-10,0)--(0,-10,0)--cycle);
draw((0,0,10)--(10,0,10)--(10,-10,10)--(0,-10,10)--cycle);
draw((0,0,0)--(0,0,10));
draw((10,0,0)--(10,0,10));
draw((10,-10,0)... | 27 | 1 | 295.375 | 295.375 | -1 | |
Given positive numbers $x$, $y$ satisfying $xy= \frac{x-y}{x+3y}$, find the maximum value of $y$. | \frac{1}{3} | 0.75 | 5,473.875 | 4,920.833333 | 7,133 | |
The nonzero roots of the equation $x^2 + 6x + k = 0$ are in the ratio $2:1$. What is the value of $k$? | 8 | 1 | 3,410.25 | 3,410.25 | -1 | |
Given the function $f(x)= \frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$ where $N^{}$ represents the set of positive integers, vector $ \overrightarrow{i}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$, determine the value of $\fra... | \frac{2017}{2018} | 0.75 | 2,590.125 | 2,381.333333 | 3,216.5 | |
Let $\triangle DEF$ be an isosceles triangle with $DE = DF$. Three circles are defined as follows: the circle $\Omega$ with its center at the centroid of $\triangle DEF$, and two circles $\Omega_1$ and $\Omega_2$, where $\Omega_1$ is tangent to $\overline{EF}$ and externally tangent to the other sides extended, while $... | 12 | 0 | 8,192 | -1 | 8,192 | |
8 people attend a party. During the party everyone shakes hands with everyone else. How many handshakes take place at the party? | 28 | 0.9375 | 2,313.8125 | 1,921.933333 | 8,192 | |
If $x$ is doubled, increased by $3$, and then divided by $5$, the result is $11$. What is the value of $x$? | 26 | 1 | 2,005.6875 | 2,005.6875 | -1 | |
Determine the probability that two edges selected at random from the twelve edges of a cube with side length 1 are skew lines (i.e., non-intersecting and not in the same plane). | \frac{4}{11} | 0.3125 | 7,143.125 | 5,014.6 | 8,110.636364 | |
Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this const... | \frac{1}{8} | 0 | 8,192 | -1 | 8,192 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.3125 | 2,287.0625 | 2,203.8 | 2,324.909091 | |
Given that $\textstyle\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer $k$. | 13 | 0.5 | 7,367.125 | 6,542.25 | 8,192 | |
Determine how many 4-digit numbers are mountain numbers, where mountain numbers are defined as having their middle two digits larger than any other digits in the number. For example, 3942 and 5732 are mountain numbers. | 240 | 0 | 8,018.8125 | -1 | 8,018.8125 | |
Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin $O$, and then ending at points $P_{1}(-1,0)$, $P_{2}(-1,1)$, $P... | \dfrac{3}{10} | 1 | 3,627.5 | 3,627.5 | -1 | |
The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$. What is the $x$-coordinate of the vertex of the parabola? | 2 | 1 | 2,633.6875 | 2,633.6875 | -1 | |
Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that
$(6-a)(6-b)(6-c)(6-d)(6-e)=45$
What is $a+b+c+d+e$? | 25 | 1. **Identify the factors of 45**: We start by noting that the equation $(6-a)(6-b)(6-c)(6-d)(6-e)=45$ implies that the expressions $(6-a), (6-b), (6-c), (6-d), (6-e)$ are integers whose product is 45. We need to find distinct integers $a, b, c, d, e$ such that this condition is satisfied.
2. **Factorize 45**: The int... | 0.9375 | 4,451.0625 | 4,201.666667 | 8,192 |
Given that the point F(0,1) is the focus of the parabola $x^2=2py$,
(1) Find the equation of the parabola C;
(2) Points A, B, and C are three points on the parabola such that $\overrightarrow{FA} + \overrightarrow{FB} + \overrightarrow{FC} = \overrightarrow{0}$, find the maximum value of the area of triangle ABC. | \frac{3\sqrt{6}}{2} | 0 | 8,047.1875 | -1 | 8,047.1875 | |
Consider a modified function $C' = \frac{2en}{R + 2nr}$, where $e = 4$, $R = 6$, and $r = 3$. Determine the behavior of $C'$ as $n$ increases from a positive starting point. | \frac{4}{3} | 0.9375 | 2,705.875 | 2,749.133333 | 2,057 | |
Given the function f(x) = $\sqrt {2}$sin $\frac {x}{2}$cos $\frac {x}{2}$ - $\sqrt {2}$sin<sup>2</sup> $\frac {x}{2}$,
(1) Find the smallest positive period of f(x);
(2) Find the minimum value of f(x) in the interval [-π, 0]. | -1 - \frac { \sqrt {2}}{2} | 0 | 5,110.75 | -1 | 5,110.75 | |
How many hits does "3.1415" get on Google? Quotes are for clarity only, and not part of the search phrase. Also note that Google does not search substrings, so a webpage with 3.14159 on it will not match 3.1415. If $A$ is your answer, and $S$ is the correct answer, then you will get $\max (25-\mid \ln (A)-\ln (S) \mid,... | 422000 | The answer is 422000. | 0 | 5,910.3125 | -1 | 5,910.3125 |
A mathematician $M^{\prime}$ is called a descendent of mathematician $M$ if there is a sequence of mathematicians $M=M_{1}, M_{2}, \ldots, M_{k}=M^{\prime}$ such that $M_{i}$ was $M_{i+1}$ 's doctoral advisor for all $i$. Estimate the number of descendents that the mathematician who has had the largest number of descen... | 82310 | First let's estimate how many "generations" of mathematicians there have been since 1300. If we suppose that a mathematician gets his PhD around age 30 and becomes a PhD advisor around age 60, then we'll get a generation length of approximately 30 years. However, not all mathematicians will train more than one PhD. Let... | 0 | 5,283.9375 | -1 | 5,283.9375 |
Given points P and Q are on a circle of radius 7 and the length of chord PQ is 8. Point R is the midpoint of the minor arc PQ. Find the length of the line segment PR. | \sqrt{98 - 14\sqrt{33}} | 0 | 6,944.8125 | -1 | 6,944.8125 | |
If $4^x - 4^{x - 1} = 24$, then $(2x)^x$ equals: | 25\sqrt{5} | We start by simplifying the given equation:
\[ 4^x - 4^{x-1} = 24 \]
1. **Express $4^{x-1}$ in terms of $4^x$:**
\[ 4^{x-1} = \frac{4^x}{4} = \frac{1}{4} \cdot 4^x \]
2. **Substitute back into the equation:**
\[ 4^x - \frac{1}{4} \cdot 4^x = 24 \]
\[ \frac{4}{4} \cdot 4^x - \frac{1}{4} \cdot 4^x = 24 \]
\... | 1 | 2,317.9375 | 2,317.9375 | -1 |
During the past summer, 100 graduates from the city of $N$ applied to 5 different universities in our country. It turned out that during the first and second waves, each university was unable to reach exactly half of the applicants to that university. In addition, representatives of at least three universities were una... | 83 | 0 | 8,142.6875 | -1 | 8,142.6875 | |
Two positive integers \( x \) and \( y \) have \( xy=24 \) and \( x-y=5 \). What is the value of \( x+y \)? | 11 | The positive integer divisors of 24 are \( 1,2,3,4,6,8,12,24 \). The pairs of divisors that give a product of 24 are \( 24 \times 1,12 \times 2,8 \times 3 \), and \( 6 \times 4 \). We want to find two positive integers \( x \) and \( y \) whose product is 24 and whose difference is 5. Since \( 8 \times 3=24 \) and \( 8... | 1 | 1,871.25 | 1,871.25 | -1 |
The function $f(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $f(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $f(x) \le f(y).$
(iii) $f(1 - x) = 1 - f(x)$ for all $0 \le x \le 1.$
(iv) $f \left( \frac{x}{3} \right) = \frac{f(x)}{2}$ for $0 \le x \le 1.$
Find $f \left( \frac{2}{7} \right).$ | \frac{3}{8} | 0 | 8,139.25 | -1 | 8,139.25 | |
What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees? | 12 | 0 | 8,192 | -1 | 8,192 | |
How many four-digit whole numbers are there such that the leftmost digit is a prime number, the second digit is even, and all four digits are different? | 1064 | 0.625 | 5,994.375 | 5,539.8 | 6,752 | |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 4 - 2\sqrt{2} | 0.375 | 7,385.125 | 6,040.333333 | 8,192 | |
Points $A, B, C$ lie on a circle \omega such that $B C$ is a diameter. $A B$ is extended past $B$ to point $B^{\prime}$ and $A C$ is extended past $C$ to point $C^{\prime}$ such that line $B^{\prime} C^{\prime}$ is parallel to $B C$ and tangent to \omega at point $D$. If $B^{\prime} D=4$ and $C^{\prime} D=6$, compute $... | \frac{24}{5} | Let $x=A B$ and $y=A C$, and define $t>0$ such that $B B^{\prime}=t x$ and $C C^{\prime}=t y$. Then $10=B^{\prime} C^{\prime}=(1+t) \sqrt{x^{2}+y^{2}}, 4^{2}=t(1+t) x^{2}$, and $6^{2}=t(1+t) y^{2}$ (by power of a point), so $52=4^{2}+6^{2}=t(1+t)\left(x^{2}+y^{2}\right)$ gives $\frac{13}{25}=\frac{52}{10^{2}}=\frac{t(1... | 0.125 | 7,844.4375 | 5,411.5 | 8,192 |
Let $a$ and $b$ be nonzero real numbers such that
\[(2 - 7i)(a + bi)\]is pure imaginary. Find $\frac{a}{b}.$ | -\frac{7}{2} | 1 | 1,681.5 | 1,681.5 | -1 | |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$ | 6 | 1 | 1,634.5625 | 1,634.5625 | -1 | |
Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. | 453 | As before, there are 4 functions with a range of size 1, 84 with a range of size 2, and 144 with a range of size 3. If the range of $f$ has size $k$, the codomain of $g$ is restricted to a set of size $4 - k$. Any function from $A$ into this codomain will do, so there are $(4 - k)^4$ possibilities for $g$ given a funct... | 0.0625 | 8,192 | 8,192 | 8,192 |
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | -2 | 0.0625 | 7,984 | 8,192 | 7,970.133333 | |
The surface area of a cube is 24. What is the volume of the cube? | 8 | A cube has six identical faces. If the surface area of a cube is 24, the area of each face is $\frac{24}{6}=4$. Since each face of this cube is a square with area 4, the edge length of the cube is $\sqrt{4}=2$. Thus, the volume of the cube is $2^{3}$ which equals 8. | 1 | 945.125 | 945.125 | -1 |
Given the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$. | 4045 | 0 | 5,634.5625 | -1 | 5,634.5625 | |
Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condition? | 62 | 0.5 | 5,829.5 | 4,382.875 | 7,276.125 | |
From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers, and calculate the probability that these $2$ numbers are coprime. | \frac{2}{3} | 0.25 | 7,976.875 | 7,331.5 | 8,192 | |
Let $\Gamma_{1}$ and $\Gamma_{2}$ be concentric circles with radii 1 and 2, respectively. Four points are chosen on the circumference of $\Gamma_{2}$ independently and uniformly at random, and are then connected to form a convex quadrilateral. What is the probability that the perimeter of this quadrilateral intersects ... | \frac{22}{27} | Define a triplet as three points on $\Gamma_{2}$ that form the vertices of an equilateral triangle. Note that due to the radii being 1 and 2, the sides of a triplet are all tangent to $\Gamma_{1}$. Rather than choosing four points on $\Gamma_{2}$ uniformly at random, we will choose four triplets of $\Gamma_{2}$ uniform... | 0 | 7,699.75 | -1 | 7,699.75 |
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 24.\] | 6\sqrt{2} | 0 | 6,440.875 | -1 | 6,440.875 | |
Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$. | 71 | 0.625 | 4,383.875 | 2,966.7 | 6,745.833333 | |
A line passes through $(2,2,1)$ and $(5,1,-2).$ A point on this line has an $x$-coordinate of 4. Find the $z$-coordinate of the point. | -1 | 0.9375 | 2,364.5 | 1,976 | 8,192 | |
The number of revolutions of a wheel, with fixed center and with an outside diameter of $6$ feet, required to cause a point on the rim to go one mile is: | \frac{880}{\pi} | 1. **Identify the radius of the wheel**: Given the diameter of the wheel is $6$ feet, the radius $r$ is half of the diameter:
\[
r = \frac{6}{2} = 3 \text{ feet}
\]
2. **Calculate the circumference of the wheel**: The circumference $C$ of a circle is given by the formula $C = 2\pi r$. Substituting the radius ... | 0.6875 | 4,698.125 | 4,215.727273 | 5,759.4 |
Among 51 consecutive odd numbers $1, 3, 5, \cdots, 101$, select $\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\mathrm{k}$? | 44 | 0.125 | 8,154.8125 | 7,894.5 | 8,192 | |
Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\%$ profit and then buys it back from Mr. $Y$ at a $15\%$ loss. | 2160 | 0.4375 | 5,651.375 | 4,428.142857 | 6,602.777778 | |
Find the area in square feet of a square with a perimeter of 32ft. | 64 | 0.9375 | 773.0625 | 801.266667 | 350 | |
How many positive integers less than $800$ are either a perfect cube or a perfect square? | 35 | 0 | 4,621.875 | -1 | 4,621.875 | |
In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the x-axis as the polar axis. Given circle C: ρ = 2cosθ - 2sinθ, and the parametric equation of line l is x = t, y = -1 + 2√2t (t is the parameter). Line l intersects with circle C at... | \frac{10\sqrt{5}}{9} | 0 | 6,443.875 | -1 | 6,443.875 | |
Kelly is attempting to unlock her electronic device with a four-digit password. She remembers that she only used digits from 1 to 6, each digit possibly being repeated, and that each odd digit must be followed by an even digit, with no specific rule for the sequences following even digits. How many combinations might K... | 648 | 0.0625 | 8,046.125 | 5,858 | 8,192 | |
A sphere is inscribed in a cube, and the cube has a surface area of 54 square meters. A second cube is then inscribed within the sphere. A third, smaller sphere is then inscribed within this second cube. What is the surface area of the second cube and the volume of the third sphere? | \frac{\sqrt{3}\pi}{2} | 0 | 2,441.5 | -1 | 2,441.5 | |
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $... | 50 | 0.0625 | 8,166.1875 | 7,779 | 8,192 | |
Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$.
Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$. | 86 | 0.6875 | 6,491.875 | 5,790.636364 | 8,034.6 | |
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$? | 8 | 1. **Identify Symmetry Requirements**: The problem states that the set $S$ is symmetric about the origin, both coordinate axes, and the line $y=x$. This implies:
- Symmetry about the origin: If $(a, b) \in S$, then $(-a, -b) \in S$.
- Symmetry about the $x$-axis: If $(a, b) \in S$, then $(a, -b) \in S$.
- Symm... | 1 | 3,977.9375 | 3,977.9375 | -1 |
In the Cartesian coordinate system $xoy$, point $P(0, \sqrt{3})$ is given. The parametric equation of curve $C$ is $\begin{cases} x = \sqrt{2} \cos \varphi \\ y = 2 \sin \varphi \end{cases}$ (where $\varphi$ is the parameter). A polar coordinate system is established with the origin as the pole and the positive half-ax... | \sqrt{14} | 0.6875 | 6,677.3125 | 6,313.727273 | 7,477.2 | |
The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes. | 4002001 | Let $x=2000$, so the numerator is $$x^{5}+x^{4}+1=\left(x^{2}+x+1\right)\left(x^{3}-x+1\right)$$ (This latter factorization can be noted by the fact that plugging in $\omega$ or $\omega^{2}$ into $x^{5}+x^{4}+1$ gives 0 .) Then $x^{2}+x+1=4002001$ divides the numerator. However, it can easily by checked that 101 doesn'... | 0 | 8,090.5625 | -1 | 8,090.5625 |
There are $2022$ grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action: $\bullet$ One should se... | 1011 | 0.1875 | 7,975.875 | 7,039.333333 | 8,192 | |
Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains? | 5/84 | 0.5 | 6,711.4375 | 5,230.875 | 8,192 | |
How many integers between $1500$ and $2500$ have the property that their units digit is the sum of the other digits?
**A)** $76$
**B)** $81$
**C)** $85$
**D)** $91$
**E)** $96$ | 81 | 0 | 8,187.375 | -1 | 8,187.375 | |
Ostap Bender organized an elephant distribution for the residents in the city of Fuks. 28 members of a union and 37 non-union members came for the distribution. Ostap distributed the elephants equally among all union members and also equally among all non-union members.
It turned out that there was only one possible w... | 1036 | 0.5 | 6,998.1875 | 5,804.375 | 8,192 | |
Let $a$ and $b$ be the solutions of the equation $2x^2+6x-14=0$. What is the value of $(2a-3)(4b-6)$? | -2 | 1 | 2,560.9375 | 2,560.9375 | -1 | |
Chewbacca has 20 pieces of cherry gum and 30 pieces of grape gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $x$ pieces of gum. If Chewbacca loses one pack of cherry gum, then the ratio of the number of pieces of cherry gum he has to the number of pieces of grape ... | 14 | 1 | 2,564.5 | 2,564.5 | -1 | |
Two particles move along the edges of equilateral $\triangle ABC$ in the direction $A\Rightarrow B\Rightarrow C\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles trac... | \frac{1}{16} | 1. **Setup and Coordinate Assignment**:
Without loss of generality (WLOG), let's place point $A$ at the origin of a coordinate system. Assume $\overline{AB}$ lies on the $x$-axis, and let point $B$ be at $(1, 0)$. Since $\triangle ABC$ is equilateral, point $C$ will be at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right... | 0 | 8,192 | -1 | 8,192 |
The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$? | 13 | 1. **Convert to slope-intercept form**:
The given equations are $ax-2y=c$ and $2x+by=-c$. We can rearrange these to solve for $y$:
\[
y = \frac{a}{2}x - \frac{c}{2}
\]
\[
y = -\frac{2}{b}x - \frac{c}{b}
\]
The slopes of the lines are $\frac{a}{2}$ and $-\frac{2}{b}$, respectively.
2. **Conditi... | 1 | 2,443 | 2,443 | -1 |
Calculate the limit of the function:
$$
\lim_{x \rightarrow 1} (2-x)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}
$$ | e | 0.3125 | 6,855.75 | 4,663.4 | 7,852.272727 | |
Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction. | \frac{1}{3} | 0.9375 | 2,556.3125 | 2,180.6 | 8,192 | |
The squares of a chessboard are labelled with numbers, as shown below.
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(i + 8 - j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eig... | 1 | 0 | 8,192 | -1 | 8,192 | |
The cheetah takes strides of 2 meters each and the fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. Given that the distance between the cheetah and the fox is 30 meters, find the distance the cheetah must run to catch up... | 120 | 0.4375 | 5,679.125 | 4,661.857143 | 6,470.333333 | |
What is the maximum number of consecutive positive integers starting from 10 that can be added together before the sum exceeds 500? | 23 | 0.9375 | 4,199.8125 | 4,371.533333 | 1,624 | |
According to the definition of the Richter scale, the relationship between the relative energy $E$ released by an earthquake and the earthquake magnitude $n$ is: $E=10^n$. What is the multiple of the relative energy released by a magnitude 9 earthquake compared to a magnitude 7 earthquake? | 100 | 1 | 1,212.0625 | 1,212.0625 | -1 | |
A fair 6-sided die is rolled. If I roll $n$, then I win $n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent. | \$15.17 | 1 | 2,588.125 | 2,588.125 | -1 | |
A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in the display. | 24 | 0.3125 | 7,343.9375 | 6,598.4 | 7,682.818182 | |
What is the greatest number of consecutive integers whose sum is $136$? | 272 | 0.5625 | 6,338.375 | 6,089.777778 | 6,658 | |
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | 0 | 0.5 | 7,673.0625 | 7,154.125 | 8,192 | |
Let $\alpha$ be a nonreal root of $x^4 = 1.$ Compute
\[(1 - \alpha + \alpha^2 - \alpha^3)^4 + (1 + \alpha - \alpha^2 + \alpha^3)^4.\] | 32 | 0 | 6,333 | -1 | 6,333 | |
Consider a function \( g \) that maps nonnegative integers to real numbers, with \( g(1) = 1 \), and for all nonnegative integers \( m \ge n \),
\[ g(m + n) + g(m - n) = \frac{g(3m) + g(3n)}{3} \]
Find the sum of all possible values of \( g(10) \). | 100 | 0 | 8,192 | -1 | 8,192 | |
Given any number a from the set {1, 2, 3, ..., 99, 100} and any number b from the same set, calculate the probability that the last digit of 3^a + 7^b is 8. | \frac{3}{16} | 0.6875 | 5,077.9375 | 4,115.545455 | 7,195.2 | |
Let $A B C D$ be a cyclic quadrilateral, and let segments $A C$ and $B D$ intersect at $E$. Let $W$ and $Y$ be the feet of the altitudes from $E$ to sides $D A$ and $B C$, respectively, and let $X$ and $Z$ be the midpoints of sides $A B$ and $C D$, respectively. Given that the area of $A E D$ is 9, the area of $B E C$ ... | 17+\frac{15}{2} \sqrt{3} | Reflect $E$ across $D A$ to $E_{W}$, and across $B C$ to $E_{Y}$. As $A B C D$ is cyclic, $\triangle A E D$ and $\triangle B E C$ are similar. Thus $E_{W} A E D$ and $E B E_{Y} C$ are similar too. Now since $W$ is the midpoint of $E_{W} E, X$ is the midpoint of $A B, Y$ is the midpoint of $E E_{Y}$, and $Z$ is the midp... | 0 | 8,192 | -1 | 8,192 |
Given $$\frac {\pi}{2} < \alpha < \pi$$, $$0 < \beta < \frac {\pi}{2}$$, $$\tan\alpha = -\frac {3}{4}$$, and $$\cos(\beta-\alpha) = \frac {5}{13}$$, find the value of $\sin\beta$. | \frac {63}{65} | 0.625 | 6,144 | 5,207 | 7,705.666667 | |
Given $f(x)= \sqrt {3}\sin \dfrac {x}{4}\cos \dfrac {x}{4}+ \cos ^{2} \dfrac {x}{4}+ \dfrac {1}{2}$.
(1) Find the period of $f(x)$;
(2) In $\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the value of $f(B)$. | \dfrac{\sqrt{3}}{2} + 1 | 0 | 5,239.625 | -1 | 5,239.625 |
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