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 |
|---|---|---|---|---|---|---|
In the Cartesian coordinate system \( xOy \), the area of the region corresponding to the set of points \( K = \{(x, y) \mid (|x| + |3y| - 6)(|3x| + |y| - 6) \leq 0 \} \) is ________. | 24 | 0.0625 | 7,682.0625 | 8,192 | 7,648.066667 | |
The sides of rectangle $ABCD$ have lengths $12$ and $14$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. | 36\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
James and his sister each spin a spinner once. The modified spinner has six congruent sectors numbered from 1 to 6. If the absolute difference of their numbers is 2 or less, James wins. Otherwise, his sister wins. What is the probability that James wins? | \frac{2}{3} | 0.5 | 6,684.75 | 5,476.125 | 7,893.375 | |
A four-digit number satisfies the following conditions:
(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;
(2) When divided by 9, the remainder is 8.
Find the smallest odd four-digit number that satisfies these conditions.
(Shando... | 1979 | 0.0625 | 7,984.1875 | 4,867 | 8,192 | |
Calculate the sum of the coefficients of $P(x)$ if $\left(20 x^{27}+2 x^{2}+1\right) P(x)=2001 x^{2001}$. | 87 | The sum of coefficients of $f(x)$ is the value of $f(1)$ for any polynomial $f$. Plugging in 1 to the above equation, $P(1)=\frac{2001}{23}=87$. | 1 | 2,305.1875 | 2,305.1875 | -1 |
The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______. | \frac{5}{4} | 0.6875 | 6,181.125 | 5,494.818182 | 7,691 | |
Dolly, Molly and Polly each can walk at $6 \mathrm{~km} / \mathrm{h}$. Their one motorcycle, which travels at $90 \mathrm{~km} / \mathrm{h}$, can accommodate at most two of them at once (and cannot drive by itself!). Let $t$ hours be the time taken for all three of them to reach a point 135 km away. Ignoring the time r... | t<3.9 | First, we note that the three people are interchangeable in this problem, so it does not matter who rides and who walks at any given moment. We abbreviate the three people as D, M and P. We call their starting point $A$ and their ending point $B$. Here is a strategy where all three people are moving at all times and al... | 0 | 8,192 | -1 | 8,192 |
1. Given $\sin\alpha + \cos\alpha = \frac{7}{13}$, with $\alpha \in (0, \pi)$, find the value of $\tan\alpha$.
2. Find the minimum value for $y=\sin 2x + 2\sqrt{2}\cos\left(\frac{\pi}{4}+x\right)+3$. | 2 - 2\sqrt{2} | 0.3125 | 7,650 | 7,237.6 | 7,837.454545 | |
The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begin with either G or K, end with T, cannot c... | 1008 | 0.3125 | 5,363.6875 | 4,126.8 | 5,925.909091 | |
Real numbers \(x\) and \(y\) satisfy the following equations: \(x=\log_{10}(10^{y-1}+1)-1\) and \(y=\log_{10}(10^{x}+1)-1\). Compute \(10^{x-y}\). | \frac{101}{110} | Taking 10 to the power of both sides in each equation, these equations become: \(10^{x}=\left(10^{y-1}+1\right) \cdot 10^{-1}\) and \(10^{y}=\left(10^{x}+1\right) \cdot 10^{-1}\). Let \(a=10^{x}\) and \(b=10^{y}\). Our equations become: \(10a=b/10+1\) and \(10b=a+1\) and we are asked to compute \(a/b\). Subtracting the... | 0.9375 | 5,035.375 | 4,824.933333 | 8,192 |
A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end? | 4:40 a.m. | 1. **Convert the contest duration to hours and minutes**: The contest lasted for 1000 minutes. To convert this into hours, we divide by 60 (since there are 60 minutes in an hour):
\[
\frac{1000}{60} = 16 \text{ hours and } 40 \text{ minutes}
\]
This means the contest lasted for 16 hours and 40 minutes.
2. ... | 0 | 5,411.4375 | -1 | 5,411.4375 |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 0 | 8,151.1875 | -1 | 8,151.1875 | |
Let the area of the regular octagon $A B C D E F G H$ be $n$, and the area of the quadrilateral $A C E G$ be $m$. Calculate the value of $\frac{m}{n}$. | \frac{\sqrt{2}}{2} | 0 | 7,343.5625 | -1 | 7,343.5625 | |
Every day at noon, a scheduled ship departs from Moscow to Astrakhan and from Astrakhan to Moscow. The ship traveling from Moscow takes exactly four days to reach Astrakhan, then stays there for two days, and at noon two days after its arrival in Astrakhan, it departs back to Moscow. The ship traveling from Astrakhan t... | 13 | 0 | 8,090.3125 | -1 | 8,090.3125 | |
Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$. | 3059 | 0 | 7,676.25 | -1 | 7,676.25 | |
Define $F(x, y, z) = x \times y^z$. What positive value of $s$ is the solution to the equation $F(s, s, 2) = 1024$? | 8 \cdot \sqrt[3]{2} | 0 | 4,879.9375 | -1 | 4,879.9375 | |
Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term. | -2 | 0.3125 | 2,079.25 | 2,758.8 | 1,770.363636 | |
Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$ . Given that $\frac{a}{b}+\frac{b}{a}=98$ , compute $\sqrt c$ . | 202 | 1 | 2,041.875 | 2,041.875 | -1 | |
What is the area and perimeter of the smallest square that can contain a circle with a radius of 6? | 48 | 0.75 | 3,720.125 | 2,758.333333 | 6,605.5 | |
The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS. | 24 | 0.0625 | 7,795.4375 | 1,965 | 8,184.133333 | |
For how many three-digit positive integers is the sum of the digits equal to $5?$ | 15 | 0.9375 | 4,109.4375 | 3,837.266667 | 8,192 | |
Given that the center of circle $C$ lies on the $x$-axis and circle $C$ is tangent to the line $x + \sqrt{3}y + n = 0$ at the point $(\frac{3}{2}, \frac{\sqrt{3}}{2})$, find:
1. The value of $n$ and the equation of circle $C$.
2. If circle $M: x^2 + (y - \sqrt{15})^2 = r^2 (r > 0)$ is tangent to circle $C$, find the l... | 2\sqrt{19} | 0.4375 | 6,742 | 6,251.428571 | 7,123.555556 | |
At a nursery, 2006 babies sit in a circle. Suddenly each baby pokes the baby immediately to either its left or its right, with equal probability. What is the expected number of unpoked babies? | \frac{1003}{2} | The probability that any given baby goes unpoked is $1 / 4$. So the answer is $2006 / 4=1003 / 2$. | 0.0625 | 7,620 | 8,192 | 7,581.866667 |
What is the least positive integer that is divisible by three distinct primes? | 30 | 1 | 1,655.875 | 1,655.875 | -1 | |
In the rhombus \(ABCD\), the angle \(BCD\) is \(135^{\circ}\), and the sides are 8. A circle touches the line \(CD\) and intersects side \(AB\) at two points located 1 unit away from \(A\) and \(B\). Find the radius of this circle. | \frac{41 \sqrt{2}}{16} | 0 | 7,794.1875 | -1 | 7,794.1875 | |
Let the operation $\#$ be defined as $\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$ and $c$. What is the value of $\#(1, 2, 3)$? | -8 | 1 | 1,596.5625 | 1,596.5625 | -1 | |
Evaluate $\cfrac{\left\lceil\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil\right\rceil}{\left\lceil\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil\right\rceil}$ | \frac{1}{9} | 1 | 2,483.75 | 2,483.75 | -1 | |
Given that the three sides of triangle $\triangle ABC$ are $a$, $a+3$, and $a+6$, and the largest angle is twice the smallest angle, calculate the cosine value of the smallest angle. | \frac{3}{4} | 0.625 | 6,266.3125 | 5,110.9 | 8,192 | |
For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\). | 312 | 0.25 | 7,760 | 6,464 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a+c=8$, $\cos B= \frac{1}{4}$.
(1) If $\overrightarrow{BA}\cdot \overrightarrow{BC}=4$, find the value of $b$;
(2) If $\sin A= \frac{\sqrt{6}}{4}$, find the value of $\sin C$. | \frac{3\sqrt{6}}{8} | 0 | 6,674.375 | -1 | 6,674.375 | |
A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servin... | 8 \frac{3}{4} | To determine the maximum number of servings Jordan can make, we need to calculate the number of servings each ingredient can produce and then find the minimum of these values, as this will be the limiting factor.
1. **Chocolate**:
- The recipe requires $2$ squares of chocolate for $5$ servings.
- Jordan has $5$... | 0 | 6,493.5625 | -1 | 6,493.5625 |
Given the hyperbola $C$: $\frac{x^2}{a^2} - y^2 = 1$ $(a > 0)$ and the line $l$: $x + y = 1$, which intersect at two distinct points $A$ and $B$.
1. Find the range of values for $a$.
2. Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\overrightarrow{PA} = \frac{5}{12}\overrightarrow{PB}$. Find the... | \frac{17}{13} | 0.6875 | 6,477.9375 | 5,698.818182 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$ and $|3\overrightarrow{a} - 2\overrightarrow{b}| = \sqrt{7}$,
(I) Find the magnitude of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$;
(II) Find the value of $|3\overrightarrow{... | \sqrt{13} | 0.9375 | 2,533.125 | 2,564.133333 | 2,068 | |
The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? | \sqrt{3} | 0.5625 | 6,198 | 4,695.444444 | 8,129.857143 | |
In the equation $|x-7| -3 = -2$, what is the product of all possible values of $x$? | 48 | 1 | 1,034.5 | 1,034.5 | -1 | |
$ABCD$ is a rectangle with $AB = CD = 2$ . A circle centered at $O$ is tangent to $BC$ , $CD$ , and $AD$ (and hence has radius $1$ ). Another circle, centered at $P$ , is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$ . If line $AT$ is tangent to both circles at $T$ , find ... | 3 - 2\sqrt{2} | 0.125 | 7,851.5 | 5,468 | 8,192 | |
Given the function $f(x)=x\ln x-x$, find the monotonic intervals and the extreme values of the function $f(x)$. | -1 | 0.75 | 2,517.125 | 2,686.166667 | 2,010 | |
Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\] | \frac{2008}{1005} | 0.1875 | 7,742.4375 | 5,794.333333 | 8,192 | |
$\triangle ABC$ is similar to $\triangle DEF$ . What is the number of centimeters in the length of $\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(8,-2)--(5,4)--cycle);
label("8cm",(2.5,2),NW);
label("5cm",(6.1,1),NE);
draw((12,0)--(18,-1.5)--(15.7,2.5)--cycle);
label("$A$",... | 4.8 | 0.0625 | 5,404.5625 | 2,464 | 5,600.6 | |
One year ago, the number of years in Jane's age was a perfect square, and one year from now, her age will be a perfect cube. How many years old is Jane? | 26 | 1 | 4,279.875 | 4,279.875 | -1 | |
It is known that each side and diagonal of a regular polygon is colored in one of exactly 2018 different colors, and not all sides and diagonals are the same color. If a regular polygon contains no two-colored triangles (i.e., a triangle whose three sides are precisely colored with two colors), then the coloring of the... | 2017^2 | 0 | 7,751.4375 | -1 | 7,751.4375 | |
Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 | 1 | 1,940.625 | 1,940.625 | -1 | |
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?
[asy]
size(4cm,4cm);
path... | 4 | 1 | 2,983.5625 | 2,983.5625 | -1 | |
An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$. | 144 | First, we find the probability both are green, then the probability both are blue, and add the two probabilities. The sum should be equal to $0.58$.
The probability both are green is $\frac{4}{10}\cdot\frac{16}{16+N}$, and the probability both are blue is $\frac{6}{10}\cdot\frac{N}{16+N}$, so \[\frac{4}{10}\cdot\frac{... | 1 | 2,730.625 | 2,730.625 | -1 |
A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the aver... | 5 | 1. **Understanding the problem**: A lemming starts at a corner of a square with side length $10$ meters. It moves $6.2$ meters along a diagonal towards the opposite corner, then makes a $90^{\circ}$ right turn and runs $2$ meters. We need to find the average of the shortest distances from the lemming to each side of th... | 1 | 3,581.4375 | 3,581.4375 | -1 |
Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Find the value of $(1+p)(1+q)(1+r)$. | 51 | 0.9375 | 2,719.25 | 2,354.4 | 8,192 | |
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden? | 600 | 1. **Convert steps to feet**:
Mr. Green's garden measures $15$ steps by $20$ steps. Given that each step is $2$ feet long, we convert the dimensions from steps to feet:
\[
15 \text{ steps} \times 2 \text{ feet/step} = 30 \text{ feet}
\]
\[
20 \text{ steps} \times 2 \text{ feet/step} = 40 \text{ feet}... | 1 | 1,128 | 1,128 | -1 |
If the inequality $x^{2}+ax+1 \geqslant 0$ holds for all $x \in (0, \frac{1}{2}]$, find the minimum value of $a$. | -\frac{5}{2} | 0.6875 | 6,621.9375 | 6,113.818182 | 7,739.8 | |
The equation $y = -16t^2 + 34t + 25$ describes the height (in feet) of a ball thrown upwards at $34$ feet per second from $25$ feet above the ground. Determine the time (in seconds) when the ball will hit the ground. | \frac{25}{8} | 0 | 7,806.125 | -1 | 7,806.125 | |
If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$. | 23 | 0.9375 | 4,085.4375 | 3,811.666667 | 8,192 | |
Let \( f(x) \) be the polynomial \( (x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) \) where \( a_1, a_2, a_3, a_4, \) and \( a_5 \) are distinct integers. Given that \( f(104) = 2012 \), evaluate \( a_1 + a_2 + a_3 + a_4 + a_5 \). | 17 | 0.5 | 7,161.875 | 6,488.5 | 7,835.25 | |
Mark has $\frac{3}{4}$ of a dollar and Carolyn has $\frac{3}{10}$ of a dollar. How many dollars do they have altogether? (Give your answer as a decimal.) | \$1.05 | 1 | 1,332.1875 | 1,332.1875 | -1 | |
If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: | -3 | 1. **Identify the values of \(y_1\) and \(y_2\) using the given quadratic equation**:
Given the quadratic equation \(y = ax^2 + bx + c\), we substitute the x-values of the points into the equation:
- For the point \((1, y_1)\), substituting \(x = 1\) gives:
\[
y_1 = a(1)^2 + b(1) + c = a + b + c
\... | 1 | 1,656.625 | 1,656.625 | -1 |
What percent of square $PQRS$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.8));
xaxis(0,7,Ticks(1.0,NoZero));
yaxis(0,7,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,... | 67.35\% | 0.125 | 8,078.5 | 7,284 | 8,192 | |
How many total days were there in the years 2001 through 2004? | 1461 | 0.8125 | 1,412.125 | 1,623.615385 | 495.666667 | |
A region $S$ in the complex plane is defined by \begin{align*}
S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.
\end{align*}A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$? | \frac 79 | 0 | 8,192 | -1 | 8,192 | |
On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90? | 15180 | 0 | 7,486.1875 | -1 | 7,486.1875 | |
23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am o... | 14 | 0.125 | 7,904.875 | 6,751 | 8,069.714286 | |
From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers. The probability that these $2$ numbers are coprime is ______. | \frac{2}{3} | 0.25 | 7,333.3125 | 4,757.25 | 8,192 | |
In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visi... | 11 | 0.0625 | 4,772.125 | 6,087 | 4,684.466667 | |
Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $4$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $9$ times the length of $\overline{CD}$. The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}$? | \frac{1}{10} | 1. **Setting up the relationships:**
- Given that $\overline{AB} = 4\overline{BD}$, we can express $\overline{AB}$ and $\overline{BD}$ in terms of a common variable, say $x$. Thus, $\overline{BD} = x$ and $\overline{AB} = 4x$.
- Since $\overline{AB} + \overline{BD} = \overline{AD}$, substituting the expressions f... | 1 | 3,585.0625 | 3,585.0625 | -1 |
Given the polynomial $f(x) = 4x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$, find the value of $V_1$ when calculating $f(5)$ using the Horner's Method. | 22 | 1 | 2,560.3125 | 2,560.3125 | -1 | |
Claire begins with 40 sweets. Amy gives one third of her sweets to Beth, Beth gives one third of all the sweets she now has to Claire, and then Claire gives one third of all the sweets she now has to Amy. Given that all the girls end up having the same number of sweets, determine the number of sweets Beth had originall... | 50 | 0.3125 | 7,475.25 | 6,036.4 | 8,129.272727 | |
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______. | \sqrt{17}-1 | 0 | 8,192 | -1 | 8,192 | |
If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\leqslant x\leqslant n\left(m \lt n\right)$, and satisfies $M-N=2$, then the function is called the "range function" on $m\leqslant x\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}$, of which function ______ is the "range function"... | \frac{1}{8} | 0.1875 | 8,020.875 | 7,562.666667 | 8,126.615385 | |
Let \((a,b,c,d)\) be an ordered quadruple of integers, each in the set \(\{-2, -1, 0, 1, 2\}\). Determine the count of such quadruples for which \(a\cdot d - b\cdot c\) is divisible by 4. | 81 | 0 | 7,971.75 | -1 | 7,971.75 | |
Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______. | -8 | 0.9375 | 3,091 | 2,750.933333 | 8,192 | |
Determine if there exists a positive integer \( m \) such that the equation
\[
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=\frac{m}{a+b+c}
\]
has infinitely many solutions in positive integers \( (a, b, c) \). | 12 | 0 | 8,192 | -1 | 8,192 | |
Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer $i$ from $1$ to $12$ , inclusive, and swapping the positions of $a_i$ and $a_{i+1}$ . Define the *weight* of a permutation to be ... | 13703 | 0 | 8,053.3125 | -1 | 8,053.3125 | |
What is the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[5]{16}|=3$? | 6 | 1 | 1,489.6875 | 1,489.6875 | -1 | |
In the expansion of $(x^{4}+y^{2}+\frac{1}{2xy})^{7}$, the constant term is ______. | \frac{105}{16} | 1 | 3,964.9375 | 3,964.9375 | -1 | |
In triangle \(ABC\), the angle bisector \(BL\) is drawn. Find the area of the triangle, given that \(AL = 2\), \(BL = \sqrt{30}\), and \(CL = 5\). | \frac{7\sqrt{39}}{4} | 0 | 6,100.4375 | -1 | 6,100.4375 | |
A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______. | 12 | 0.25 | 3,893.8125 | 2,821.75 | 4,251.166667 | |
Find the ones digit of $22^{22(11^{11})}$ | 4 | 1 | 3,633.5 | 3,633.5 | -1 | |
One piece of string is 1.5 inches long and another piece of string is 4.5 inches long. What is the average length, in inches, of these two pieces of string? | 3 | 0.9375 | 1,231.3125 | 1,211.733333 | 1,525 | |
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at | \text{4:10 P.M.} | 1. **Calculate the time worked before lunch:** Maria starts working at 7:25 A.M. and takes her lunch break at noon. The time interval from 7:25 A.M. to 12:00 P.M. is calculated as follows:
\[
12:00 - 7:25 = 4 \text{ hours and } 35 \text{ minutes}
\]
2. **Account for the lunch break:** Maria's lunch break last... | 0 | 5,559.0625 | -1 | 5,559.0625 |
Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$... | 83 | We let $x = EP = FQ$, $y = EQ$, $k = PQ$. Since $AE = \frac {1}{2}AB$ and $AD = \frac {1}{2}AC$, $\triangle AED \sim \triangle ABC$ and $ED \parallel BC$.
By alternate interior angles, we have $\angle PEQ = \angle BFQ$ and $\angle EPQ = \angle FBQ$. By vertical angles, $\angle EQP = \angle FQB$.
Thus $\triangle EQP \... | 0.0625 | 8,144.1875 | 7,955 | 8,156.8 |
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle. | \frac{2}{3} | 0 | 2,823.5 | -1 | 2,823.5 | |
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of t... | 25 | 0.0625 | 7,941.75 | 5,472 | 8,106.4 | |
A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?
A) 12
B) 15
C) 18
D) 20
E) 24 | 20 | 0.125 | 5,047.4375 | 583.5 | 5,685.142857 | |
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differ... | 2 |
Let us consider a \(100 \times 100\) chessboard and the placement of 2500 kings such that:
1. No king can capture another king, meaning no two kings can be placed on squares that share a common vertex.
2. Each row and each column contains exactly 25 kings.
The primary challenge is to ensure that each king is placed ... | 0 | 7,999.8125 | -1 | 7,999.8125 |
Calculate:
$$
202.2 \times 89.8 - 20.22 \times 186 + 2.022 \times 3570 - 0.2022 \times 16900
$$ | 18198 | 0.5625 | 6,195.0625 | 4,724.888889 | 8,085.285714 | |
During the "Cool Summer Happy Shopping" promotion held in a certain shopping mall, Xiao Yang bought $m$ items of type A goods priced at $5$ yuan each, and $n$ items of type B goods priced at $17 yuan each, spending a total of $203$ yuan. Then the maximum value of $m+n$ is ______. | 31 | 1 | 3,947.875 | 3,947.875 | -1 | |
Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by
\[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\
n \minus{} b, & \text{if } n >M. \end{cases}
\]
Let $ f^1(n) \equal{} f(n),$ $ f_{i ... | \frac {a + b}{\gcd(a,b)} |
Let \( a, b \in \mathbb{N} \) with \( 1 \leq a \leq b \), and let \( M = \left\lfloor \frac{a + b}{2} \right\rfloor \). The function \( f: \mathbb{Z} \to \mathbb{Z} \) is defined as:
\[
f(n) =
\begin{cases}
n + a, & \text{if } n \leq M, \\
n - b, & \text{if } n > M.
\end{cases}
\]
We are required to find the small... | 0 | 8,192 | -1 | 8,192 |
When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | \frac{1}{36} | 0 | 7,860.1875 | -1 | 7,860.1875 | |
Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^{y}=9$, compute $x y$. | 16 \sqrt[4]{3} | Note that $$27^{\sqrt{y}}=\left(x^{\sqrt{y}}\right)^{\sqrt{y}}=x^{y}=(\sqrt{x})^{2 y}=81$$ so $\sqrt{y}=4 / 3$ or $y=16 / 9$. It follows that $x^{4 / 3}=27$ or $x=9 \sqrt[4]{3}$. The final answer is $9 \sqrt[4]{3} \cdot 16 / 9=16 \sqrt[4]{3}$. | 0.625 | 7,586.8125 | 7,240.9 | 8,163.333333 |
When a number is tripled and then decreased by 5, the result is 16. What is the original number? | 7 | To get back to the original number, we undo the given operations. We add 5 to 16 to obtain 21 and then divide by 3 to obtain 7. These are the 'inverse' operations of decreasing by 5 and multiplying by 3. | 1 | 1,579.9375 | 1,579.9375 | -1 |
Out of two hundred ninth-grade students, $80\%$ received excellent grades on the first exam, $70\%$ on the second exam, and $59\%$ on the third exam. What is the minimum number of students who could have received excellent grades on all three exams?
| 18 | 0.25 | 6,285.75 | 3,911.75 | 7,077.083333 | |
Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \in \{-1,0,1,2,3\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions. | 17 | 0.75 | 5,716.5 | 5,439.166667 | 6,548.5 | |
Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? | 8654232 | 0 | 8,192 | -1 | 8,192 | |
In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\left\{{\begin{array}{l}{x=3-\frac{{\sqrt{3}}}{2}t,}\\{y=\sqrt{3}-\frac{1}{2}t}\end{array}}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of... | \frac{\sqrt{3}}{2} | 0 | 6,068.1875 | -1 | 6,068.1875 | |
In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $ \begin{cases} x=-\frac{1}{2}t \\ y=2+\frac{\sqrt{3}}{2}t \end{cases} (t\text{ is the parameter})$, and a circle $C$ with polar coordinate equation $\rho=4\cos\theta$ is established with the origin $O$ as the pole and the pos... | 4+2\sqrt{3} | 0 | 7,713.5 | -1 | 7,713.5 | |
Find the number of sets $A$ that satisfy the three conditions: $\star$ $A$ is a set of two positive integers $\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\star$ $A$ contains the number $30.$ | 129 | 0.25 | 7,719.125 | 6,639.5 | 8,079 | |
A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to do this (answer with a number)? | 30 | 0 | 7,897.3125 | -1 | 7,897.3125 | |
Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2? | \frac{1}{4} | 0.875 | 1,718.5 | 1,742 | 1,554 | |
Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the s... | 0.64 | 0.5 | 5,814.5625 | 5,044.875 | 6,584.25 | |
A numerical sequence is defined by the conditions: \( a_{1}=1 \), \( a_{n+1}=a_{n}+\left \lfloor \sqrt{a_{n}} \right \rfloor \).
How many perfect squares are among the first terms of this sequence that do not exceed 1,000,000? | 10 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Trapezoid $A B C D$, with bases $A B$ and $C D$, has side lengths $A B=28, B C=13, C D=14$, and $D A=15$. Let diagonals $A C$ and $B D$ intersect at $P$, and let $E$ and $F$ be the midpoints of $A P$ and $B P$, respectively. Find the area of quadrilateral $C D E F$. | 112 | Note that $E F$ is a midline of triangle $A P B$, so $E F$ is parallel to $A B$ and $E F=\frac{1}{2} A B=14=C D$. We also have that $E F$ is parallel to $C D$, and so $C D E F$ is a parallelogram. From this, we have $E P=P C$ as well, so $\frac{C E}{C A}=\frac{2}{3}$. It follows that the height from $C$ to $E F$ is $\f... | 0.75 | 6,750.125 | 6,269.5 | 8,192 |
A positive integer \( N \) and \( N^2 \) end with the same sequence of digits \(\overline{abcd}\), where \( a \) is a non-zero digit. Find \(\overline{abc}\). | 937 | 1 | 4,857.0625 | 4,857.0625 | -1 | |
Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{5}{7},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}.$ | \frac{25}{49} | 0.9375 | 4,753.625 | 4,524.4 | 8,192 |
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