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 |
|---|---|---|---|---|---|---|
For how many integers $a$ with $|a| \leq 2005$ , does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions? | 90 | 0 | 8,192 | -1 | 8,192 | |
If a computer executes the following program:
1. Initial values \( x = 3 \), \( S = 0 \)
2. \( x = x + 2 \)
3. \( S = S + x \)
4. If \( S \geq 10000 \), go to step 5; otherwise, repeat from step 2
5. Print \( x \)
6. Stop
What value will be printed by step 5? | 201 | 0.4375 | 5,882 | 4,553.571429 | 6,915.222222 | |
Compute the number of even positive integers $n \leq 2024$ such that $1,2, \ldots, n$ can be split into $\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of 3. | 675 | There have to be an even number of multiples of 3 at most $n$, so this means that $n \equiv 0,2 \pmod{6}$. We claim that all these work. We know there are an even number of multiples of 3, so we can pair them; then we can pair $3k+1$ and $3k+2$ for all $k$. This means the answer is $\frac{2022}{3}+1=675$. | 0.125 | 8,029.5625 | 7,290.5 | 8,135.142857 |
Given a monotonically increasing sequence of positive integers $\left\{a_{n}\right\}$ that satisfies the recurrence relation $a_{n+2}=3 a_{n+1}-a_{n}$, with $a_{6}=280$, find the value of $a_{7}$. | 733 | 0.875 | 4,878.25 | 4,404.857143 | 8,192 | |
In right $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 12$, $BC = 35$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\o... | 11 | 0 | 8,192 | -1 | 8,192 | |
Find $x$, where $x$ is a square number which is divisible by four, and between 39 and 80. | 64 | 0.9375 | 1,261.1875 | 1,320.266667 | 375 | |
What is the units digit when $5^4$ is multiplied out? | 5 | 1 | 1,492.125 | 1,492.125 | -1 | |
The ancient Chinese mathematical classic "The Nine Chapters on the Mathematical Art" contains a problem called "Rice and Grain Separation". During the collection of grain in a granary, 1524 "shi" (a unit of weight) of rice was received, but it was found to contain grains of another type mixed in. A sample of rice was t... | 168 | 0.6875 | 5,222.3125 | 3,872.454545 | 8,192 | |
What is the value of $f(-1)$ if $f(x)=x^{2}-2x$? | 3 | 0.9375 | 398.625 | 405.866667 | 290 | |
The area of a square plot of land is 325 square meters. What is the perimeter of the square, in meters? Express your answer in simplest radical form. | 20\sqrt{13} | 1 | 1,044.375 | 1,044.375 | -1 | |
Fold a piece of graph paper once so that the point (0, 2) coincides with the point (4, 0), and the point (9, 5) coincides with the point (m, n). The value of m+n is ______. | 10 | 0.9375 | 3,237.1875 | 2,906.866667 | 8,192 | |
What is the greatest possible value of the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\), where \(a, b, c\) are distinct non-zero digits? | 1/203 | 0 | 8,192 | -1 | 8,192 | |
Simplify $1-(1+(1-(1+(1-x))))$. | 1-x | 0.625 | 5,241.125 | 3,470.6 | 8,192 | |
Walking is a form of exercise that falls between walking and racewalking. It is a simple and safe aerobic exercise that can enhance lung capacity and promote heart health. A sports physiologist conducted a large number of surveys on the body fat percentage ($X$) of people engaged in walking activities and found that th... | 0.03 | 0.875 | 4,162 | 4,058.642857 | 4,885.5 | |
A bag contains five identical balls except for the numbers on them, which are labeled from 1 to 5. Three balls are randomly drawn from this bag one at a time without replacement, and the highest number among the drawn balls is denoted as $X$. Calculate the probability $P(X=4)$. | 0.3 | 0 | 5,042.8125 | -1 | 5,042.8125 | |
A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? | $\frac{15}{8}$ | 1. **Identify the Triangle Type**: Given the side lengths $3, 4, 5$, we recognize $\triangle ABC$ as a right triangle (by the Pythagorean theorem, $3^2 + 4^2 = 5^2$).
2. **Folding Point A to Point B**: When the triangle is folded such that point $A$ falls on point $B$, the crease will be a line segment that is equidis... | 0 | 5,447.6875 | -1 | 5,447.6875 |
Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.
- At 12:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "82."
- At 13:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "71."
- At 15:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "46.... | 34 | 0 | 7,888.875 | -1 | 7,888.875 | |
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | 21 | 0.1875 | 7,792.875 | 6,063.333333 | 8,192 | |
Let \( a_1, a_2, \ldots, a_{2020} \) be the roots of the polynomial
\[ x^{2020} + x^{2019} + \cdots + x^2 + x - 2022 = 0. \]
Compute
\[ \sum_{n = 1}^{2020} \frac{1}{1 - a_n}. \] | 2041210 | 0 | 6,565.6875 | -1 | 6,565.6875 | |
Given the four digits 2, 4, 6, and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? | 12 | 1 | 1,857.5 | 1,857.5 | -1 | |
The cafeteria is tiled with the same floor tiles, and it takes 630 tiles to cover an area with 18 square decimeters of tiles. How many tiles will it take if we switch to square tiles with a side length of 6 decimeters? | 315 | 0.0625 | 498.8125 | 466 | 501 | |
Consider the given functions: $$\begin{array}{ccc}
f(x) & = & 5x^2 - \frac{1}{x}+ 3\\
g(x) & = & x^2-k
\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? | k = \frac{-33}{2} | 0 | 2,243.25 | -1 | 2,243.25 | |
Let $ABCD$ be a quadrilateral with side lengths $AB=2, BC=3, CD=5$, and $DA=4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$? | \frac{2\sqrt{30}}{7} | Let the tangent lengths be $a, b, c, d$ so that $a+b=2, b+c=3, c+d=5, d+a=4$. Then $b=2-a$ and $c=1+a$ and $d=4-a$. The radius of the inscribed circle of quadrilateral $ABCD$ is given by $\sqrt{\frac{abc+abd+acd+bcd}{a+b+c+d}}=\sqrt{\frac{-7a^{2}+16a+8}{7}}$. This is clearly maximized when $a=\frac{8}{7}$ which leads t... | 0 | 6,940.4375 | -1 | 6,940.4375 |
How many ways can you tile the white squares of the following \(2 \times 24\) grid with dominoes? (A domino covers two adjacent squares, and a tiling is a non-overlapping arrangement of dominoes that covers every white square and does not intersect any black square.) | 27 | 0 | 5,774.4375 | -1 | 5,774.4375 | |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in... | 3001 | 0 | 8,192 | -1 | 8,192 | |
Suppose that $3$ is a factor of $a$, $a$ is a divisor of $12$, and $a$ is positive. What is the number of possible values of $a$? | 3 | 1 | 1,751.5 | 1,751.5 | -1 | |
If $512^{x}=64^{240}$, what is the value of $x$? | 160 | We note that $64=2^{6}$ and $512=2^{9}$. Therefore, the equation $512^{x}=64^{240}$ can be rewritten as $(2^{9})^{x}=(2^{6})^{240}$ or $2^{9x}=2^{6(240)}$. Since the bases in this last equation are equal, then the exponents are equal, so $9x=6(240)$ or $x=\frac{1440}{9}=160$. | 1 | 1,620.5625 | 1,620.5625 | -1 |
In the rectangular coordinate system xOy, the parametric equation of curve C1 is given by $$\begin{cases} x=5cos\alpha \\ y=5+5sin\alpha \end{cases}$$ (where α is the parameter). Point M is a moving point on curve C1. When the line segment OM is rotated counterclockwise by 90° around point O, line segment ON is obtaine... | 15-5 \sqrt {3} | 0 | 7,311.25 | -1 | 7,311.25 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{55} | 1 | 4,043.75 | 4,043.75 | -1 | |
Let acute triangle $ABC$ have circumcenter $O$, and let $M$ be the midpoint of $BC$. Let $P$ be the unique point such that $\angle BAP=\angle CAM, \angle CAP=\angle BAM$, and $\angle APO=90^{\circ}$. If $AO=53, OM=28$, and $AM=75$, compute the perimeter of $\triangle BPC$. | 192 | The point $P$ has many well-known properties, including the property that $\angle BAP=\angle ACP$ and $\angle CAP=\angle BAP$. We prove this for completeness. Invert at $A$ with radius $\sqrt{AB \cdot AC}$ and reflect about the $A$-angle bisector. Let $P^{\prime}$ be the image of $P$. The angle conditions translate to ... | 0 | 8,192 | -1 | 8,192 |
A basketball team has 20 players. The coach needs to choose a starting lineup consisting of one point guard and 7 other players (these players are interchangeable). How many different starting lineups can the coach choose? | 1007760 | 0.75 | 5,133.4375 | 4,376.416667 | 7,404.5 | |
Calculate
\[\prod_{n = 1}^{13} \frac{n(n + 2)}{(n + 4)^2}.\] | \frac{3}{161840} | 0 | 8,192 | -1 | 8,192 | |
If the Cesaro sum of the 50-term sequence \((b_1,\dots,b_{50})\) is 500, what is the Cesaro sum of the 51-term sequence \((2,b_1,\dots,b_{50})\)? | 492 | 0 | 7,613.5 | -1 | 7,613.5 | |
Given that out of 6 products, 2 are defective and the rest are qualified, calculate the probability of selecting exactly one defective product from these 6 products. | \frac{8}{15} | 0.5 | 3,736.25 | 3,287.125 | 4,185.375 | |
Find the integer $n$ such that $-150 < n < 150$ and $\tan n^\circ = \tan 286^\circ$. | -74 | 0.5625 | 7,138.75 | 6,760.777778 | 7,624.714286 | |
Simplify $\dfrac{12}{11}\cdot\dfrac{15}{28}\cdot\dfrac{44}{45}$. | \frac{4}{7} | 0.8125 | 5,244.25 | 4,564 | 8,192 | |
A triangle with vertices at \((1003,0), (1004,3),\) and \((1005,1)\) in the \(xy\)-plane is revolved all the way around the \(y\)-axis. Find the volume of the solid thus obtained. | 5020 \pi | 0.0625 | 8,192 | 8,192 | 8,192 | |
Convert $1011001_2$ to base 4. | 1121_4 | 0.5625 | 6,070.0625 | 4,419.666667 | 8,192 | |
Determine how many prime dates occurred in a leap year, such as 2012, where both the month and the day are prime numbers. | 53 | 0 | 8,192 | -1 | 8,192 | |
In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY / YZ$, in this rectangl... | 5 | 1. **Identify the dimensions of the square and the pieces**: The square has a side length of $2$. Points $E$ and $F$ are midpoints of opposite sides, so $EF = 2$. Since $AG$ is perpendicular to $BF$, and $E$ and $F$ are midpoints, $AE = EB = 1$ and $BF = 2$.
2. **Calculate the length of $BF$ using the Pythagorean theo... | 0 | 7,718.625 | -1 | 7,718.625 |
A city does not have electric lighting yet, so candles are used in houses at night. In João's house, one candle is used per night without burning it completely, and with four of these candle stubs, João makes a new candle. How many nights can João light up his house with 43 candles? | 57 | 0.1875 | 7,843.1875 | 6,331.666667 | 8,192 | |
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\] | -7 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Solve for $n$: $2^n\cdot 4^n=64^{n-36}$. | 72 | 1 | 1,708.125 | 1,708.125 | -1 | |
Four mice: White, Gray, Fat, and Thin were dividing a cheese wheel. They cut it into 4 apparently identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's slice, and White's slice weighed 8 grams less than Gray's slice. However, White wasn't upset because his slice weighed exactly ... | 14 | 0 | 8,153.375 | -1 | 8,153.375 | |
Let $f(x)=x^2+3x+2$ and let $S$ be the set of integers $\{0, 1, 2, \dots , 25 \}$.
The number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by $6$ is: | 17 | 1. **Understanding the Polynomial Function**:
Given the polynomial function $f(x) = x^2 + 3x + 2$. We need to find the number of integers $s$ in the set $S = \{0, 1, 2, \dots, 25\}$ such that $f(s)$ is divisible by $6$.
2. **Periodicity of Polynomial Functions Modulo $6$**:
We observe that for any polynomial $f... | 0.9375 | 4,213.1875 | 3,947.933333 | 8,192 |
If \( p = \frac{21^{3}-11^{3}}{21^{2}+21 \times 11+11^{2}} \), find \( p \).
If \( p \) men can do a job in 6 days and 4 men can do the same job in \( q \) days, find \( q \).
If the \( q \)-th day of March in a year is Wednesday and the \( r \)-th day of March in the same year is Friday, where \( 18 < r < 26 \), fin... | 27 | 0 | 7,252.6875 | -1 | 7,252.6875 | |
In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid, one in each quadrant. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$? | 180 | 0.75 | 2,857.9375 | 2,927.75 | 2,648.5 | |
If infinitely many values of $y$ satisfy the equation $2(4+cy) = 12y+8$, then what is the value of $c$? | 6 | 1 | 1,184.1875 | 1,184.1875 | -1 | |
Add 47.2189 to 34.0076 and round to the nearest hundredth. | 81.23 | 1 | 419.0625 | 419.0625 | -1 | |
A farmer has a rectangular field with dimensions $2m+7$ and $m-2$. If the field has an area of 51 square units, what is the value of $m$? | 5 | 1 | 1,832.75 | 1,832.75 | -1 | |
There are 11 seats, and now we need to arrange for 2 people to sit down. It is stipulated that the middle seat (the 6th seat) cannot be occupied, and the two people must not sit next to each other. How many different seating arrangements are there? | 84 | 0 | 7,979.125 | -1 | 7,979.125 | |
Find the minimum value of
\[\sqrt{x^2 + (x-2)^2} + \sqrt{(x-2)^2 + (x+2)^2}\] over all real numbers $x$. | 2\sqrt{5} | 0.625 | 6,900.8125 | 6,305.4 | 7,893.166667 | |
Twenty percent less than 60 is one-third more than what number? | 36 | 1. **Calculate 20% less than 60**:
To find 20% less than a number, we subtract 20% of the number from itself.
\[
60 - 0.20 \times 60 = 60 - 12 = 48
\]
Alternatively, we can calculate this as 80% of 60:
\[
0.80 \times 60 = 48
\]
So, twenty percent less than 60 is 48.
2. **Set up the equatio... | 1 | 1,603.4375 | 1,603.4375 | -1 |
A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents? | 105 | 1 | 1,604 | 1,604 | -1 | |
A student, Theo, needs to earn a total of 30 homework points. For the first six homework points, he has to do one assignment each; for the next six points, he needs to do two assignments each; and so on, such that for every subsequent set of six points, the number of assignments he needs to complete doubles the previou... | 186 | 0.0625 | 7,757.3125 | 6,245 | 7,858.133333 | |
What is the positive difference between $\frac{6^2 + 6^2}{6}$ and $\frac{6^2 \times 6^2}{6}$? | 204 | 1 | 1,299.3125 | 1,299.3125 | -1 | |
Given the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $|\varphi| < \frac{\pi}{2}$) has a minimum positive period of $\pi$, and its graph is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function $g(x) = \sin(\omega x)$, determine the value of $\varphi$. | \frac{\pi}{3} | 1 | 3,544.1875 | 3,544.1875 | -1 | |
$A, B, C, D$ attended a meeting, and each of them received the same positive integer. Each person made three statements about this integer, with at least one statement being true and at least one being false. Their statements are as follows:
$A:\left(A_{1}\right)$ The number is less than 12;
$\left(A_{2}\right)$ 7 can... | 89 | 0 | 8,192 | -1 | 8,192 | |
When $0.73\overline{864}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$? | 737910 | 0 | 7,933.75 | -1 | 7,933.75 | |
An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? | \frac{243}{1024} | 1. **Understanding the Problem**: We start with an equilateral triangle that is entirely black. Each time a change occurs, the middle fourth of each black triangle turns white. We need to determine the fraction of the original triangle that remains black after five changes.
2. **Analyzing the Change Process**: Each bl... | 1 | 3,001.5625 | 3,001.5625 | -1 |
In a Cartesian coordinate system, the "rectangular distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1}-x_{2}\right| + \left|y_{1}-y_{2}\right|$. If the "rectangular distance" from point $C(x, y)$ to points $A(1,3)$ and $B(6,9)$ is equal, where real... | 5(\sqrt{2} + 1) | 0 | 7,739.625 | -1 | 7,739.625 | |
Three faces $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ of a unit cube share a common vertex. Suppose the projections of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ onto a fixed plane $\mathcal{P}$ have areas $x, y, z$, respectively. If $x: y: z=6: 10: 15$, then $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positiv... | 3119 | Introduce coordinates so that $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ are normal to $(1,0,0),(0,1,0)$, and $(0,0,1)$, respectively. Also, suppose that $\mathcal{P}$ is normal to unit vector $(\alpha, \beta, \gamma)$ with $\alpha, \beta, \gamma \geq 0$. Since the area of $\mathcal{X}$ is 1, the area of its projection is... | 0.875 | 4,852.6875 | 4,375.642857 | 8,192 |
What number corresponds to the point \( P \) indicated on the given scale? | 12.50 | 0 | 7,031.25 | -1 | 7,031.25 | |
The sequence $(a_n)$ satisfies
\[a_1 + a_2 + a_3 + \dots + a_n = n^2 a_n\]for all $n \ge 2.$ If $a_{63} = 1,$ find $a_1.$ | 2016 | 0.9375 | 4,438.1875 | 4,253.133333 | 7,214 | |
$ABC$ is an isosceles triangle with base $AC$. $CD$ is the bisector of angle $C$, and $\angle ADC = 150^\circ$. Find $\angle B$. | 140 | 0.4375 | 7,034.3125 | 5,984 | 7,851.222222 | |
If $x$ is a positive integer such that $1^{x+2} + 2^{x+1} + 3^{x-1} + 4^x = 1170$, what is the value of $x$? | 5 | 1 | 1,953.5625 | 1,953.5625 | -1 | |
The graph of the function $y=\sin(\omega x+ \frac {5\pi}{6})$ where $0<\omega<\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \frac {1}{2})$ and $( \frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-axis. | -1 | 0.1875 | 7,545.1875 | 4,987.333333 | 8,135.461538 | |
Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$. | 96 | 0.8125 | 5,465.4375 | 4,836.230769 | 8,192 | |
In equilateral triangle $ABC,$ let points $D$ and $E$ trisect $\overline{BC}$. Find $\sin \angle DAE.$ | \frac{3 \sqrt{3}}{14} | 0 | 3,846.875 | -1 | 3,846.875 | |
How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$ ? | 42 | 0 | 8,192 | -1 | 8,192 | |
Points $\boldsymbol{A}$ and $\boldsymbol{B}$ are located on a straight highway running from west to east. Point B is 9 km east of A. A car departs from point A heading east at a speed of 40 km/h. Simultaneously, from point B, a motorcycle starts traveling in the same direction with a constant acceleration of 32 km/h². ... | 16 | 0.5 | 6,884.5625 | 6,063.625 | 7,705.5 | |
Simplify first, then evaluate.
$(1) 3x^{3}-[x^{3}+(6x^{2}-7x)]-2(x^{3}-3x^{2}-4x)$, where $x=-1$;
$(2) 2(ab^{2}-2a^{2}b)-3(ab^{2}-a^{2}b)+(2ab^{2}-2a^{2}b)$, where $a=2$, $b=1$. | -10 | 1 | 2,857.125 | 2,857.125 | -1 | |
If $5a+2b=0$ and $a$ is two less than $b$, what is $7b$? | 10 | 1 | 1,660.5625 | 1,660.5625 | -1 | |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]*Proposed by Evan Chen* | 365 | 0.5 | 7,261.4375 | 6,470.875 | 8,052 | |
If the function $f(x)$ satisfies $f(3x) = f\left(3x - \frac{3}{2}\right)$ for all $x \in \mathbb{R}$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_. | \frac{1}{2} | 0 | 6,754.375 | -1 | 6,754.375 | |
Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability $\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes... | 167 | 0.8125 | 4,179.5625 | 4,041.615385 | 4,777.333333 | |
Given a right triangle \( ABC \) with leg lengths equal to 1, a point \( P \) is chosen on one of the sides of the triangle. Find the maximum value of \( PA \cdot PB \cdot PC \). | \frac{\sqrt{2}}{4} | 0 | 8,192 | -1 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = 1$, $| \overrightarrow{b}| = 6$, and $\overrightarrow{a} \cdot (\overrightarrow{b} - \overrightarrow{a}) = 2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{\pi}{3} | 0.25 | 1,745.5625 | 2,427.75 | 1,518.166667 | |
Compute $1011_2 + 101_2 - 1100_2 + 1101_2$. Express your answer in base 2. | 10001_2 | 0.4375 | 7,029.5 | 5,534.857143 | 8,192 | |
Person A can only be in the first or second position, and person B can only be in the second or third position. Find the total number of different possible arrangements of five people in a row. | 18 | 0.25 | 7,376.4375 | 5,848.75 | 7,885.666667 | |
Points $ K$ , $ L$ , $ M$ , and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$ , $ BLC$ , $ CMD$ , and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$ , find the area of $ KLMN$. | 32 + 16\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Let $S$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{2}$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$. | \sqrt{2} | 0.6875 | 5,132.8125 | 4,684.727273 | 6,118.6 | |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
| 913 | 0 | 7,791.6875 | -1 | 7,791.6875 | |
Let \[f(x) = \left\{
\begin{array}{cl}
2x + 7 & \text{if } x < -2, \\
-x^2 - x + 1 & \text{if } x \ge -2.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = -5.$ | -4 | 1 | 1,729.9375 | 1,729.9375 | -1 | |
If non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}| = |\overrightarrow{b}|$ and $(\sqrt{3}\overrightarrow{a} - 2\overrightarrow{b}) \cdot \overrightarrow{a} = 0$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________. | \frac{\pi}{6} | 0.625 | 2,470 | 2,612.7 | 2,232.166667 | |
There are two types of tables in a restaurant: a square table can seat 4 people, and a round table can seat 9 people. The restaurant manager calls a number a "wealth number" if the total number of diners can exactly fill a certain number of tables. How many "wealth numbers" are there among the numbers from 1 to 100? | 88 | 0.0625 | 7,568.8125 | 5,339 | 7,717.466667 | |
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$
and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area.
(Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$
The ratio $\frac{AD}{AB}$ is | \frac{1}{\sqrt[4]{12}} | 1. **Identify the position of $E$**: We start by assuming $E = C$ and analyze the consequences. Drop an altitude from $C$ to $AB$, meeting $AB$ at $F$. Assume $CF = 1$ for simplicity.
2. **Calculate the ratio of areas assuming $E = C$**: The area of $\triangle ABC$ is proportional to the base $AB$ and the height $CF$... | 0 | 8,192 | -1 | 8,192 |
Fully factor the following expression: $2x^2-8$ | (2) (x+2) (x-2) | 0 | 1,442.125 | -1 | 1,442.125 | |
Private Petrov took a bucket of unpeeled potatoes and peeled them in 1 hour. During this process, 25% of the potatoes went to peels. How long did it take him to collect half a bucket of peeled potatoes? | 40 | 0.25 | 1,527.6875 | 592 | 1,839.583333 | |
Translate the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$ to the left by $\varphi$ ($\varphi > 0$) units. If the resulting graph is symmetric about the y-axis, then the minimum value of $\varphi$ is \_\_\_\_\_. | \frac{\pi}{12} | 0.625 | 6,178.5 | 4,970.4 | 8,192 | |
What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, -1, 3, 5, 20\}$, where one of the numbers must be negative? | -0.05 | 0.0625 | 6,443.9375 | 8,192 | 6,327.4 | |
Given a triangle $\triangle ABC$ whose side lengths form an arithmetic sequence with a common difference of $2$, and the sine of its largest angle is $\frac{\sqrt{3}}{2}$, find the perimeter of this triangle. | 15 | 1 | 2,988.8125 | 2,988.8125 | -1 | |
Given that there are 6 balls of each of the four colors: red, blue, yellow, and green, each numbered from 1 to 6, calculate the number of ways to select 3 balls with distinct numbers, such that no two balls have the same color or consecutive numbers. | 96 | 0.625 | 6,995.9375 | 6,278.3 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\sin C + \sin(B - A) = \sqrt{2} \sin 2A$, and $A \neq \frac{\pi}{2}$.
(I) Find the range of values for angle $A$;
(II) If $a = 1$, the area of $\triangle ABC$ is $S = \frac{\sqrt{3} + 1}{4}$, and $C$ is an... | \frac{\pi}{6} | 0.375 | 7,638.5 | 6,716 | 8,192 | |
Factor the expression $x(x+2)+(x+2)$. | (x+1)(x+2) | 0.8125 | 1,730.6875 | 1,919.615385 | 912 | |
Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote? | 1113 | 0.125 | 8,087.75 | 7,358 | 8,192 | |
Four normal students, A, B, C, and D, are to be assigned to work at three schools, School A, School B, and School C, with at least one person at each school. It is known that A is assigned to School A. What is the probability that B is assigned to School B? | \dfrac{5}{12} | 0 | 7,900.375 | -1 | 7,900.375 | |
Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length 3. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A, C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $AJG$ is an equilateral triangle, then the area of $... | 48 | The fact that $EHGF$ and $IHJK$ have side length $n/6$ ends up being irrelevant. Since $A$ and $H$ are both equidistant from $G$ and $J$, we conclude that the line $ACHM$ is the perpendicular bisector of $GJ$. Now, define the point $C^{\prime}$ so that the spiral similarity centered at $J$ sends $M$ and $H$ to $C^{\pri... | 0 | 8,192 | -1 | 8,192 |
What is the remainder when $2007$ is divided by $81$? | 63 | 0.8125 | 6,367.375 | 5,946.307692 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\cos(B-C)+a\cos A=2\sqrt{3}b\sin C\cos A$.
$(1)$ Find angle $A$;
$(2)$ If the perimeter of $\triangle ABC$ is $8$ and the radius of the circumcircle is $\sqrt{3}$, find the area of $\triangle AB... | \frac{4\sqrt{3}}{3} | 0 | 7,321.875 | -1 | 7,321.875 |
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