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 |
|---|---|---|---|---|---|---|
Matt has somewhere between 1000 and 2000 pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2,3,4,5,6,7$, and 8 piles but ends up with one sheet left over each time. How many piles does he need? | 41 | The number of sheets will leave a remainder of 1 when divided by the least common multiple of $2,3,4,5,6,7$, and 8, which is $8 \cdot 3 \cdot 5 \cdot 7=840$. Since the number of sheets is between 1000 and 2000, the only possibility is 1681. The number of piles must be a divisor of $1681=41^{2}$, hence it must be 41. | 1 | 2,619.375 | 2,619.375 | -1 |
What is the smallest prime whose digits sum to \(28\)? | 1999 | 0.4375 | 7,347.5625 | 6,261.857143 | 8,192 | |
When Joyce counts the pennies in her bank by fives, she has one left over. When she counts them by threes, there are two left over. What is the least possible number of pennies in the bank? | 11 | 1 | 2,895.375 | 2,895.375 | -1 | |
Add 24.567 to 38.924, then multiply the sum by 2.5, and round the result to the nearest hundredth. | 158.73 | 1 | 339.6875 | 339.6875 | -1 | |
At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian aga... | 100 | 0.125 | 7,323 | 5,980.5 | 7,514.785714 | |
Let $a,$ $b,$ $c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7 = 0.$ Given that $P(x)$ is a cubic polynomial such that $P(a) = b + c,$ $P(b) = a + c,$ $P(c) = a + b,$ and $P(a + b + c) = -16,$ find $P(x).$ | 2x^3 + 6x^2 + 9x + 11 | 0.875 | 4,029.25 | 3,782.071429 | 5,759.5 | |
In a region, three villages \(A, B\), and \(C\) are connected by rural roads, with more than one road between any two villages. The roads are bidirectional. A path from one village to another is defined as either a direct connecting road or a chain of two roads passing through a third village. It is known that there a... | 106 | 0.1875 | 7,533.3125 | 7,195.333333 | 7,611.307692 | |
A box contains a total of 400 tickets that come in five colours: blue, green, red, yellow and orange. The ratio of blue to green to red tickets is $1: 2: 4$. The ratio of green to yellow to orange tickets is $1: 3: 6$. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one c... | 196 | 0.375 | 7,348.25 | 6,602.666667 | 7,795.6 | |
Given that $\theta$ and $\phi$ are acute angles such that $\tan \theta = \frac{1}{7}$ and $\sin \phi = \frac{1}{\sqrt{10}},$ find $\theta + 2 \phi,$ measured in radians. | \frac{\pi}{4} | 0.9375 | 4,117.125 | 3,845.466667 | 8,192 | |
The crafty rabbit and the foolish fox made an agreement: every time the fox crosses the bridge in front of the rabbit's house, the rabbit would double the fox's money. However, each time the fox crosses the bridge, he has to pay the rabbit a toll of 40 cents. Hearing that his money would double each time he crossed the... | 35 | 0.0625 | 5,183.4375 | 5,125 | 5,187.333333 | |
Suppose $\cos S = 0.5$ in the diagram below. What is $ST$?
[asy]
pair P,S,T;
P = (0,0);
S = (6,0);
T = (0,6*tan(acos(0.5)));
draw(P--S--T--P);
draw(rightanglemark(S,P,T,18));
label("$P$",P,SW);
label("$S$",S,SE);
label("$T$",T,N);
label("$10$",S/2,S);
[/asy] | 20 | 0.3125 | 3,686.0625 | 2,457 | 4,244.727273 | |
Find the constant $t$ such that \[(5x^2 - 6x + 7)(4x^2 +tx + 10) = 20x^4 -54x^3 +114x^2 -102x +70.\] | -6 | 1 | 2,559.125 | 2,559.125 | -1 | |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | 6 | 1. **Given Information and Assumptions**:
- Convex quadrilateral $ABCD$ has $AB = 9$, $CD = 12$, $AC = 14$.
- Diagonals $AC$ and $BD$ intersect at $E$.
- $\triangle AED$ and $\triangle BEC$ have equal areas.
2. **Using Equal Areas to Infer Ratios**:
- Since $\triangle AED$ and $\triangle BEC$ have equal ar... | 0.5625 | 5,309.625 | 4,929.444444 | 5,798.428571 |
What is the nearest integer to $(2+\sqrt3)^4$? | 194 | 0.9375 | 5,367 | 5,178.666667 | 8,192 | |
Four people, A, B, C, and D, participated in an exam. The combined scores of A and B are 17 points higher than the combined scores of C and D. A scored 4 points less than B, and C scored 5 points more than D. How many points higher is the highest score compared to the lowest score among the four? | 13 | 0.625 | 3,469.125 | 2,750.6 | 4,666.666667 | |
Determine the largest value of $S$ such that any finite collection of small squares with a total area $S$ can always be placed inside a unit square $T$ in such a way that no two of the small squares share an interior point. | \frac{1}{2} | 0.25 | 7,992.5625 | 7,609.75 | 8,120.166667 | |
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s.$ | 380 | [asy] size(220); pointpen = black; pathpen = black + linewidth(0.7); pair O=(0,0),A=(-15,0),B=(-6,0),C=(15,0),D=(0,8); D(D(MP("A",A))--D(MP("C",C))--D(MP("D",D,NE))--cycle); D(D(MP("B",B))--D); D((0,-4)--(0,12),linetype("4 4")+linewidth(0.7)); MP("6",B/2); MP("15",C/2); MP("9",(A+B)/2); [/asy]
Denote the height of $\t... | 0.9375 | 4,021.0625 | 4,063.066667 | 3,391 |
Two of the altitudes of an acute triangle divide the sides into segments of lengths $5,3,2$ and $x$ units, as shown. What is the value of $x$? [asy]
defaultpen(linewidth(0.7)); size(75);
pair A = (0,0);
pair B = (1,0);
pair C = (74/136,119/136);
pair D = foot(B, A, C);
pair E = /*foot(A,B,C)*/ (52*B+(119-52)*C)/(119);
... | 10 | 0.0625 | 7,415.6875 | 7,559 | 7,406.133333 | |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b... | 22 | 0.1875 | 7,994.375 | 7,205.666667 | 8,176.384615 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2c-2a\cos B=b$.
$(1)$ Find the size of angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}}{4}$, and $c^{2}+ab\cos C+a^{2}=4$, find $a$. | \frac{\sqrt{7}}{2} | 0 | 5,563.1875 | -1 | 5,563.1875 | |
Two 8-sided dice are tossed (each die has faces numbered 1 to 8). What is the probability that the sum of the numbers shown on the dice is either a prime or a multiple of 4? | \frac{39}{64} | 0.5625 | 5,055.3125 | 4,435.111111 | 5,852.714286 | |
What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | 15 | 1 | 4,403.125 | 4,403.125 | -1 | |
Find the smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,
\[\sqrt{\frac{x}{y + z + w}} + \sqrt{\frac{y}{x + z + w}} + \sqrt{\frac{z}{x + y + w}} + \sqrt{\frac{w}{x + y + z}} < M.\] | \frac{4}{\sqrt{3}} | 0 | 8,192 | -1 | 8,192 | |
Positive integer $n$ cannot be divided by $2$ and $3$ , there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$ . Find the minimum value of $n$ . | 35 | 0.0625 | 8,192 | 8,192 | 8,192 | |
In the given configuration, triangle $ABC$ has a right angle at $C$, with $AC=4$ and $BC=3$. Triangle $ABE$ has a right angle at $A$ where $AE=5$. The line through $E$ parallel to $\overline{AC}$ meets $\overline{BC}$ extended at $D$. Calculate the ratio $\frac{ED}{EB}$. | \frac{4}{5} | 0 | 7,665.1875 | -1 | 7,665.1875 | |
In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion spri... | 43 |
To determine the number of races required to find the champion sprinter, we need to consider how the competition progresses. Each race has 6 sprinters, and only the winner of each race advances to the next round, while the other 5 are eliminated.
#### Detailed Analysis:
1. **Initial Setup:**
- Total number of spr... | 0.75 | 5,566.5 | 4,789.916667 | 7,896.25 |
Given that \( x_{1}, x_{2}, \cdots, x_{n} \) are real numbers, find the minimum value of
\[
E(x_{1}, x_{2}, \cdots, x_{n}) = \sum_{i=1}^{n} x_{i}^{2} + \sum_{i=1}^{n-1} x_{i} x_{i+1} + \sum_{i=1}^{n} x_{i}
\] | -1/2 | 0 | 8,192 | -1 | 8,192 | |
The distance from $A$ to $B$ is covered 3 hours and 12 minutes faster by a passenger train compared to a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train covers 288 km more. If the speed of each train is increased by $10 \mathrm{~km} / \mathrm{h}$, the passenger trai... | 360 | 0.375 | 6,932.375 | 4,833 | 8,192 | |
The arithmetic mean of a set of $60$ numbers is $42$. If three numbers from the set, $48$, $58$, and $52$, are removed, find the arithmetic mean of the remaining set of numbers. | 41.4 | 0 | 7,411.375 | -1 | 7,411.375 | |
A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possibl... | 27.1 | 0 | 8,174.0625 | -1 | 8,174.0625 | |
In the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; (a > b > 0)$, a line with slope $k(k > 0)$ intersects the ellipse at the left vertex $A$ and another point $B$. The projection of point $B$ on the $x$-axis is exactly the right focus $F$. If the eccentricity of the ellipse $e= \frac{1}{3}$, then the value ... | \frac{2}{3} | 1 | 4,766.625 | 4,766.625 | -1 | |
A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots? | 3/4 | 0.125 | 7,403.9375 | 5,740 | 7,641.642857 | |
Given a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.
A) $3\pi$
B) $6\pi$
C) $9\pi$
D) $12\pi$
E) $15\pi$ | 9\pi | 0 | 8,174.875 | -1 | 8,174.875 | |
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is... | \frac{17}{8} | To solve this problem, we need to understand how the iterative average process works and how the order of numbers affects the final result. The iterative average process can be thought of as a weighted average where numbers added later in the sequence have a greater influence on the final average.
#### Step 1: Underst... | 0.5625 | 6,980.5625 | 6,038.333333 | 8,192 |
Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$. The diagonals have length $10\sqrt {21}$, and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitud... | 32 | Extend $\overline {AB}$ through $B$, to meet $\overline {DC}$ (extended through $C$) at $G$. $ADG$ is an equilateral triangle because of the angle conditions on the base.
If $\overline {GC} = x$ then $\overline {CD} = 40\sqrt{7}-x$, because $\overline{AD}$ and therefore $\overline{GD}$ $= 40\sqrt{7}$.
By simple angle... | 0.0625 | 8,069.0625 | 6,225 | 8,192 |
Given that \(\alpha\) and \(\beta\) are both acute angles, and
$$(1+\tan \alpha)(1+\tan \beta)=2.$$
Find \(\alpha + \beta =\). | \frac{\pi}{4} | 0.625 | 3,342.1875 | 3,158.9 | 3,647.666667 | |
Let ellipse $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a>b>0)$ pass through the point $\left(0,4\right)$, with eccentricity $\frac{3}{5}$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ If a line $l$ passing through the point $\left(3,0\right)$ with a slope of $\frac{4}{5}$ intersects the ellipse $C$ at points... | \frac{41}{5} | 1 | 4,322.9375 | 4,322.9375 | -1 | |
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$ . In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absol... | 1014049 | 0.375 | 7,774.375 | 7,078.333333 | 8,192 | |
What is the probability that the same number will be facing up on each of five standard six-sided dice that are tossed simultaneously? Express your answer as a simplified common fraction. | \frac{1}{1296} | 0.9375 | 2,439.875 | 2,056.4 | 8,192 | |
If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to | 89 | Given the equations:
1. \(\log_2(\log_3(\log_4 x)) = 0\)
2. \(\log_3(\log_4(\log_2 y)) = 0\)
3. \(\log_4(\log_2(\log_3 z)) = 0\)
From the property of logarithms, if \(\log_b(a) = 0\), then \(a = 1\). Applying this property to each equation:
1. \(\log_3(\log_4 x) = 1\)
- By the definition of logarithms, \(\log_4 x ... | 1 | 2,462.9375 | 2,462.9375 | -1 |
Given that
\[
2^{-\frac{5}{3} + \sin 2\theta} + 2 = 2^{\frac{1}{3} + \sin \theta},
\]
compute \(\cos 2\theta.\) | -1 | 0 | 8,192 | -1 | 8,192 | |
Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest $a$ million dollars in the dorms, the students will spend an additional $5a$... | 34 | 0.8125 | 5,234.4375 | 4,551.923077 | 8,192 | |
If the operation symbol "△" is defined as: $a \triangle b = a + b + ab - 1$, and the operation symbol "⊗" is defined as: $a \otimes b = a^2 - ab + b^2$, evaluate the value of $3 \triangle (2 \otimes 4)$. | 50 | 1 | 1,539.5 | 1,539.5 | -1 | |
Angela has deposited $\$8,\!000$ into an account that pays $6\%$ interest compounded annually.
Bob has deposited $\$10,\!000$ into an account that pays $7\%$ simple annual interest.
In $20$ years Angela and Bob compare their respective balances. To the nearest dollar, what is the positive difference between their bal... | \$1,\!657 | 0.0625 | 5,722.25 | 734 | 6,054.8 | |
Eight positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $2107$, what is the sum of the numbers written on the faces? | 57 | 0.6875 | 4,209.0625 | 2,943.727273 | 6,992.8 | |
The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term? | 315 | 1 | 1,546.875 | 1,546.875 | -1 | |
Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, w... | 298 | Consider the cross section of the cones and sphere by a plane that contains the two axes of symmetry of the cones as shown below. The sphere with maximum radius will be tangent to the sides of each of the cones. The center of that sphere must be on the axis of symmetry of each of the cones and thus must be at the inter... | 0.0625 | 8,190 | 8,160 | 8,192 |
In the Cartesian coordinate plane $xOy$, the equation of the ellipse $C$ is $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1$. Let $F$ be the upper focus of $C$, $A$ be the right vertex of $C$, and $P$ be a moving point on $C$ located in the first quadrant. Find the maximum value of the area of the quadrilateral $OAPF$. | \frac{3 \sqrt{11}}{2} | 0 | 6,760.4375 | -1 | 6,760.4375 | |
Given that a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola. | \sqrt{5} | 0 | 8,100.1875 | -1 | 8,100.1875 | |
In right triangle $DEF$ with $\angle D = 90^\circ$, side $DE = 9$ cm and side $EF = 15$ cm. Find $\sin F$. | \frac{3\sqrt{34}}{34} | 0 | 1,787.125 | -1 | 1,787.125 | |
How many integers $n$ satisfy the condition $100 < n < 200$ and the condition $n$ has the same remainder whether it is divided by $6$ or by $8$? | 25 | 0.25 | 6,941.5 | 7,084.25 | 6,893.916667 | |
Given an ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \left( a > b > 0 \right)$ with eccentricity $\dfrac{1}{2}$, and it passes through the point $\left( 1,\dfrac{3}{2} \right)$. If point $M\left( x_{0},y_{0} \right)$ is on the ellipse $C$, then the point $N\left( \dfrac{x_{0}}{a},\dfrac{y_{0}}{b} \right)$ is called ... | \sqrt{3} | 0.8125 | 6,228.5 | 6,079.923077 | 6,872.333333 | |
Given a geometric sequence $\{a_n\}$ where the sum of the first $n$ terms $S_n = a \cdot 3^n - 2$, find the value of $a_2$. | 12 | 0.75 | 5,445.0625 | 4,644.416667 | 7,847 | |
Given a line segment of length $6$ is divided into three line segments of lengths that are positive integers, calculate the probability that these three line segments can form a triangle. | \frac {1}{10} | 0.0625 | 7,628.4375 | 8,192 | 7,590.866667 | |
For how many positive integers $x$ is $100 \leq x^2 \leq 200$? | 5 | 1 | 2,001.375 | 2,001.375 | -1 | |
The graph of \[\sqrt{(x-1)^2+(y+2)^2} - \sqrt{(x-5)^2+(y+2)^2} = 3\]consists of one branch of a hyperbola. Compute the positive value for the slope of an asymptote of the hyperbola. | \frac{\sqrt7}{3} | 0 | 4,261.4375 | -1 | 4,261.4375 | |
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 187 | Note that rolling a 1/4 will not affect whether or not the product is a perfect square. This means that in order for the product to be a perfect square, all non 1/4 numbers rolled must come in pairs, with the only exception being the triplet 2,3, 6. Now we can do casework:
If there are four 1/4's, then there are $2^4=... | 0 | 7,954.125 | -1 | 7,954.125 |
From the four numbers $0,1,2,3$, we want to select $3$ digits to form a three-digit number with no repeating digits. What is the probability that this three-digit number is divisible by $3$? | \dfrac{5}{9} | 0.875 | 5,824.8125 | 5,486.642857 | 8,192 | |
Given two congruent squares $ABCD$ and $EFGH$, each with a side length of $12$, they overlap to form a rectangle $AEHD$ with dimensions $12$ by $20$. Calculate the percent of the area of rectangle $AEHD$ that is shaded. | 20\% | 1 | 4,327.125 | 4,327.125 | -1 | |
Given Angela has 4 files that are each 1.2 MB, 8 files that are 0.9 MB each, and 10 files that are 0.6 MB each, calculate the minimum number of disks Angela will need to store all her files. | 11 | 0 | 754.5 | -1 | 754.5 | |
In triangle \( \triangle ABC \), point \( D \) is on side \( BC \) such that \( AD \perp BC \) and \( AD = BC = a \). Find the maximum value of \( \frac{b}{c} + \frac{c}{b} \). | \frac{3}{2} \sqrt{2} | 0 | 8,113 | -1 | 8,113 | |
Distribute 10 volunteer positions among 4 schools, with the requirement that each school receives at least one position. How many different ways can the positions be distributed? (Answer with a number.) | 84 | 0.3125 | 6,782.125 | 4,608.4 | 7,770.181818 | |
Given that $x<1$ and \[(\log_{10} x)^2 - \log_{10}(x^2) = 48,\]compute the value of \[(\log_{10}x)^3 - \log_{10}(x^3).\] | -198 | 0.875 | 2,316.125 | 2,415.642857 | 1,619.5 | |
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor$? | \frac{1}{3} | 1. **Understanding the Problem**: We need to find the probability that $\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor$ where $x$ and $y$ are chosen independently and uniformly from the interval $(0,1)$.
2. **Interpreting the Floor Function and Logarithm**: The expression $\lfloor\log_2x\rfloor$ represents the greatest i... | 0.9375 | 5,083.875 | 4,876.666667 | 8,192 |
Regular octagon \( CH I L D R E N \) has area 1. Determine the area of quadrilateral \( L I N E \). | 1/2 | 0.25 | 7,694.5625 | 6,403.25 | 8,125 | |
For how many integers $n$ between 1 and 11 (inclusive) is $\frac{n}{12}$ a repeating decimal? | 8 | 0.9375 | 3,416.875 | 3,098.533333 | 8,192 | |
Given a point $P$ on the curve $y= \frac{1}{2}e^{x}$ and a point $Q$ on the curve $y=\ln (2x)$, determine the minimum value of $|PQ|$. | \sqrt{2}(1-\ln 2) | 0.1875 | 8,047.8125 | 7,423 | 8,192 | |
Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = \omega$, the area of $PQRS$ can be expressed as the quadratic polynomial\[Area(PQRS) = \a... | 161 | 0.5 | 6,782.6875 | 5,373.375 | 8,192 | |
From a square with a side length of $6 \text{ cm}$, identical isosceles right triangles are cut off from each corner so that the area of the square is reduced by $32\%$. What is the length of the legs of these triangles? | 2.4 | 0.5625 | 6,708.9375 | 5,555.444444 | 8,192 | |
Jolene and Tia are playing a two-player game at a carnival. In one bin, there are five red balls numbered 5, 10, 15, 20, and 25. In another bin, there are 25 green balls numbered 1 through 25. In the first stage of the game, Jolene chooses one of the red balls at random. Next, the carnival worker removes the green ball... | 13/40 | 0.75 | 2,766.5625 | 3,510.583333 | 534.5 | |
How many positive multiples of $7$ that are less than $1000$ end with the digit $3$? | 14 | 0.9375 | 3,952.625 | 3,670 | 8,192 | |
The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probability that one person quits is independent of the probability that any other person quits, wha... | \frac{7}{15} | 0.9375 | 2,912.75 | 2,560.8 | 8,192 | |
In a sequence of positive integers starting from 1, certain numbers are painted red according to the following rules: First paint 1, then the next 2 even numbers $2, 4$; then the next 3 consecutive odd numbers after 4, which are $5, 7, 9$; then the next 4 consecutive even numbers after 9, which are $10, 12, 14, 16$; th... | 3943 | 0.3125 | 7,306.6875 | 5,606.4 | 8,079.545455 | |
When the greatest common divisor and least common multiple of two integers are multiplied, their product is 200. How many different values could be the greatest common divisor of the two integers? | 4 | 0.9375 | 5,399.4375 | 5,213.266667 | 8,192 | |
Lucas wants to buy a book that costs $28.50. He has two $10 bills, five $1 bills, and six quarters in his wallet. What is the minimum number of nickels that must be in his wallet so he can afford the book? | 40 | 0.5625 | 4,922.5 | 2,980 | 7,420 | |
Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), and \( M \) and \( I \) are the centroid and incenter of \(\triangle P F_{1} F_{2}\) respectively, if \( M I \) is perpendicular to the \( x \)-axis, then the r... | \sqrt{6} | 0.375 | 7,765.25 | 7,393.5 | 7,988.3 | |
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text... | 13 | 0 | 7,788 | -1 | 7,788 | |
Simplify $90r - 44r$. | 46r | 1 | 929.25 | 929.25 | -1 | |
What is the remainder when $3x^7-x^6-7x^5+2x^3+4x^2-11$ is divided by $2x-4$? | 117 | 0.8125 | 4,545.5 | 4,094.615385 | 6,499.333333 | |
In triangle $XYZ,$ $\angle X = 60^\circ,$ $\angle Y = 75^\circ,$ and $YZ = 6.$ Find $XZ.$ | 3\sqrt{2} + \sqrt{6} | 0.875 | 5,192.5 | 4,764 | 8,192 | |
A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows:
\begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108}... | 1 | 0.4375 | 7,320.3125 | 7,047.428571 | 7,532.555556 | |
Given the power function $y=(m^2-5m-5)x^{2m+1}$ is a decreasing function on $(0, +\infty)$, then the real number $m=$ . | -1 | 0.0625 | 8,172.8125 | 7,942 | 8,188.2 | |
Three friends are driving to New York City and splitting the gas cost equally. At the last minute, 2 more friends joined the trip. The cost of gas was then evenly redistributed among all of the friends. The cost for each of the original 3 decreased by $\$$11.00. What was the total cost of the gas, in dollars? | 82.50 | 1 | 2,556 | 2,556 | -1 | |
Determine the period of the function $y = \tan(2x) + \cot(2x)$. | \frac{\pi}{2} | 0.8125 | 4,542.3125 | 3,700.076923 | 8,192 | |
Starting with $10,000,000$, Esha forms a sequence by alternatively dividing by 2 and multiplying by 3. If she continues this process, what is the form of her sequence after 8 steps? Express your answer in the form $a^b$, where $a$ and $b$ are integers and $a$ is as small as possible. | (2^3)(3^4)(5^7) | 0 | 8,192 | -1 | 8,192 | |
Given the sequence ${a_n}$ where $a_{1}= \frac {3}{2}$, and $a_{n}=a_{n-1}+ \frac {9}{2}(- \frac {1}{2})^{n-1}$ (for $n\geq2$).
(I) Find the general term formula $a_n$ and the sum of the first $n$ terms $S_n$;
(II) Let $T_{n}=S_{n}- \frac {1}{S_{n}}$ ($n\in\mathbb{N}^*$), find the maximum and minimum terms of the seque... | -\frac{7}{12} | 0.4375 | 7,576 | 6,909.857143 | 8,094.111111 | |
My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team without restrictions? | 3003 | 0.875 | 4,676 | 4,173.714286 | 8,192 | |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0, S_{2017} < 0$. For any positive integer $n$, it holds that $|a_n| \geqslant |a_k|$, find the value of $k$. | 1009 | 0.125 | 7,373.25 | 6,043.5 | 7,563.214286 | |
The school plans to purchase 8 projectors and 32 computers, with each projector costing 7500 yuan and each computer costing 3600 yuan. How much money is needed in total? | 175200 | 0.75 | 394.5 | 383.25 | 428.25 | |
If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\$32$? | 96 | 1 | 2,255.375 | 2,255.375 | -1 | |
What is the sum of $1+2+4+8+16+ \cdots + 1024$? | 2047 | 1 | 2,352.875 | 2,352.875 | -1 | |
A puppy and two cats together weigh 24 pounds. The puppy and the larger cat together weigh exactly twice as much as the smaller cat, and the puppy and the smaller cat together weigh exactly the same as the larger cat. How many pounds does the puppy weigh? | 4 | 1 | 2,015.125 | 2,015.125 | -1 | |
How many ways are there to arrange the letters of the word $\text{B}_1\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{B}_2$, where three A's, two N's, and two B's are all considered different within each letter group but identical between groups? | 210 | 0.5 | 5,135.875 | 3,988.375 | 6,283.375 | |
A right triangle $ABC$ is inscribed in a circle. From the vertex $C$ of the right angle, a chord $CM$ is drawn, intersecting the hypotenuse at point $K$. Find the area of triangle $ABM$ if $BK: AB = 3:4$, $BC=2\sqrt{2}$, $AC=4$. | \frac{36}{19} \sqrt{2} | 0 | 6,684.25 | -1 | 6,684.25 | |
If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A} \mathbf{B}).$ | 24 | 1 | 2,108.3125 | 2,108.3125 | -1 | |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | 118 | 0.8125 | 4,466.5 | 3,606.769231 | 8,192 | |
From a set of integers $\{1, 2, 3, \ldots, 12\}$, eight distinct integers are chosen at random. What is the probability that, among those selected, the third smallest number is $4$? | \frac{56}{165} | 0.6875 | 3,939.4375 | 3,771.636364 | 4,308.6 | |
Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, then neither wins. What is the probability that Amina wins? | \frac{21}{32} | If Amina wins, she can win on her first turn, on her second turn, or on her third turn. If she wins on her first turn, then she went first and tossed tails. This occurs with probability $\frac{1}{2}$. If she wins on her second turn, then she tossed heads, then Bert tossed heads, then Amina tossed tails. This gives the ... | 0 | 8,025.0625 | -1 | 8,025.0625 |
Let $\mathbf{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -7 \\ 3 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$ | 13 | 0 | 5,285.875 | -1 | 5,285.875 | |
A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed? | 16 | 0.0625 | 7,805.0625 | 4,773 | 8,007.2 |
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