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 |
|---|---|---|---|---|---|---|
A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there? | 36 | 0 | 3,623 | -1 | 3,623 | |
Consider a rectangle \( ABCD \) where the side lengths are \( \overline{AB}=4 \) and \( \overline{BC}=8 \). Points \( M \) and \( N \) are fixed on sides \( BC \) and \( AD \), respectively, such that the quadrilateral \( BMDN \) is a rhombus. Calculate the area of this rhombus. | 20 | 0.9375 | 4,348.75 | 4,092.533333 | 8,192 | |
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals... | (m, n) = (36, 6) |
To solve this problem, we have to determine the total number of medals, \( m \), and the number of days, \( n \), based on the distribution given over the days.
Let's denote the number of remaining medals after each day as \( R_i \) for day \( i \). Initially, we have all \( m \) medals, so \( R_0 = m \).
The given ... | 0 | 7,780.25 | -1 | 7,780.25 |
Suppose $\csc y + \cot y = \frac{25}{7}$ and $\sec y + \tan y = \frac{p}{q}$, where $\frac{p}{q}$ is in lowest terms. Find $p+q$. | 29517 | 0 | 4,959.3125 | -1 | 4,959.3125 | |
How many paths are there from point $A$ to point $B$, if every step must be up or to the right in a grid where $A$ is at the bottom left corner and $B$ is at the top right corner of a 7x7 grid? | 3432 | 0.9375 | 3,568.6875 | 3,539.066667 | 4,013 | |
A certain organism begins as three cells. Each cell splits and becomes two cells at the end of two days. At the end of another two days, every cell of the organism splits and becomes two cells. This process lasts for a total of 8 days, and no cells die during this time. How many cells are there at the end of the $8^\te... | 48 | 1 | 2,708.25 | 2,708.25 | -1 | |
If altitude $CD$ is $\sqrt3$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?
[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(3)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$30^{\circ}$",A-(0.05,0.4),E);
label("$A$",A,N);label("$B$",B,E... | 2\sqrt{3} | 0.4375 | 7,004.4375 | 5,477.571429 | 8,192 | |
Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]
draw((0,0)--(30,0)--(12,8)--(22,8)--(0,0));
draw((10,0)--(12,8));
draw((20,0)--(22,8));
label("$I$",(0,0),W);
label("$C$",(10,0),S)... | 36 | 0 | 8,052.0625 | -1 | 8,052.0625 | |
The base of a right prism is an isosceles trapezoid \(ABCD\) with \(AB = CD = 13\), \(BC = 11\), and \(AD = 21\). The area of the diagonal cross-section of the prism is 180. Find the total surface area of the prism. | 906 | 0.25 | 7,006.5625 | 5,223.5 | 7,600.916667 | |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $$\begin{cases} x=3\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is the parameter}),$$ and the parametric equation of the line $l$ is $$\begin{cases} x=t-1 \\ y=2t-a-1 \end{cases} (t \text{ is the parameter}).$$
(Ⅰ) If $a=1$,... | 2\sqrt{5}-2\sqrt{2} | 0.1875 | 7,958 | 6,944 | 8,192 | |
Points $ A$ and $ B$ lie on a circle centered at $ O$ , and $ \angle AOB=60^\circ$ . A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$ . What is the ratio of the area of the smaller circle to that of the larger circle? | \frac{1}{9} | 0.9375 | 4,373.75 | 4,119.2 | 8,192 | |
An integer $n$ is decreased by 2 and then multiplied by 5. If the result is 85, what is the value of $n$? | 19 | We undo each of the operations in reverse order. The final result, 85, was obtained by multiplying a number by 5. This number was $85 \div 5=17$. The number 17 was obtained by decreasing $n$ by 2. Thus, $n=17+2=19$. | 1 | 1,608.0625 | 1,608.0625 | -1 |
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale.) [asy]
/* AMC8 1998 #13P */
size(1inch,1inch);
pair r1c1=(0,0), r1c2=(10,0), r1c3=(20,0), r1c4=(30, 0), r1c5=(40, 0);
pair r2c1=(0,10), r2c2=(10,10), r2c3=(20,10), r2c4=(30, 10), r2c5=(40, 10);
pair r3c1=(... | \frac{1}{8} | 0.75 | 5,988.625 | 5,254.166667 | 8,192 | |
Comparing two rectangular parallelepiped bars, it was found that the length, width, and height of the second bar are each 1 cm greater than those of the first bar, and the volume and total surface area of the second bar are 18 cm³ and 30 cm² greater, respectively, than those of the first one. What is the total surface ... | 22 | 0.8125 | 4,283.5625 | 3,381.615385 | 8,192 | |
For a permutation $\sigma$ of $1,2, \ldots, 7$, a transposition is a swapping of two elements. Let $f(\sigma)$ be the minimum number of transpositions necessary to turn $\sigma$ into the permutation $1,2,3,4,5,6,7$. Find the sum of $f(\sigma)$ over all permutations $\sigma$ of $1,2, \ldots, 7$. | 22212 | To solve this problem, we use the idea of a cycle in a permutation. If $\sigma$ is a permutation, we say that $\left(a_{1} a_{2} \cdots a_{k}\right)$ is a cycle if $\sigma\left(a_{i}\right)=\sigma\left(a_{i+1}\right)$ for $1 \leq i \leq k-1$ and $\sigma\left(a_{k}\right)=a_{1}$. Any permutation can be decomposed into d... | 0.375 | 6,650.875 | 5,319.166667 | 7,449.9 |
One of the angles of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and also circumscribed around this trapezoid. | 1:3 | 0 | 6,443.875 | -1 | 6,443.875 | |
A group of schoolchildren, heading to a school camp, was planned to be seated in buses so that there would be an equal number of passengers in each bus. Initially, 22 people were seated in each bus, but it turned out that three schoolchildren could not be seated. However, when one bus left empty, all the remaining sch... | 135 | 0.1875 | 6,726.5 | 3,848.666667 | 7,390.615385 | |
In $\Delta ABC$, $c=2a$, $B={120}^{\circ}$, and the area of $\Delta ABC$ is $\frac{\sqrt{3}}{2}$.
(I) Find the value of $b$;
(II) Find the value of $\tan A$. | \frac{\sqrt{3}}{5} | 0 | 4,092.9375 | -1 | 4,092.9375 | |
Al's age is $16$ more than the sum of Bob's age and Carl's age, and the square of Al's age is $1632$ more than the square of the sum of Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl? | 102 | 1. **Define Variables:**
Let $a$ be Al's age, $b$ be Bob's age, and $c$ be Carl's age.
2. **Set Up Equations:**
From the problem, we have two equations:
\[ a = 16 + b + c \]
\[ a^2 = 1632 + (b+c)^2 \]
3. **Substitute and Expand:**
Substitute $a = 16 + b + c$ into the second equation:
\[ (16 + b + c... | 1 | 1,752.6875 | 1,752.6875 | -1 |
Given $\begin{vmatrix} p & q \\ r & s \end{vmatrix} = 6,$ find
\[\begin{vmatrix} p & 9p + 4q \\ r & 9r + 4s \end{vmatrix}.\] | 24 | 0.875 | 3,789.1875 | 3,160.214286 | 8,192 | |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | 28 | 0.8125 | 4,119.8125 | 3,180.076923 | 8,192 | |
When randomly selecting three line segments from the lengths 1, 3, 5, 7, and 9, calculate the probability that the selected line segments cannot form a triangle. | \frac{7}{10} | 0.3125 | 5,833.6875 | 5,901.8 | 5,802.727273 | |
There is a box containing red, blue, green, and yellow balls. It is known that the number of red balls is twice the number of blue balls, the number of blue balls is twice the number of green balls, and the number of yellow balls is more than seven. How many yellow balls are in the box if there are 27 balls in total? | 20 | 0.3125 | 1,090.125 | 751.4 | 1,244.090909 | |
Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form. | 15\sqrt{3} | 0.9375 | 4,201.125 | 3,935.066667 | 8,192 | |
A $30\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\equiv0,-1,1 \pmod {30}$ and $(i,j)\neq(x,y)$ . Therefore, each square has exactly $8$ neighbors)
What is the ... | 300 | 0 | 8,187.25 | -1 | 8,187.25 | |
To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 19 | 0.1875 | 8,098.4375 | 7,693 | 8,192 | |
How many different four-digit numbers can be formed by arranging the four digits in 2004? | 6 | 1 | 2,336.25 | 2,336.25 | -1 | |
Six students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Aaron ate the most apples and Zeb ate the fewest. How many more apples than Zeb did Aaron eat?
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[8];
for( int i = 0 ; i <... | 5 | 0.625 | 2,100.6875 | 1,556.6 | 3,007.5 | |
If $\cos \theta = \frac{2}{3},$ then find $\cos 2 \theta.$ | -\frac{1}{9} | 1 | 2,124.3125 | 2,124.3125 | -1 | |
The integer $n$ is the smallest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$. | 592 | Any multiple of 15 is a multiple of 5 and a multiple of 3.
Any multiple of 5 ends in 0 or 5; since $n$ only contains the digits 0 and 8, the units digit of $n$ must be 0.
The sum of the digits of any multiple of 3 must be divisible by 3. If $n$ has $a$ digits equal to 8, the sum of the digits of $n$ is $8a$. For this... | 0.9375 | 4,464.9375 | 4,216.466667 | 8,192 |
Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\frac{2}{3}$ as fast as Eugene, and Katie runs $\frac{7}{5}$ as fast as Brianna, how fast does Katie run? | \frac{56}{15} | 1 | 2,414 | 2,414 | -1 | |
Given a basketball player made 8 baskets during a game, each worth 1, 2, or 3 points, calculate the total number of different numbers that could represent the total points scored by the player. | 17 | 0.375 | 6,489.6875 | 5,734.666667 | 6,942.7 | |
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$ . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$ , where $m$ , $n$ are relatively prime integers. Find $m + n$ . | 2224 | 0.125 | 8,026.625 | 7,453.5 | 8,108.5 | |
Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\] | 3 |
To find the minimum value of the expression
\[
A = \frac{2-a^3}{a} + \frac{2-b^3}{b} + \frac{2-c^3}{c},
\]
given that \( a, b, c \) are positive real numbers and \( a + b + c = 3 \), we proceed as follows:
First, we rewrite the expression:
\[
A = \frac{2}{a} - a^2 + \frac{2}{b} - b^2 + \frac{2}{c} - c^2.
\]
Consi... | 0.5 | 6,863.9375 | 5,535.875 | 8,192 |
Let the base areas of two cylinders be $S_1$ and $S_2$, and their volumes be $\upsilon_1$ and $\upsilon_2$, respectively. If their lateral areas are equal, and $$\frac {S_{1}}{S_{2}}= \frac {16}{9},$$ then the value of $$\frac {\upsilon_{1}}{\upsilon_{2}}$$ is \_\_\_\_\_\_. | \frac {4}{3} | 1 | 3,282.75 | 3,282.75 | -1 | |
Three noncollinear points and a line $\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\ell$ (or $\ell$ itself). There are exactly $n$ lines perpendicular to $\ell$ with the following property: the three circles with centers at the given points and tangent to the line all concur at ... | 1 | The condition for the line is that each of the three points lies at an equal distance from the line as from some fixed point; in other words, the line is the directrix of a parabola containing the three points. Three noncollinear points in the coordinate plane determine a quadratic polynomial in $x$ unless two of the p... | 0 | 8,192 | -1 | 8,192 |
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any ci... | 200 | 0.125 | 7,885.75 | 7,951 | 7,876.428571 | |
Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
| 682 | 0.125 | 7,811.875 | 5,151 | 8,192 | |
John has 15 marbles of different colors, including one red, one green, one blue, and three yellow marbles. In how many ways can he choose 5 marbles, if he must choose exactly one marble that is red, green, blue, or yellow? | 756 | 0.4375 | 4,796.625 | 3,542.714286 | 5,771.888889 | |
Two 24-sided dice have the following configurations: 5 purple sides, 8 blue sides, 10 red sides, and 1 gold side. What is the probability that when both dice are rolled, they will show the same color? | \dfrac{95}{288} | 0.8125 | 3,701.875 | 3,147.769231 | 6,103 | |
Given a line $l$ passes through the foci of the ellipse $\frac {y^{2}}{2}+x^{2}=1$ and intersects the ellipse at points P and Q. The perpendicular bisector of segment PQ intersects the x-axis at point M. The maximum area of $\triangle MPQ$ is __________. | \frac {3 \sqrt {6}}{8} | 0 | 8,067.125 | -1 | 8,067.125 | |
The electronic clock on the International Space Station displayed time in the format HH:MM. Due to an electromagnetic storm, the device malfunctioned, and each digit on the display either increased by 1 or decreased by 1. What was the actual time of the storm if the clock displayed 20:09 immediately after it? | 11:18 | 0 | 8,093.4375 | -1 | 8,093.4375 | |
In a unit cube \(ABCDA_1B_1C_1D_1\), eight planes \(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\) and \(D_1AB_1\) intersect the cube. What is the volume of the part that contains the center of the cube? | 1/6 | 0 | 8,192 | -1 | 8,192 | |
In convex quadrilateral $ABCD, \angle A \cong \angle C, AB = CD = 180,$ and $AD \neq BC.$ The perimeter of $ABCD$ is $640$. Find $\lfloor 1000 \cos A \rfloor.$ (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x.$) | 777 | [asy] real x = 1.60; /* arbitrary */ pointpen = black; pathpen = black+linewidth(0.7); size(180); real BD = x*x + 1.80*1.80 - 2 * 1.80 * x * 7 / 9; pair A=(0,0),B=(1.8,0),D=IP(CR(A,x),CR(B,BD)),C=OP(CR(D,1.8),CR(B,2.80 - x)); D(MP("A",A)--MP("B",B)--MP("C",C)--MP("D",D,N)--B--A--D); MP("180",(A+B)/2); MP("180",(C+D)/2,... | 0.8125 | 5,031.375 | 4,302 | 8,192 |
Let $\mathbf{M} = \begin{pmatrix} 2 & 7 \\ -3 & -1 \end{pmatrix}.$ There exist scalars $p$ and $q$ such that
\[\mathbf{M}^2 = p \mathbf{M} + q \mathbf{I}.\]Enter the ordered pair $(p,q).$ | (1,-19) | 1 | 2,761.4375 | 2,761.4375 | -1 | |
If $2^{1998} - 2^{1997} - 2^{1996} + 2^{1995} = k \cdot 2^{1995},$ what is the value of $k$? | 3 | 1. **Rewrite the equation with a common factor:**
Start by factoring out the smallest power of 2 from each term in the equation:
\[
2^{1998} - 2^{1997} - 2^{1996} + 2^{1995} = k \cdot 2^{1995}
\]
Factor out $2^{1995}$:
\[
2^{1995}(2^3 - 2^2 - 2^1 + 1) = k \cdot 2^{1995}
\]
2. **Simplify the e... | 0.9375 | 2,313.375 | 1,921.466667 | 8,192 |
Given \( f(x) = 2^x \) and \( g(x) = \log_{\sqrt{2}} (8x) \), find the value of \( x \) that satisfies \( f[g(x)] = g[f(x)] \). | \frac{1 + \sqrt{385}}{64} | 0 | 6,301.1875 | -1 | 6,301.1875 | |
Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$
[asy] import graph; size(5cm); real l... | 616 | Let $x=\angle CAM$, so $3x=\angle CDM$. Then, $\frac{\tan 3x}{\tan x}=\frac{CM/1}{CM/11}=11$. Expanding $\tan 3x$ using the angle sum identity gives \[\tan 3x=\tan(2x+x)=\frac{3\tan x-\tan^3x}{1-3\tan^2x}.\] Thus, $\frac{3-\tan^2x}{1-3\tan^2x}=11$. Solving, we get $\tan x= \frac 12$. Hence, $CM=\frac{11}2$ and $AC= \fr... | 0.25 | 8,148.375 | 8,017.5 | 8,192 |
From the set {1, 2, 3, ..., 14}, select \(a_{1}, a_{2}, a_{3}\) in increasing order such that \(a_{2} - a_{1} \geq 3\) and \(a_{3} - a_{2} \geq 3\). Find the total number of different ways to select these numbers. | 120 | 0.4375 | 7,452 | 6,500.571429 | 8,192 | |
How many numbers in the set $\{3,13,23,33, \ldots\}$ can be written as the difference of two primes? | 1 | 0.625 | 6,689.25 | 5,787.6 | 8,192 | |
The remainder when the product $1492 \cdot 1776 \cdot 1812 \cdot 1996$ is divided by 5 is | 4 | 1. **Identify the last digits of each number**:
- The last digit of $1492$ is $2$.
- The last digit of $1776$ is $6$.
- The last digit of $1812$ is $2$.
- The last digit of $1996$ is $6$.
2. **Calculate the product of the last digits**:
- Compute the product of these last digits: $2 \cdot 6 \cdot 2 \cd... | 0.9375 | 4,606.1875 | 4,367.133333 | 8,192 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $b\sin 2C=c\sin B$.
$(1)$ Find the value of angle $C$;
$(2)$ If $\sin \left(B- \frac {\pi}{3}\right)= \frac {3}{5}$, find the value of $\sin A$. | \frac {4 \sqrt {3}-3}{10} | 0 | 5,842.4375 | -1 | 5,842.4375 | |
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$ . | 10 | 0.0625 | 8,111.0625 | 6,897 | 8,192 | |
What is the probability that at most two of the cars have a deceased passenger, given that there are 4 deceased passengers among the travelers on a metro train consisting of 6 cars? | 1/6 | 0.125 | 7,478.375 | 7,718 | 7,444.142857 | |
On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is ... | \frac{1}{6} | 0.0625 | 7,979.625 | 4,794 | 8,192 | |
Triangle $ABC$ has $\angle C = 60^{\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\tan{\angle BAD}$?
$\mathrm{(A)}\ \frac{\sqrt{3}}{6}\qquad\mathrm{(B)}\ \frac{\sqrt{3}}{3}\qquad\mathrm{(C)}\ \frac{\sqrt{3}}{2\sqrt{2}}\qquad\mathrm{(D)}\ \frac{\sqrt{3}}{4\sqrt{2}-3}\qqua... | \frac{\sqrt{3}}{4\sqrt{2}-3} | 0 | 6,562.3125 | -1 | 6,562.3125 | |
If $x$, $y$, and $z$ are positive numbers satisfying \[
x+\frac{1}{y}=4,\ \ \ y+\frac{1}{z}=1,\text{ and }z+\frac{1}{x}=\frac{7}{3},
\]find the value of $xyz$. | 1 | 0.875 | 3,891.5 | 3,277.142857 | 8,192 | |
On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$.
| 810 | 0.6875 | 5,648.9375 | 4,493 | 8,192 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.375 | 1,747.4375 | 1,788.5 | 1,722.8 | |
A toy store sells a type of building block set: each starship is priced at 8 yuan, and each mech is priced at 26 yuan. A starship and a mech can be combined to form an ultimate mech, which sells for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in one week, earning 370 yuan, how many starsh... | 20 | 0.1875 | 5,230.9375 | 3,538.333333 | 5,621.538462 | |
A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed ... | 68 | 0 | 6,799.9375 | -1 | 6,799.9375 | |
Given that A and B can only take on the first three roles, and the other three volunteers (C, D, and E) can take on all four roles, calculate the total number of different selection schemes for four people from five volunteers. | 72 | 0.1875 | 7,546.625 | 4,750 | 8,192 | |
Diana is three years younger than Eduardo. Eduardo is four years older than Chad. Faye is three years older than Chad. Diana is 14 years old. How many years old is Faye? | 16 | 0.5625 | 1,292 | 1,241.333333 | 1,357.142857 | |
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? | 61917364224 | The number of such colorings is $2^{20} 3^{10} = 61917364224$. Identify the three colors red, white, and blue with (in some order) the elements of the field \mathbb{F}_3 of three elements (i.e., the ring of integers mod 3). The set of colorings may then be identified with the \mathbb{F}_3-vector space \mathbb{F}_3^E ge... | 0 | 8,137.1875 | -1 | 8,137.1875 |
Let \( p(x) = 2x^3 - 3x^2 + 1 \). How many squares of integers are there among the numbers \( p(1), p(2), \ldots, p(2016) \)? | 32 | 0.0625 | 7,867.8125 | 7,642 | 7,882.866667 | |
Find the sum of all real roots \( x \) of the equation \( \left(2^{x}-4\right)^{3}+\left(4^{x}-2\right)^{3}=\left(4^{x}+2^{x}-6\right)^{3} \). | 3.5 | 0 | 4,519.8125 | -1 | 4,519.8125 | |
If \( A \) is a positive integer such that \( \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \cdots + \frac{1}{(A+1)(A+3)} = \frac{12}{25} \), find the value of \( A \). | 22 | 0.375 | 6,755.375 | 4,361 | 8,192 | |
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\sqrt{2}$, what is the volume of the solid?
[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(... | 288 | 0.0625 | 8,083.5625 | 6,548 | 8,185.933333 | |
How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct? | 5040 | 0.1875 | 6,022.125 | 4,947.666667 | 6,270.076923 | |
In triangle $ABC$, $AB = 5$, $BC = 12$, and $AC = 13$. Let $BM$ be the median from vertex $B$ to side $AC$. If $BM = m \sqrt{2}$, then find $m$. | \frac{13}{2} | 0 | 3,299.125 | -1 | 3,299.125 | |
If $y=f(x)=\frac{x+2}{x-1}$, then it is incorrect to say: | $f(1)=0$ | We are given the function $y = f(x) = \frac{x+2}{x-1}$ and need to determine which statement about this function is incorrect.
1. **Checking option (A)**: We need to solve for $x$ in terms of $y$:
\[
y = \frac{x+2}{x-1} \implies y(x-1) = x+2 \implies yx - y = x + 2 \implies yx - x = y + 2 \implies x(y-1) = y + 2... | 0 | 6,696.1875 | -1 | 6,696.1875 |
In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\begin{cases} x=5\cos \alpha \\ y=\sin \alpha \end{cases}$ where $\alpha$ is a parameter, and point $P$ has coordinates $(3 \sqrt {2},0)$.
(1) Determine the shape of curve $C$;
(2) Given that line $l$ passes through poi... | \frac{7}{13} | 0.6875 | 6,277.8125 | 5,407.727273 | 8,192 | |
Given the equations $y = 2x^2$ and $y = 4x + c$, determine the value(s) of $c$ for which these two equations have precisely two identical solutions. | -2 | 1 | 3,061.75 | 3,061.75 | -1 | |
Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\overline{ABC}_7$, and in base $11$, then $n$ can be expressed as $\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. | 247 | 0.875 | 5,489.9375 | 5,103.928571 | 8,192 | |
Given vectors $\overrightarrow{a} = (x, -3)$, $\overrightarrow{b} = (-2, 1)$, $\overrightarrow{c} = (1, y)$ on a plane. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b} - \overrightarrow{c}$, and $\overrightarrow{b}$ is parallel to $\overrightarrow{a} + \overrightarrow{c}$, find the projection of $\overr... | -\sqrt{5} | 0 | 5,098.9375 | -1 | 5,098.9375 | |
How many times does the digit 9 appear in the list of all integers from 1 to 500? (The number $ 99 $, for example, is counted twice, because $9$ appears two times in it.) | 100 | 0.375 | 7,218.5625 | 5,596.166667 | 8,192 | |
Find $(1_8 + 2_8 + 3_8 + \cdots + 30_8) \times 3_{10}$, expressed in base $8$. | 1604_8 | 0.625 | 5,919.9375 | 4,556.7 | 8,192 | |
Suppose $\sqrt{1 + \sqrt{2y-3}} = \sqrt{6}$; find $y$. | 14 | 1 | 1,810.4375 | 1,810.4375 | -1 | |
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac ... | 25 | This problem just requires a good diagram and strong 3D visualization.
The region in $(x,y,z)$ where $x \ge \frac{1}{2}, y \ge \frac{1}{3}$ is that of a little triangle on the bottom of the above diagram, of $y \ge \frac{1}{3}, z \ge \frac{1}{6}$ is the triangle at the right, and $x \ge \frac 12, z \ge \frac 16$ the t... | 0 | 8,137 | -1 | 8,137 |
In how many ways can the number 5 be expressed as the sum of one or more positive integers? | 16 | 0.0625 | 2,971.5 | 1,863 | 3,045.4 | |
A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction. | \frac{4}{3} | 0.9375 | 4,880.8125 | 4,984.6 | 3,324 | |
The eccentricity of the ellipse given by the parametric equations $\begin{cases} x=3\cos\theta \\ y=4\sin\theta\end{cases}$ is $\frac{\sqrt{7}}{\sqrt{3^2+4^2}}$, calculate this value. | \frac { \sqrt {7}}{4} | 0 | 8,005.375 | -1 | 8,005.375 | |
Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\textdollar10500$ for both taxes. How many dollars was his inheritance? | 37500 | 1. **Define the variable**: Let $x$ be the total amount of Mr. Jon Q. Public's inheritance.
2. **Calculate federal taxes**: He pays $20\%$ in federal taxes, which is $0.2x$.
3. **Calculate remaining amount after federal taxes**: After paying federal taxes, he has $x - 0.2x = 0.8x$ left.
4. **Calculate state taxes**:... | 1 | 1,967.375 | 1,967.375 | -1 |
Calculate:<br/>$(1)-7+13-6+20$;<br/>$(2)-2^{3}+\left(2-3\right)-2\times \left(-1\right)^{2023}$. | -7 | 0.8125 | 601.125 | 610.769231 | 559.333333 | |
A three-digit $\overline{abc}$ number is called *Ecuadorian* if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$ . $\bullet$ $\overline{abc}$ is a multiple of $36$ . $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$ .
Determine all the Ecuadori... | 864 | 0.6875 | 6,754.4375 | 6,101 | 8,192 | |
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$ , find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$ . | 15 | 0 | 8,192 | -1 | 8,192 | |
Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{3}}{2}$, and it passes through point $A(2,1)$.
(Ⅰ) Find the equation of ellipse $C$;
(Ⅱ) If $P$, $Q$ are two points on ellipse $C$, and the angle bisector of $\angle PAQ$ always perpendicular to the x-ax... | \dfrac{1}{2} | 0 | 7,904.625 | -1 | 7,904.625 | |
Given that the point \( P(x, y) \) satisfies the equation \( (x-4 \cos \theta)^{2}+(y-4 \sin \theta)^{2}=4(\theta \in \mathbf{R}) \), find the area of the region where the point \( P(x, y) \) can be located. | 32\pi | 0.8125 | 4,608.875 | 3,794.461538 | 8,138 | |
Let $a, b$, and $c$ be real numbers. Consider the system of simultaneous equations in variables $x$ and $y:$ $a x+b y =c-1$ and $(a+5) x+(b+3) y =c+1$. Determine the value(s) of $c$ in terms of $a$ such that the system always has a solution for any $a$ and $b$. | 2a/5 + 1 \text{ or } \frac{2a+5}{5} | We have to only consider when the determinant of $\begin{pmatrix}a & b \\ a+5 & b+3\end{pmatrix}$ is zero. That is, when $b=3 a / 5$. Plugging in $b=3 a / 5$, we find that $(a+5)(c-1)=a(c+1)$ or that $c=2 a / 5+1$. | 0 | 6,526.125 | -1 | 6,526.125 |
How many multiples of 5 are between 100 and 400? | 59 | 0.625 | 5,169.9375 | 4,737.8 | 5,890.166667 | |
When the repeating decimal $0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is: | 15 | 1. Let $x = 0.\overline{36}$, which means $x$ is the repeating decimal $0.363636\ldots$.
2. To eliminate the repeating part, multiply $x$ by 100 (since the repeating part, 36, has two digits):
\[
100x = 36.363636\ldots
\]
3. Subtract the original $x$ from $100x$ to isolate the repeating decimal:
\[
100... | 1 | 1,263.1875 | 1,263.1875 | -1 |
Engineers use the formula $L=\frac{25T^4}{H^2}$ to find the crushing load for square pillars. If $T = 4$ and $H = 8$, what is the value of $L$? | 100 | 1 | 1,840.625 | 1,840.625 | -1 | |
Let \( a_{1}, a_{2}, \cdots, a_{105} \) be a permutation of \( 1, 2, \cdots, 105 \), satisfying the condition that for any \( m \in \{3, 5, 7\} \), for all \( n \) such that \( 1 \leqslant n < n+m \leqslant 105 \), we have \( m \mid (a_{n+m}-a_{n}) \). How many such distinct permutations exist? (Provide the answer as a... | 3628800 | 0.0625 | 7,658.5 | 7,414 | 7,674.8 | |
Given that $\overrightarrow{OA}=(1,0)$, $\overrightarrow{OB}=(1,1)$, and $(x,y)=λ \overrightarrow{OA}+μ \overrightarrow{OB}$, if $0\leqslant λ\leqslant 1\leqslant μ\leqslant 2$, then the maximum value of $z= \frac {x}{m}+ \frac{y}{n}(m > 0,n > 0)$ is $2$. Find the minimum value of $m+n$. | \frac{5}{2}+ \sqrt{6} | 0 | 7,323.6875 | -1 | 7,323.6875 | |
The diagram below shows an isosceles triangle with base $21$ and height $28$ . Inscribed in the triangle is a square. Find the area of the shaded region inside the triangle and outside of the square.
[asy]
size(170);
defaultpen(linewidth(0.8));
draw((0,0)--(1,1));
pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-... | 150 | 0.875 | 5,446.25 | 5,054 | 8,192 | |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, and their angle is $120^{\circ}$, calculate the magnitude of the vector $|\overrightarrow{a}-2\overrightarrow{b}|$. | \sqrt{7} | 0.9375 | 3,042.125 | 2,698.8 | 8,192 | |
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ | 100 | 1. **Identify the range and conditions**: We are looking for 4-digit integers between 1000 and 9999, inclusive, that have only even digits and are divisible by 5.
2. **Digits must be even**: The possible even digits are 0, 2, 4, 6, and 8.
3. **Divisibility by 5**: For a number to be divisible by 5, its units digit mu... | 1 | 1,836.1875 | 1,836.1875 | -1 |
The lengths of the three sides of a triangle are $7$, $x+4$ and $2x+1$. The perimeter of the triangle is 36. What is the length of the longest side of the triangle? | 17 | 1 | 1,403.9375 | 1,403.9375 | -1 | |
A contest has six problems worth seven points each. On any given problem, a contestant can score either 0,1 , or 7 points. How many possible total scores can a contestant achieve over all six problems? | 28 | For $0 \leq k \leq 6$, to obtain a score that is $k(\bmod 6)$ exactly $k$ problems must get a score of 1 . The remaining $6-k$ problems can generate any multiple of 7 from 0 to $7(6-k)$, of which there are $7-k$. So the total number of possible scores is $\sum_{k=0}^{6}(7-k)=28$. | 0.125 | 8,058.375 | 7,123 | 8,192 |
In how many ways can 9 identical items be distributed into four boxes? | 220 | 1 | 2,445.3125 | 2,445.3125 | -1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.