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 |
|---|---|---|---|---|---|---|
Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = TB = 10$ and $TC = 9$, then what is the volume of pyramid $TABC$? | 150 | 1 | 2,487.25 | 2,487.25 | -1 | |
Given that $α$ and $β$ are both acute angles, $cosα= \frac {3}{5}$, and $cos(α+β)=- \frac {5}{13}$, find the value of $sinβ$. | \frac {56}{65} | 0.875 | 5,349.625 | 4,943.571429 | 8,192 | |
A math class has fewer than 50 students. When the students try to sit in rows of 8, 5 students are left in the last row. When the students try to sit in rows of 6, 3 students remain in the last row. How many students are in this class? | 45 | 0.1875 | 7,463.75 | 6,446 | 7,698.615385 | |
The area of the orthogonal projection of a circle with a radius of 1 on the plane $\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\alpha$. | \frac{2\sqrt{\pi^2 - 1}}{\pi} | 0 | 7,257.6875 | -1 | 7,257.6875 | |
Given the function f(x) = $\sqrt{|x+2|+|6-x|-m}$, whose domain is R,
(I) Find the range of the real number m;
(II) If the maximum value of the real number m is n, and the positive numbers a and b satisfy $\frac{8}{3a+b}$ + $\frac{2}{a+2b}$ = n, find the minimum value of 2a + $\frac{3}{2}$b. | \frac{9}{8} | 0.8125 | 5,577.4375 | 4,974.076923 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c=2$, $C=\dfrac{\pi }{3}$, and $\sin B=2\sin A$, find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} | 0 | 5,035.8125 | -1 | 5,035.8125 | |
In triangle $ABC$, let $AB = 4$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD$. | \sqrt{\frac{5}{14}} | 0 | 6,526.6875 | -1 | 6,526.6875 | |
The total number of toothpicks used to build a rectangular grid 15 toothpicks high and 12 toothpicks wide, with internal diagonal toothpicks, is calculated by finding the sum of the toothpicks. | 567 | 0.0625 | 7,011.8125 | 7,496 | 6,979.533333 | |
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? | 17 | 1. **Define Variables:**
Let $b$ represent the number of boys and $g$ represent the number of girls in Mr. Neatkin's class.
2. **Set Up the Equation:**
According to the problem, two-thirds of the boys and three-fourths of the girls passed the test, and the number of boys and girls who passed the test are equal. ... | 1 | 2,375.6875 | 2,375.6875 | -1 |
In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then what is the value of $C$? | 1 | To solve this problem, we need to analyze the given cryptarithmetic puzzle step by step. We are given that different letters represent different digits, and we need to find the value of $C$.
1. **Understanding the Problem**:
We are given the equation:
\[
A + B + C = 10
\]
and from a carry over, we have:... | 0.0625 | 8,043.9375 | 5,823 | 8,192 |
What is the smallest positive integer $x$ that, when multiplied by $400$, produces a product that is a multiple of $576$? | 36 | 0.875 | 4,150.125 | 3,572.714286 | 8,192 | |
Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$ th positive integer which is not a perfect square.
*Proposed by David Stoner* | 6077248 | 0 | 8,192 | -1 | 8,192 | |
Today is December 19, 2010. What is the integer part of the sum $\frac{2010}{1000}+\frac{1219}{100}+\frac{27}{10}$? | 16 | 0.9375 | 4,970.5625 | 4,755.8 | 8,192 | |
Nine identical marbles weigh the same as five identical model cars. If four of the model cars weigh a total of 120 kilograms, how many kilograms does one marble weigh? | \frac{50}{3} | 0.25 | 668.375 | 1,260.5 | 471 | |
A square board with three rows and three columns contains nine cells. In how many different ways can we write the three letters A, B, and C in three different cells, so that exactly one of these three letters is written in each row? | 162 | 0.25 | 6,366.375 | 7,092 | 6,124.5 | |
Pirate Pete shares his treasure with Pirate Paul in an interesting way. Pete first says, ``One for me, one for you,'' giving himself one coin and starting Paul's pile with one coin. Then Pete says, ``Two for me, and two for you,'' giving himself two more coins but making Paul's pile two coins in total. Next Pete says, ... | 35 | 0.75 | 5,159.9375 | 4,149.25 | 8,192 | |
In a class of 50 students, each student must be paired with another student for a project. Due to prior groupings in other classes, 10 students already have assigned partners. If the pairing for the remaining students is done randomly, what is the probability that Alex is paired with her best friend, Jamie? Express you... | \frac{1}{29} | 0.3125 | 4,159.75 | 2,782.2 | 4,785.909091 | |
Given the function $f(x)=\cos (\sqrt{3}x+\varphi)$, where $\varphi \in (-\pi, 0)$. If the function $g(x)=f(x)+f'(x)$ (where $f'(x)$ is the derivative of $f(x)$) is an even function, determine the value of $\varphi$. | -\frac{\pi }{3} | 0.75 | 5,255.375 | 4,794.166667 | 6,639 | |
Tessa picks three real numbers $x, y, z$ and computes the values of the eight expressions of the form $\pm x \pm y \pm z$. She notices that the eight values are all distinct, so she writes the expressions down in increasing order. How many possible orders are there? | 96 | There are $2^{3}=8$ ways to choose the sign for each of $x, y$, and $z$. Furthermore, we can order $|x|,|y|$, and $|z|$ in $3!=6$ different ways. Now assume without loss of generality that $0<x<y<z$. Then there are only two possible orders depending on the sign of $x+y-z$: $-x-y-z,+x-y-z,-x+y-z,-x-y+z, x+y-z, x-y+z,-x+... | 0 | 6,379.875 | -1 | 6,379.875 |
When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 33. If one integer is 45, what is the smallest possible value of the other integer? | 165 | 0.9375 | 5,015.375 | 4,803.6 | 8,192 | |
A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must cons... | 70 | 0 | 8,192 | -1 | 8,192 | |
Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself? | 14 | Let the people be $A, B, C, D, E$ so that their heights are in that order, with $A$ the tallest and $E$ the shortest. We will do casework based on the position of $C$. - Case 1: $C$ is in the middle. Then, $B$ must be on one of the two ends, for two choices. This leaves only one choice for $D$-the other end. Then, we k... | 0 | 8,192 | -1 | 8,192 |
Find the number of positive integer solutions \((x, y, z, w)\) to the equation \(x + y + z + w = 25\) that satisfy \(x < y\). | 946 | 0 | 8,189.9375 | -1 | 8,189.9375 | |
A polynomial with integer coefficients is of the form
\[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 15 = 0.\]Find the number of different possible rational roots of this polynomial. | 16 | 0.5625 | 4,675.375 | 4,322.444444 | 5,129.142857 | |
Four cars $A, B, C$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counter-clockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, $A$ meets $C$ for the first time, and at that same moment,... | 371 | 0.25 | 7,645.125 | 6,019.75 | 8,186.916667 | |
Let \( x \in \left(-\frac{3\pi}{4}, \frac{\pi}{4}\right) \), and \( \cos \left(\frac{\pi}{4} - x\right) = -\frac{3}{5} \). Find the value of \( \cos 2x \). | -\frac{24}{25} | 0.4375 | 7,178.5 | 6,673.571429 | 7,571.222222 | |
What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ? | 18 | 0.625 | 5,784.875 | 4,340.6 | 8,192 | |
If the function $f(x)=\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+ax+4$ is strictly decreasing on the interval $[-1,4]$, then the value of the real number $a$ is ______. | -4 | 0.125 | 8,180.75 | 8,102 | 8,192 | |
A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.
What is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?
A $3: 5$
B $2: 3$
C $5: 8$
D $1: 2$
E $1: 1$ | 3:5 | 0 | 8,093.4375 | -1 | 8,093.4375 | |
Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have? | 905 | We first show that we can choose at most 5 numbers from $\{1, 2, \ldots , 11\}$ such that no two numbers have a difference of $4$ or $7$. We take the smallest number to be $1$, which rules out $5,8$. Now we can take at most one from each of the pairs: $[2,9]$, $[3,7]$, $[4,11]$, $[6,10]$. Now, $1989 = 180\cdot 11 + 9$.... | 0 | 8,192 | -1 | 8,192 |
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day? | \frac{4}{9} | 1. **Calculate the amount of paint remaining after the first day:**
Initially, there is 1 gallon of paint. On the first day, one third of the paint is used. The amount of paint used on the first day is:
\[
\frac{1}{3} \times 1 = \frac{1}{3} \text{ gallons}
\]
Therefore, the amount of paint remaining afte... | 1 | 1,356.625 | 1,356.625 | -1 |
Two cars, Car A and Car B, start simultaneously from points A and B, respectively, and move towards each other. After 6 hours, the distance they have traveled is 3/5 of the distance between points A and B. Car A travels at a speed of 42 km per hour, which is 1/7 less than Car B's hourly speed. Find the distance between... | 780 | 0 | 2,023.9375 | -1 | 2,023.9375 | |
Given a small cube block, each face is painted with a different color. If you want to carve 1, 2, 3, 4, 5, 6 small dots on the faces of the block, and the dots 1 and 6, 2 and 5, 3 and 4 are carved on opposite faces respectively, determine the number of different carving methods. | 48 | 0.125 | 7,498.4375 | 6,039 | 7,706.928571 | |
Let \( n \in \mathbf{Z}_{+} \). When \( n > 100 \), the first two digits of the decimal part of \( \sqrt{n^{2}+3n+1} \) are ______. | 50 | 0 | 8,192 | -1 | 8,192 | |
Find the value of $b$ that satisfies the equation $161_{b}+134_{b}=315_{b}$. | 8 | 1 | 2,175.375 | 2,175.375 | -1 | |
Define $E(a,b,c) = a \cdot b^2 + c$. What value of $a$ is the solution to the equation $E(a,4,5) = E(a,6,7)$? | -\frac{1}{10} | 1 | 1,456.8125 | 1,456.8125 | -1 | |
18.14 People are participating in a round-robin Japanese chess tournament. Each person plays against 13 others, with no draws in the matches. Find the maximum number of "circular triples" (where each of the three participants wins against one and loses to another) in the tournament. | 112 | 0 | 7,921.125 | -1 | 7,921.125 | |
A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines. | 16 |
Let us consider a broken line made up of 31 segments with no self-intersections, where the start and end points are distinct. Each segment of the broken line can be extended indefinitely to form a straight line. The problem asks us to find the least possible number of distinct straight lines that can be created from t... | 0.0625 | 7,626.1875 | 7,501 | 7,634.533333 |
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[\begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline\end{tabular}\] | $y=100-5x-5x^{2}$ | To find the correct formula that expresses the relationship between $x$ and $y$, we will substitute the given points into each of the proposed formulas and check which one satisfies all the points.
#### Step 1: Test the point $(0, 100)$
- **Option A:** $y = 100 - 10x \implies y = 100 - 10(0) = 100$
- **Option B:** $y ... | 0 | 3,423.1875 | -1 | 3,423.1875 |
If $2137^{753}$ is multiplied out, the units' digit in the final product is: | 7 | 1. **Identify the units digit of the base number**: The units digit of $2137$ is $7$. Therefore, the units digit of $2137^{753}$ will be the same as the units digit of $7^{753}$.
2. **Determine the pattern of units digits for powers of $7$**:
- $7^1$ has a units digit of $7$.
- $7^2 = 49$, which has a units digi... | 1 | 1,789.9375 | 1,789.9375 | -1 |
A $3$ by $3$ determinant has three entries equal to $2$ , three entries equal to $5$ , and three entries equal to $8$ . Find the maximum possible value of the determinant. | 405 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers? | 21 | 0.9375 | 1,673.8125 | 1,707.933333 | 1,162 | |
This year is 2017, and the sum of the digits of the year is 10. Find the sum of all the years in this century whose digits sum to 10. | 18396 | 0.5 | 6,523.4375 | 4,854.875 | 8,192 | |
For each positive integer $n$, let $f(n)$ denote the last digit of the sum $1+2+3+\ldots+n$.
For example: $f(1)=1$, $f(2)=3$ (the last digit of $1+2$), $f(5)=5$ (the last digit of $1+2+3+4+5$), $f(7)=8$ (the last digit of $1+2+3+4+5+6+7$)
Then, the value of $f(1)+f(2)+f(3)+\ldots+f(2005)$ is . | 7015 | 0.8125 | 6,537.1875 | 6,155.307692 | 8,192 | |
What is the value in simplest form of the following expression?
\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7} | 10 | 1. **Identify the pattern in the expression**: The expression given is \(\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\). We need to simplify each term under the square root.
2. **Simplify each term**:
- The first term is \(\sqrt{1}\).
- The second term simplifies as \(\sqrt{1+3} = \sqrt{4}\).
- The t... | 1 | 422.625 | 422.625 | -1 |
In triangle $XYZ$, where $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3\cos Y$. What is the length of $XY$? | \frac{40\sqrt{2}}{3} | 0 | 5,719.0625 | -1 | 5,719.0625 | |
The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\qquad$ | 10829 | 0.9375 | 3,819.8125 | 3,528.333333 | 8,192 | |
A number $n$ has $3$ divisors. How many divisors does $n^2$ have? | 5 | 1 | 1,759.0625 | 1,759.0625 | -1 | |
Consider the set $\{0.34,0.304,0.034,0.43\}$. Find the sum of the smallest and largest numbers in the set. | 0.464 | 1 | 1,760.4375 | 1,760.4375 | -1 | |
In the Cartesian coordinate system $xOy$, the graph of the parabola $y=ax^2 - 3x + 3 \ (a \neq 0)$ is symmetric with the graph of the parabola $y^2 = 2px \ (p > 0)$ with respect to the line $y = x + m$. Find the product of the real numbers $a$, $p$, and $m$. | -3 | 0.8125 | 5,475.375 | 4,848.461538 | 8,192 | |
There are 4 willow trees and 4 poplar trees planted in a row. How many ways can they be planted alternately? | 1152 | 0.875 | 4,443.4375 | 4,255.142857 | 5,761.5 | |
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square. | 38 | When $n \geq 12$, we have \[(n-10)^2 < n^2 -19n + 99 < (n-8)^2.\]
So if $n \geq 12$ and $n^2 -19n + 99$ is a perfect square, then \[n^2 -19n + 99 = (n-9)^2\]
or $n = 18$.
For $1 \leq n < 12$, it is easy to check that $n^2 -19n + 99$ is a perfect square when $n = 1, 9$ and $10$ ( using the identity $n^2 -19n + 99 = (... | 0.875 | 4,471.8125 | 3,970.642857 | 7,980 |
Given the function $f(x)=\sin \omega x+\cos \left(\omega x+\dfrac{\pi }{6}\right)$, where $x\in R$, $\omega >0$.
(1) When $\omega =1$, find the value of $f\left(\dfrac{\pi }{3}\right)$;
(2) When the smallest positive period of $f(x)$ is $\pi $, find the value of $x$ when $f(x)$ reaches the maximum value in $\left[0,\... | \dfrac{\pi }{12} | 1 | 3,654.375 | 3,654.375 | -1 | |
Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,$ and $BC=6$. What is $AC$? | 6 | 1. **Identify Angles Subtended by the Same Arc**:
Since $\angle ADB$ and $\angle ACB$ are both subtended by the same arc $AB$ in the circle, by the Inscribed Angle Theorem, we have:
\[
\angle ACB = \angle ADB = 40^\circ.
\]
2. **Calculate $\angle ABC$ in $\triangle ABC$**:
We know that the sum of an... | 0.1875 | 7,601.375 | 5,042 | 8,192 |
In triangle $XYZ$, medians $XM$ and $YN$ intersect at $Q$, $QN=3$, $QM=4$, and $MN=5$. What is the area of $XMYN$? | 54 | 0 | 7,945.6875 | -1 | 7,945.6875 | |
Let \(a, b, c, d, e\) be positive integers. Their sum is 2345. Let \(M = \max (a+b, b+c, c+d, d+e)\). Find the smallest possible value of \(M\). | 782 | 0.1875 | 7,754.0625 | 6,657.666667 | 8,007.076923 | |
Given a hyperbola $C$ with an eccentricity of $\sqrt {3}$, foci $F\_1$ and $F\_2$, and a point $A$ on the curve $C$. If $|F\_1A|=3|F\_2A|$, then $\cos \angle AF\_2F\_1=$ \_\_\_\_\_\_. | \frac{\sqrt{3}}{3} | 0 | 5,037.375 | -1 | 5,037.375 | |
In triangular prism \( P-ABC \), \( PA \perp \) plane \( ABC \), and \( AC \perp BC \). Given \( AC = 2 \), the dihedral angle \( P-BC-A \) is \( 60^\circ \), and the volume of the triangular prism \( P-ABC \) is \( \frac{4\sqrt{6}}{3} \). Find the sine value of the angle between line \( PB \) and plane \( PAC \). | \frac{\sqrt{3}}{3} | 0 | 7,829.1875 | -1 | 7,829.1875 | |
A coin is tossed 10 times. Find the probability that no two heads appear consecutively. | 9/64 | 0.875 | 4,555.1875 | 4,035.642857 | 8,192 | |
Given $2$ red and $2$ white balls, a total of $4$ balls are randomly arranged in a row. The probability that balls of the same color are adjacent to each other is $\_\_\_\_\_\_$. | \frac{1}{3} | 0.6875 | 6,007 | 5,702.545455 | 6,676.8 | |
Given a random variable $0.4987X \sim N\left( 9, \sigma^2 \right)$, and $P(X < 6) = 0.2$, determine the probability that $9 < X < 12$. | 0.3 | 0 | 7,947.75 | -1 | 7,947.75 | |
A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter? | \frac{1}{12} | 0.1875 | 6,388.3125 | 3,778.666667 | 6,990.538462 | |
In triangle \(ABC\), angle bisectors \(AA_{1}\), \(BB_{1}\), and \(CC_{1}\) are drawn. \(L\) is the intersection point of segments \(B_{1}C_{1}\) and \(AA_{1}\), \(K\) is the intersection point of segments \(B_{1}A_{1}\) and \(CC_{1}\). Find the ratio \(LM: MK\) if \(M\) is the intersection point of angle bisector \(BB... | 11/12 | 0 | 8,192 | -1 | 8,192 | |
The three sides of a triangle are 25, 39, and 40. Find the diameter of its circumscribed circle. | \frac{125}{3} | 0.875 | 5,177.75 | 4,747.142857 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $a=6$ and $b\sin\frac{B+C}{2}=a\sin B$. Find:
1. The measure of angle $A$.
2. Let $M$ be a point inside triangle $\triangle ABC$. Extend $AM$ to intersect $BC$ at point $D$. _______. Find t... | 9\sqrt{3} | 0.6875 | 6,758.5625 | 6,107 | 8,192 | |
The distance between Ivan's house and his grandmother's house is 12 km. Exactly at 12:00, Ivan left his house and walked along the straight road to his grandmother's house at a speed of 1 m/s. At 12:30, Ivan's parents called his grandmother, informed her that Ivan was coming to visit, and she released her dog Tuzik to ... | 12:47 | 0.5 | 4,644.25 | 2,563.75 | 6,724.75 | |
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\] | 4 |
To determine the smallest positive integer \( t \) such that there exist integers \( x_1, x_2, \ldots, x_t \) satisfying
\[
x_1^3 + x_2^3 + \cdots + x_t^3 = 2002^{2002},
\]
we will apply Fermat's Last Theorem and results regarding sums of cubes.
### Step 1: Understanding the Sum of Cubes
The problem requires expres... | 0.1875 | 7,533.5 | 4,680 | 8,192 |
Given an angle measuring $54^{\circ}$, use only a compass to divide it into three equal parts (that is, find such points that rays passing through the vertex of the given angle and these points divide the angle into three equal parts). | 18 | 0.5 | 7,463.6875 | 6,735.375 | 8,192 | |
The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. What is the diameter of the circle? | 65 | 0.5 | 6,593.25 | 4,994.5 | 8,192 | |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-102)^{2}
$$
If you obtain a non-integer number, round the result to the nearest whole number. | 46852 | 0.0625 | 7,089.8125 | 7,053 | 7,092.266667 | |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 0.375 | 4,225.625 | 3,348 | 4,752.2 | |
Let $p,$ $q,$ $r,$ and $s$ be the roots of \[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\] | \frac{10}{3} | 0.5625 | 5,306.6875 | 3,062.555556 | 8,192 | |
Find the number of real solutions of the equation
\[\frac{4x}{x^2 + x + 3} + \frac{5x}{x^2 - 5x + 3} = -\frac{3}{2}.\] | 2 | 0.625 | 6,252 | 5,363.4 | 7,733 | |
The triangle $ABC$ is isosceles with $AB=BC$ . The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$ . Fine the measure of the angle $ABC$ . | 36 | 0 | 7,770.625 | -1 | 7,770.625 | |
To complete the grid below, each of the digits 1 through 4 must occur once
in each row and once in each column. What number will occupy the lower
right-hand square?
\[\begin{tabular}{|c|c|c|c|}\hline 1 & & 2 &\ \hline 2 & 3 & &\ \hline & &&4\ \hline & &&\ \hline\end{tabular}\] | 1 | We are given a 4x4 grid where each digit from 1 through 4 must appear exactly once in each row and once in each column. The initial grid is:
\[
\begin{array}{|c|c|c|c|}
\hline
1 & & 2 & \\
\hline
2 & 3 & & \\
\hline
& & & 4 \\
\hline
& & & \\
\hline
\end{array}
\]
We need to determine the number... | 0.0625 | 7,634.625 | 6,804 | 7,690 |
Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their ori... | 5:4 | 0.0625 | 5,272.3125 | 3,769 | 5,372.533333 | |
Ten identical books cost no more than 11 rubles, whereas 11 of the same books cost more than 12 rubles. How much does one book cost? | 110 | 0 | 5,810.5625 | -1 | 5,810.5625 | |
Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
all of t... | 199 | We use the Principle of Inclusion-Exclusion (PIE).
If we join the adjacent vertices of the regular $n$-star, we get a regular $n$-gon. We number the vertices of this $n$-gon in a counterclockwise direction: $0, 1, 2, 3, \ldots, n-1.$
A regular $n$-star will be formed if we choose a vertex number $m$, where $0 \le m \... | 0.6875 | 5,052.1875 | 3,753.363636 | 7,909.6 |
In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is: | 24n | 1. **Construct the Triangle and Medians**: Begin by constructing triangle $\triangle ABC$. Let $M$, $N$, and $P$ be the midpoints of sides $\overline{BC}$, $\overline{AB}$, and $\overline{AC}$ respectively. Draw the medians $\overline{AM}$, $\overline{BP}$, and $\overline{CN}$.
2. **Intersection of Medians**: The medi... | 0.875 | 4,762.5625 | 4,272.642857 | 8,192 |
Given that the terminal side of $\alpha$ passes through the point $(a, 2a)$ (where $a < 0$),
(1) Find the values of $\cos\alpha$ and $\tan\alpha$.
(2) Simplify and find the value of $$\frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\sin(-\alpha+ \frac {3\pi}{2})}{\tan(-\alpha-\pi)\sin(-\pi-\alpha)}$$. | \frac{1}{10} | 0.9375 | 4,242 | 4,268.466667 | 3,845 | |
Given the function $f(x)=x^{2}+x-2$, determine the characteristics of the function $f(x)$ in the interval $[-1,1)$. | -\frac{9}{4} | 0.0625 | 4,037.125 | 4,265 | 4,021.933333 | |
Find the distance between the foci of the ellipse \[x^2 + 4y^2 = 400.\] | 20\sqrt3 | 1 | 1,790.1875 | 1,790.1875 | -1 | |
The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}... | -\frac{1}{40} | 1. **Definition of Remainder**: The remainder function $\text{rem}(x, y)$ for real numbers $x$ and $y$ (with $y \neq 0$) is defined as:
\[
\text{rem}(x, y) = x - y \left\lfloor \frac{x}{y} \right\rfloor
\]
where $\left\lfloor \cdot \right\rfloor$ denotes the greatest integer less than or equal to the enclos... | 0.875 | 4,247.125 | 3,960.714286 | 6,252 |
Given that events $A$ and $B$ are independent, and $P(A)=\frac{1}{2}$, $P(B)=\frac{2}{3}$, find $P(\overline{AB})$. | \frac{1}{6} | 0.25 | 2,132.375 | 2,640 | 1,963.166667 | |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 152 | 0 | 7,845.8125 | -1 | 7,845.8125 | |
What is the fifth-largest divisor of 3,640,350,000? | 227,521,875 | 0 | 7,937.0625 | -1 | 7,937.0625 | |
A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have? | 91 | 0 | 8,192 | -1 | 8,192 | |
Nasim buys trading cards in packages of 5 cards and in packages of 8 cards. He can purchase exactly 18 cards by buying two 5-packs and one 8-pack, but he cannot purchase exactly 12 cards with any combination of packages. For how many of the integers $n=24,25,26,27,28,29$ can he buy exactly $n$ cards? | 5 | Nasim can buy 24 cards by buying three 8-packs $(3 imes 8=24)$. Nasim can buy 25 cards by buying five 5-packs $(5 imes 5=25)$. Nasim can buy 26 cards by buying two 5-packs and two 8-packs $(2 imes 5+2 imes 8=26)$. Nasim can buy 28 cards by buying four 5-packs and one 8-pack $(4 imes 5+1 imes 8=28)$. Nasim can buy... | 1 | 3,845.8125 | 3,845.8125 | -1 |
In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$ , and $FE$ perpendicular to $BD$ . Suppose $BC = 8\sqrt3$ . Find $AB$ . | 24 | 0.6875 | 5,286.5625 | 3,974.818182 | 8,172.4 | |
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = ... | 919 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given the equation about $x$, $(x-2)(x^2-4x+m)=0$ has three real roots.
(1) Find the range of values for $m$.
(2) If these three real roots can exactly be the lengths of the sides of a triangle, find the range of values for $m$.
(3) If the triangle formed by these three real roots is an isosceles triangle, find t... | \sqrt{3} | 0.1875 | 7,838.125 | 6,304.666667 | 8,192 | |
Compute $\sin 45^\circ$. | \frac{\sqrt{2}}{2} | 0 | 2,435.5625 | -1 | 2,435.5625 | |
An insect has just told me that she has laid $154_6$ eggs. In base 10, how many eggs did she lay? | 70 | 0.9375 | 3,003.1875 | 2,657.266667 | 8,192 | |
Given that 5 students each specialize in one subject (Chinese, Mathematics, Physics, Chemistry, History) and there are 5 test papers (one for each subject: Chinese, Mathematics, Physics, Chemistry, History), a teacher randomly distributes one test paper to each student. Calculate the probability that at least 4 student... | 89/120 | 0.25 | 6,656.8125 | 6,011.75 | 6,871.833333 | |
Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots. | 500 | We find that the given equation has a $2000^{\text{th}}$ degree polynomial. Note that there are no multiple roots. Thus, if $\frac{1}{2} - x$ is a root, $x$ is also a root. Thus, we pair up $1000$ pairs of roots that sum to $\frac{1}{2}$ to get a sum of $\boxed{500}$. | 0.125 | 8,192 | 8,192 | 8,192 |
If three people are selected at random from a group of seven men and three women, what is the probability that at least one woman is selected? Express your answer as a common fraction. | \frac{17}{24} | 0.875 | 3,461.6875 | 2,785.928571 | 8,192 | |
A jar contains $5$ different colors of gumdrops. $30\%$ are blue, $20\%$ are brown, $15\%$ are red, $10\%$ are yellow, and other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown? | 42 | 1. **Calculate the total number of gumdrops**:
Given that $30\%$ are blue, $20\%$ are brown, $15\%$ are red, $10\%$ are yellow, and $30$ gumdrops are green. First, we calculate the percentage of gumdrops that are green:
\[
100\% - (30\% + 20\% + 15\% + 10\%) = 100\% - 75\% = 25\%
\]
Since $25\%$ of the g... | 0.875 | 1,980.0625 | 2,003.5 | 1,816 |
In rectangle \(ABCD\), \(BE = 5\), \(EC = 4\), \(CF = 4\), and \(FD = 1\), as shown in the diagram. What is the area of triangle \(\triangle AEF\)? | 42.5 | 0 | 3,744.9375 | -1 | 3,744.9375 | |
How many distinct sequences of four letters can be made from the letters in "EXAMPLE" if each letter can be used only once and each sequence must begin with X and not end with E? | 80 | 0 | 6,765.4375 | -1 | 6,765.4375 | |
Calculate the value of the polynomial f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x using the Qin Jiushao algorithm when x = 3. Find the value of V₄. | 789 | 0.875 | 4,254.6875 | 3,692.214286 | 8,192 |
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