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 |
|---|---|---|---|---|---|---|
Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice. | 800 | 0.1875 | 6,545.0625 | 2,849.333333 | 7,397.923077 | |
Find the radius of the circle inscribed in triangle $PQR$ if $PQ = 26$, $PR=10$, and $QR=18$. Express your answer in simplest radical form. | \sqrt{17} | 0 | 4,192.4375 | -1 | 4,192.4375 | |
The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer? | 6 | Suppose that the original integer has tens digit $a$ and ones (units) digit $b$. This integer is equal to $10 a+b$. When the digits are reversed, the tens digit of the new integer is $b$ and the ones digit is $a$. This new integer is equal to $10 b+a$. Since the new two-digit integer minus the original integer is 54, t... | 1 | 1,614.5625 | 1,614.5625 | -1 |
Determine how many hours it will take Carl to mow the lawn, given that the lawn measures 120 feet by 100 feet, the mower's swath is 30 inches wide with an overlap of 6 inches, and Carl walks at a rate of 4000 feet per hour. | 1.5 | 0.25 | 5,849 | 6,248.25 | 5,715.916667 | |
Massachusetts Avenue is ten blocks long. One boy and one girl live on each block. They want to form friendships such that each boy is friends with exactly one girl and vice versa. Nobody wants a friend living more than one block away (but they may be on the same block). How many pairings are possible? | 89 | 89 Let $a_{n}$ be the number of pairings if there are $n$ blocks; we have $a_{1}=$ $1, a_{2}=2$, and we claim the Fibonacci recurrence is satisfied. Indeed, if there are $n$ blocks, either the boy on block 1 is friends with the girl on block 1, leaving $a_{n-1}$ possible pairings for the people on the remaining $n-1$ b... | 0.25 | 6,930.125 | 5,521.5 | 7,399.666667 |
If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which $343x^3+64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$? | 3506 | 0.125 | 7,025.75 | 6,540.5 | 7,095.071429 | |
In regular hexagon $ABCDEF$, diagonal $AD$ is drawn. Given that each interior angle of a regular hexagon measures 120 degrees, calculate the measure of angle $DAB$. | 30 | 0 | 5,547.0625 | -1 | 5,547.0625 | |
If Anna flips 8 coins, what is the probability that she gets more heads than tails? | \dfrac{93}{256} | 1 | 3,608 | 3,608 | -1 | |
The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail... | 12.8 | 0 | 5,610.8125 | -1 | 5,610.8125 | |
The set \( S \) is given by \( S = \{1, 2, 3, 4, 5, 6\} \). A non-empty subset \( T \) of \( S \) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \( T \)? | 27 | 0 | 8,135.75 | -1 | 8,135.75 | |
Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$, $b = 4$, and an eccentricity of $\frac{3}{5}$. A line passing through $F\_1$ intersects the ellipse at points $A$ and $B$. Find the perimeter of $\triangle ABF\_2$. | 20 | 0.9375 | 4,687.8125 | 4,454.2 | 8,192 | |
Given the following matrix $$ \begin{pmatrix}
11& 17 & 25& 19& 16
24 &10 &13 & 15&3
12 &5 &14& 2&18
23 &4 &1 &8 &22
6&20&7 &21&9
\end{pmatrix}, $$ choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible. | 17 | 0.0625 | 8,135.9375 | 7,295 | 8,192 | |
Consider the function defined piecewise by
\[ f(x) = \left\{ \begin{aligned}
2x + 1 & \quad \text{if } x < 1 \\
x^2 & \quad \text{if } x \ge 1
\end{aligned} \right. \]
Determine the value of \( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \). | 1 + \sqrt{3} | 0.5 | 6,083.9375 | 5,103.125 | 7,064.75 | |
The shortest distance for an ant to crawl along the surface of a rectangular box with length and width both being $6 \mathrm{~cm}$ from vertex $A$ to vertex $B$ is $20 \mathrm{~cm}$. What is the volume of this rectangular box in $\mathrm{cm}^{3}$? | 576 | 0.3125 | 7,359 | 5,638 | 8,141.272727 | |
Let $ABCD$ be a square and $P$ be a point on the shorter arc $AB$ of the circumcircle of the square. Which values can the expression $\frac{AP+BP}{CP+DP}$ take? | \sqrt{2} - 1 | 0.0625 | 8,192 | 8,192 | 8,192 | |
How many positive divisors does 6! have? | 30 | 1 | 2,108.0625 | 2,108.0625 | -1 | |
At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many ha... | 441 | 1. **Count the total number of twins and triplets:**
- There are $9$ sets of twins, with $2$ twins per set, giving a total of $9 \times 2 = 18$ twins.
- There are $6$ sets of triplets, with $3$ triplets per set, giving a total of $6 \times 3 = 18$ triplets.
2. **Calculate handshakes among twins:**
- Each twin... | 0.0625 | 7,631.5625 | 4,245 | 7,857.333333 |
Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$ Find
\[\frac{a}{p} + \frac{b}{q}... | 2 | 0.3125 | 3,098.125 | 2,226.6 | 3,494.272727 | |
The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$. Find $m+n+r$. | 200 | Notice the original expression can be written as $2000x^6+100x^5-200x^4+200x^4+10x^3-20x^2+20x^2+x-2$.
Which equals to $(20x^2+x-2)(100x^4+10x^2+1)=0$
So our solution is to find what is the root for $20x^2+x-2=0$ since the determinant of $100t^2+10t+1<0$(Let $x^2=t$)
By solving the equation, we can get that $x = \fr... | 0.0625 | 8,129.125 | 7,186 | 8,192 |
Given that the sequence ${a_n}$ is an arithmetic sequence with first term $1$ and common difference $2$,
(1) Find the general term formula for ${a_n}$;
(2) Let $b_n = \frac{1}{a_n \cdot a_{n-1}}$. Denote the sum of the first $n$ terms of the sequence ${b_n}$ as $T_n$. Find the minimum value of $T_n$. | \frac{1}{3} | 0.125 | 7,448.9375 | 4,681 | 7,844.357143 | |
Find a positive integer that is divisible by 21 and has a square root between 30 and 30.5. | 903 | 0.375 | 3,115.4375 | 2,828.166667 | 3,287.8 | |
Compute $\dbinom{9}{8}$. | 9 | 1 | 1,893.75 | 1,893.75 | -1 | |
In Mr. Smith's class, the ratio of boys to girls is 3 boys for every 4 girls and there are 42 students in his class, calculate the percentage of students that are boys. | 42.857\% | 0 | 482.8125 | -1 | 482.8125 | |
In triangle \(ABC\), a point \(D\) is marked on side \(AC\) such that \(BC = CD\). Find \(AD\) if it is known that \(BD = 13\) and angle \(CAB\) is three times smaller than angle \(CBA\). | 13 | 0.3125 | 7,328 | 5,427.2 | 8,192 | |
Randomly color the four vertices of a tetrahedron with two colors, red and yellow. The probability that "three vertices on the same face are of the same color" is ______. | \dfrac{5}{8} | 0.25 | 7,982.75 | 7,355 | 8,192 | |
Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\cdots=x_{1500}+3000=\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{1500}x_n$. | 1500 | 0 | 5,877.625 | -1 | 5,877.625 | |
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitti... | 12 | 0 | 8,192 | -1 | 8,192 | |
In triangle $PQR$, $PQ = 12$, $QR = 16$, and $PR = 20$. Point $X$ is on $\overline{PQ}$, $Y$ is on $\overline{QR}$, and $Z$ is on $\overline{PR}$. Let $PX = u \cdot PQ$, $QY = v \cdot QR$, and $RZ = w \cdot PR$, where $u$, $v$, and $w$ are positive and satisfy $u+v+w=3/4$ and $u^2+v^2+w^2=1/2$. The ratio of the area of... | 41 | 0.4375 | 7,335.875 | 6,235.142857 | 8,192 | |
In the quadrilateral pyramid $S-ABCD$ with a right trapezoid as its base, where $\angle ABC = 90^\circ$, $SA \perp$ plane $ABCD$, $SA = AB = BC = 1$, and $AD = \frac{1}{2}$, find the tangent of the angle between plane $SCD$ and plane $SBA$. | \frac{\sqrt{2}}{2} | 0 | 7,475.375 | -1 | 7,475.375 | |
Given a function $f(x)$ that satisfies $f(x+3)=-f(x)$, when $x\in \left[-3,0\right)$, $f(x)=2^{x}+\sin \frac{πx}{3}$, determine the value of $f(2023)$. | -\frac{1}{4} + \frac{\sqrt{3}}{2} | 0 | 4,079.375 | -1 | 4,079.375 | |
In $\triangle XYZ$, the medians $\overline{XU}$ and $\overline{YV}$ intersect at right angles. If $XU = 18$ and $YV = 24$, find the area of $\triangle XYZ$. | 288 | 0.875 | 5,488.0625 | 5,101.785714 | 8,192 | |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | \text{D} | 0 | 7,478.0625 | -1 | 7,478.0625 | |
An electric company installed a total of 402 poles along both sides of the road, with a distance of 20 meters between each adjacent pole. Later, all poles were replaced, and only 202 poles were installed. The distance between each adjacent pole after the replacement is $\qquad$ meters. | 40 | 0.5 | 669 | 801.875 | 536.125 | |
The number of students in Teresa's graduating class is more than 50 and fewer than 100 and is 1 less than a multiple of 3, 2 less than a multiple of 4, and 3 less than a multiple of 5. How many students are in Teresa's graduating class? | 62 | 0.875 | 2,736.1875 | 2,592.071429 | 3,745 | |
Given two unit vectors $a$ and $b$, and $|a-2b| = \sqrt{7}$, then the angle between $a$ and $b$ is ______. | \frac{2\pi}{3} | 0.25 | 1,694.5 | 1,964.5 | 1,604.5 | |
A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM? | 5/16 | 0.125 | 7,385.0625 | 6,358.5 | 7,531.714286 | |
Find the last three digits of \(1 \times 3 \times 5 \times \cdots \times 1997\). | 375 | 0.5625 | 7,137.875 | 6,663.111111 | 7,748.285714 | |
Point $P$ lies on the line $x= -3$ and is 10 units from the point $(5,2)$. Find the product of all possible $y$-coordinates that satisfy the given conditions. | -32 | 1 | 1,738 | 1,738 | -1 | |
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor... | 929 | 1. **Define Variables**:
Let $w = \lfloor x \rfloor$ and $f = \{x\}$ denote the whole part and the fractional part of $x$, respectively. Thus, $x = w + f$ where $0 \leq f < 1$.
2. **Rewrite the Equation**:
The given equation is:
\[
\lfloor x \rfloor \cdot \{x\} = a \cdot x^2
\]
Substituting $w$ and $... | 0 | 8,192 | -1 | 8,192 |
How many binary strings of length $10$ do not contain the substrings $101$ or $010$ ? | 178 | 0 | 8,192 | -1 | 8,192 | |
Compute $$\sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)}$$ | \frac{137}{24}-8 \ln 2 | First, we manipulate using partial fractions and telescoping: $$\begin{aligned} \sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)} & =\frac{1}{2} \cdot \sum_{n=1}^{\infty} \frac{1}{2^{n}}\left(\frac{2}{n+1}-\frac{1}{n+2}-\frac{1}{n+4}\right) \\ & =\frac{1}{4}-\frac{1}{2} \sum_{n=1}^{\infty} ... | 0.0625 | 7,807.0625 | 6,630 | 7,885.533333 |
Food safety issues are increasingly attracting people's attention. The abuse of pesticides and chemical fertilizers poses certain health risks to the public. To provide consumers with safe vegetables, a rural cooperative invests 2 million yuan each year to build two pollution-free vegetable greenhouses, A and B. Each g... | 282 | 0 | 6,586 | -1 | 6,586 | |
Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$\begin{aligned} & 3a+5b \equiv 19 \quad(\bmod n+1) \\ & 4a+2b \equiv 25 \quad(\bmod n+1) \end{aligned}$$ Find $2a+6b$. | 96 | Let $m=n+1$, so that the conditions become $$\begin{align*} & 3a+5b \equiv 19 \quad(\bmod m) \tag{1}\\ & 4a+2b \equiv 25 \quad(\bmod m) \tag{2}\\ & 2a+6b \equiv-1 \quad(\bmod m) \tag{3} \end{align*}$$ We can subtract (2) from twice (3) to obtain $$10b \equiv-27 \quad(\bmod m)$$ Multiplying (1) by 2 and replacing $10b... | 0.0625 | 7,770.875 | 5,041 | 7,952.866667 |
There are 6 seats in a row and 3 people taking their seats, find the number of different ways of seating them such that there are exactly two adjacent empty seats. | 72 | 0.5 | 7,456.5625 | 6,721.125 | 8,192 | |
A right circular cone is cut into five pieces by four planes parallel to its base, each piece having equal height. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece. | \frac{37}{61} | 0.3125 | 7,461.4375 | 5,854.2 | 8,192 | |
Climbing the first flight of stairs takes Jimmy 20 seconds, and each following flight takes 5 seconds more than the preceding one. How many total seconds does it take to climb the first five flights of stairs? | 150 | 1 | 2,434.875 | 2,434.875 | -1 | |
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \] | (-1,1)(3,3)(19,-1)(53, -17) |
To solve the integer equation
\[
x^2 + xy + y^2 = \left(\frac{x+y}{3} + 1\right)^3,
\]
let us explore potential integer solutions by breaking down the equation and using substitution techniques for easier handling.
### Step 1: Simplify the Equation
The right-hand side of the equation contains a cubic term involvi... | 0 | 8,192 | -1 | 8,192 |
There are 4 college entrance examination candidates entering the school through 2 different intelligent security gates. Each security gate can only allow 1 person to pass at a time. It is required that each security gate must have someone passing through. Then there are ______ different ways for the candidates to enter... | 72 | 0 | 5,655.8125 | -1 | 5,655.8125 | |
For positive integers $i = 2, 3, \ldots, 2020$ , let \[
a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}.
\]Let $x_2$ , $\ldots$ , $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$ . Let $S$ be the maximum possible value of \[
\sum_{i=2}^{2020} a_i x_i (\sqr... | 47 | 0 | 8,192 | -1 | 8,192 | |
There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. Then, the ratio of the surface areas of these three spheres is ______. | 1:2:3 | 0.875 | 3,513 | 3,247.285714 | 5,373 | |
How many even perfect square factors does $2^4 \cdot 7^9$ have? | 10 | 1 | 2,589.75 | 2,589.75 | -1 | |
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is th... | 12 | 1. **Understanding the Problem:**
We are given a line of balls consisting of one red ball and $N$ green balls, where $N$ is a multiple of 5. We need to find the probability $P(N)$ that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball.
2. **Initial Observations:**
- For $N=5$, all gr... | 0.3125 | 7,392.0625 | 5,632.2 | 8,192 |
Given $\alpha \in \left(0,\pi \right)$, $sin\alpha+cos\alpha=\frac{\sqrt{3}}{3}$, find $\cos 2\alpha$. | -\frac{\sqrt{5}}{3} | 0 | 6,290.5625 | -1 | 6,290.5625 | |
Given that $α$ and $β$ are acute angles, and it is given that $\sin \alpha =\frac{3}{5}$ and $\cos (\alpha +\beta )=\frac{5}{13}$, calculate the value of $\cos \beta$. | \frac{56}{65} | 0.375 | 6,641.3125 | 4,056.833333 | 8,192 | |
What is the intersection of the lines given by $y=-4x$ and $y-2=12x$? Express your answer as an ordered pair, with both coordinates expressed as common fractions. | \left(-\frac{1}{8}, \frac{1}{2}\right) | 1 | 1,948.5 | 1,948.5 | -1 | |
If the matrix $\mathbf{A}$ has an inverse and $(\mathbf{A} - 2 \mathbf{I})(\mathbf{A} - 4 \mathbf{I}) = \mathbf{0},$ then find
\[\mathbf{A} + 8 \mathbf{A}^{-1}.\] | \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} | 0 | 2,464.3125 | -1 | 2,464.3125 | |
Using three different weights of 1 gram, 3 grams, and 9 grams, various weights of objects can be measured. Assuming the objects to be measured and the known weights can be placed on either side of the balance scale, how many different weights of objects can be measured? | 13 | 0.4375 | 7,389.625 | 6,358 | 8,192 | |
Jessica is tasked with placing four identical, dotless dominoes on a 4 by 5 grid to form a continuous path from the upper left-hand corner \(C\) to the lower right-hand corner \(D\). The dominoes are shaded 1 by 2 rectangles that must touch each other at their sides, not just at the corners, and cannot be placed diagon... | 35 | 0 | 8,016.9375 | -1 | 8,016.9375 | |
A right circular cone has a base with radius $600$ and height $200\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}.$... | 625 | 0.6875 | 5,856.8125 | 5,387 | 6,890.4 | |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of
\[2ab \sqrt{2} + 2bc.\] | \sqrt{3} | 0.8125 | 6,907.3125 | 6,610.846154 | 8,192 | |
Suppose that $a,b,$ and $c$ are positive integers satisfying $(a+b+c)^3 - a^3 - b^3 - c^3 = 150$. Find $a+b+c$. | 6 | 0.875 | 5,541.625 | 5,163 | 8,192 | |
Find $A$ and $B$ such that
\[\frac{5x+2}{x^2-7x-30}=\frac{A}{x-10}+\frac{B}{x+3}.\]Write your answer in the form $(A,B)$. | (4,1) | 1 | 2,098.5625 | 2,098.5625 | -1 | |
Let $ABCD$ be a square with side length $2$ , and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle? | 4 - 2\sqrt{3} | 0.3125 | 7,809.25 | 6,967.2 | 8,192 | |
The median of a set of consecutive odd integers is 138. If the greatest integer in the set is 145, what is the smallest integer in the set? | 131 | 0.375 | 6,743.375 | 4,329 | 8,192 | |
If $x+y=9$ and $xy=10$, what is the value of $x^3+y^3$? | 459 | 1 | 2,653.8125 | 2,653.8125 | -1 | |
An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die? | 4.5 | 1 | 1,503.75 | 1,503.75 | -1 | |
Given the function $f(x)=x^{2}+ax+4$, if for any $x \in (0,2]$, $f(x) \leqslant 6$ always holds, then find the maximum value of the real number $a$. | -1 | 0.9375 | 6,162.3125 | 6,027 | 8,192 | |
When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder. | 96 | 0.6875 | 6,521.25 | 5,761.818182 | 8,192 | |
Given that \(ABCD\) is a square, points \(E\) and \(F\) lie on the side \(BC\) and \(CD\) respectively, such that \(BE = CF = \frac{1}{3} AB\). \(G\) is the intersection of \(BF\) and \(DE\). If
\[
\frac{\text{Area of } ABGD}{\text{Area of } ABCD} = \frac{m}{n}
\]
is in its lowest terms, find the value of \(m+n\). | 23 | 0.75 | 6,136.0625 | 5,450.75 | 8,192 | |
Let $ABCDEF$ be a regular hexagon. Let $G, H, I, J, K,$ and $L$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DE}, \overline{EF},$ and $\overline{FA},$ each exterior to the hexagon. What is the ratio of the area of hexagon $GHIJKL$ to the area ... | \frac{10}{3} | 0 | 8,192 | -1 | 8,192 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $b= \sqrt {2}$, $c=3$, $B+C=3A$.
(1) Find the length of side $a$;
(2) Find the value of $\sin (B+ \frac {3π}{4})$. | \frac{\sqrt{10}}{10} | 0 | 4,290.4375 | -1 | 4,290.4375 | |
Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occup... | 981 | 0.6875 | 4,545.875 | 2,888.545455 | 8,192 | |
Perform the calculations.
36×17+129
320×(300-294)
25×5×4
18.45-25.6-24.4. | -31.55 | 0.9375 | 591.9375 | 589 | 636 | |
Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?
*Roland Hablutzel, Venezuela*
<details><summary>Remark</summary>The original question was: Among the positive integers that can be expressed as the sum of 2004 consec... | 2005 * 2004 * 2005 | 0 | 6,587.5 | -1 | 6,587.5 | |
Find the phase shift, vertical shift, and the maximum and minimum values of the graph of \( y = 3\sin\left(3x - \frac{\pi}{4}\right) + 1 \). | -2 | 1 | 2,346 | 2,346 | -1 | |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0. | 2014 | 0 | 7,672.3125 | -1 | 7,672.3125 | |
Two boards, one 5 inches wide and the other 7 inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are later separated, what is the area of the unpainted region on the five-inch board? Assume the holes caused by the nails are negligible. | 35\sqrt{2} | 0.125 | 7,358.9375 | 7,327.5 | 7,363.428571 | |
In a box, there are red and black socks. If two socks are randomly taken from the box, the probability that both of them are red is $1/2$.
a) What is the minimum number of socks that can be in the box?
b) What is the minimum number of socks that can be in the box, given that the number of black socks is even? | 21 | 0.5 | 7,399.6875 | 6,607.375 | 8,192 | |
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$ ) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$ . | 200 | 0.125 | 7,325 | 5,234.5 | 7,623.642857 | |
Alison is eating 2401 grains of rice for lunch. She eats the rice in a very peculiar manner: every step, if she has only one grain of rice remaining, she eats it. Otherwise, she finds the smallest positive integer $d>1$ for which she can group the rice into equal groups of size $d$ with none left over. She then groups ... | 17 | Note that $2401=7^{4}$. Also, note that the operation is equivalent to replacing $n$ grains of rice with $n \cdot \frac{p-1}{p}$ grains of rice, where $p$ is the smallest prime factor of $n$. Now, suppose that at some moment Alison has $7^{k}$ grains of rice. After each of the next four steps, she will have $6 \cdot 7^... | 0.1875 | 7,769.625 | 5,939.333333 | 8,192 |
Let \( a \) be a positive integer that is a multiple of 5 such that \( a+1 \) is a multiple of 7, \( a+2 \) is a multiple of 9, and \( a+3 \) is a multiple of 11. Determine the smallest possible value of \( a \). | 1735 | 0.875 | 6,292.75 | 6,021.428571 | 8,192 | |
Let $\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$, respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $X$ varies along $\overline{BC}$. | 126 | First note that \[\angle I_1AI_2 = \angle I_1AX + \angle XAI_2 = \frac{\angle BAX}2 + \frac{\angle CAX}2 = \frac{\angle A}2\] is a constant not depending on $X$, so by $[AI_1I_2] = \tfrac12(AI_1)(AI_2)\sin\angle I_1AI_2$ it suffices to minimize $(AI_1)(AI_2)$. Let $a = BC$, $b = AC$, $c = AB$, and $\alpha = \angle AXB$... | 0 | 8,192 | -1 | 8,192 |
Let $L$ be the intersection point of the diagonals $CE$ and $DF$ of a regular hexagon $ABCDEF$ with side length 2. Point $K$ is defined such that $\overrightarrow{LK} = \overrightarrow{AC} - 3 \overrightarrow{BC}$. Determine whether point $K$ lies inside, on the boundary, or outside of $ABCDEF$, and find the length of ... | \frac{2\sqrt{3}}{3} | 0 | 5,091.625 | -1 | 5,091.625 | |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with its right focus $F$, draw a line perpendicular to the $x$-axis passing through $F$, intersecting the two asymptotes at points $A$ and $B$, and intersecting the hyperbola at point $P$ in the first quadrant. Denote $O$ as the origin. ... | \frac{2\sqrt{3}}{3} | 0 | 5,126.3125 | -1 | 5,126.3125 | |
Factor the expression $3x(x+1) + 7(x+1)$. | (3x+7)(x+1) | 0 | 1,747.0625 | -1 | 1,747.0625 | |
Alexio has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then randomly picks a card. What is the probability that the number on the card is a multiple of 2, 4, or 6? Express your answer as a common fraction. | \frac{1}{2} | 0.625 | 5,470.8125 | 5,282.5 | 5,784.666667 | |
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$? | 60 | 1. **Identify the properties of the lines**: Given that the lines are perpendicular and intersect at $A(6,8)$, we can denote the equations of the lines as $y = m_1x + b_1$ and $y = m_2x + b_2$. Since the lines are perpendicular, their slopes satisfy $m_1 \cdot m_2 = -1$.
2. **Determine the $y$-intercepts**: The proble... | 1 | 3,389.4375 | 3,389.4375 | -1 |
The pages of a book are numbered $1_{}^{}$ through $n_{}^{}$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986_{}^{}$. What was the number of the page that was added twice? | 33 | 1 | 3,398.625 | 3,398.625 | -1 | |
There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repeats until only one student remains. What wa... | 128 | 0.125 | 8,053.0625 | 7,080.5 | 8,192 | |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 152 | 0 | 7,701.25 | -1 | 7,701.25 | |
A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \[(-\tfrac32, 1), \; (0,0), \;(0,2),\; (3,0),\; (3,2).\]is an ellipse... | \frac{4\sqrt3}{3} | 0 | 5,998.3125 | -1 | 5,998.3125 | |
An academy has $200$ students and $8$ teachers. The class sizes are as follows: $80, 40, 40, 20, 10, 5, 3, 2$. Calculate the average number of students per class as seen by a randomly picked teacher, represented by $t$, and the average number of students per class from the perspective of a randomly selected student, de... | -25.69 | 0.5 | 6,378.8125 | 6,450.875 | 6,306.75 | |
In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, $a_{n+2}$ is equal to the remainder of $a_n + a_{n+1}$ divided by 3. Find the sum of the first 89 terms of $\{a_n\}$. | 100 | 0.375 | 6,524.5 | 3,745.333333 | 8,192 | |
Define the operation "□" as: $a□b=a^2+2ab-b^2$. Let the function $f(x)=x□2$, and the equation related to $x$ is $f(x)=\lg|x+2|$ ($x\neq -2$) has exactly four distinct real roots $x_1$, $x_2$, $x_3$, $x_4$. Find the value of $x_1+x_2+x_3+x_4$. | -8 | 0.625 | 6,570.8125 | 5,598.1 | 8,192 | |
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$? | 4 | 1. **Understanding the problem**: We start with a cube of side length $n$, which is painted on all six faces. This cube is then cut into $n^3$ smaller unit cubes. Each unit cube has 6 faces. We need to find $n$ such that exactly one-fourth of the total number of faces of all the unit cubes are red.
2. **Total number o... | 0.9375 | 2,702.5 | 2,336.533333 | 8,192 |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | -7 | 1 | 2,534.4375 | 2,534.4375 | -1 | |
The lengths of the three sides of $\triangle ABC$ are 5, 7, and 8, respectively. The radius of its circumcircle is ______, and the radius of its incircle is ______. | \sqrt{3} | 1 | 1,813.25 | 1,813.25 | -1 | |
Let $\alpha \in (\pi, 2\pi)$, if $\tan\left(\alpha + \frac{\pi}{6}\right) = 2$, then the value of $\cos\left(\frac{\pi}{6} - 2\alpha\right)$ is \_\_\_\_\_. | \frac{4}{5} | 0.5625 | 6,817.125 | 5,911.222222 | 7,981.857143 | |
The fraction \(\frac{1}{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\) where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$? | 883 | To solve the problem, we first need to understand the decimal expansion of $\frac{1}{99^2}$ and then find the sum of the digits in one period of the repeating decimal.
1. **Calculate $\frac{1}{99}$:**
\[
\frac{1}{99} = 0.\overline{01}
\]
This is because $99 \times 0.01 = 0.99$ and the next digit after shif... | 0 | 8,192 | -1 | 8,192 |
How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?
| 880 | 0.8125 | 5,541 | 4,929.230769 | 8,192 |
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