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 |
|---|---|---|---|---|---|---|
Misha is the 50th best as well as the 50th worst student in her grade. How many students are in Misha's grade? | 99 | 0.9375 | 2,166.0625 | 1,764.333333 | 8,192 | |
Find the total number of solutions to the equation $(a-b)(a+b)+(a-b)(c)=(a-b)(a+b+c)=2012$ where $a, b, c$ are positive integers. | 1755 | We write this as $(a-b)(a+b)+(a-b)(c)=(a-b)(a+b+c)=2012$. Since $a, b, c$ are positive integers, $a-b<a+b+c$. So, we have three possibilities: $a-b=1$ and $a+b+c=2012$, $a-b=2$ and $a+b+c=1006$, and $a-b=4$ and $a+b+c=503$. The first solution gives $a=b+1$ and $c=2011-2b$, so $b$ can range from 1 through 1005, which de... | 0.875 | 5,562.125 | 5,186.428571 | 8,192 |
At an observation station $C$, the distances to two lighthouses $A$ and $B$ are $300$ meters and $500$ meters, respectively. Lighthouse $A$ is observed at $30^{\circ}$ north by east from station $C$, and lighthouse $B$ is due west of station $C$. Calculate the distance between the two lighthouses $A$ and $B$. | 700 | 0.25 | 7,026.1875 | 4,222 | 7,960.916667 | |
Triangle $PQR$ has sides $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$ of length 47, 14, and 50, respectively. Let $\omega$ be the circle circumscribed around $\triangle PQR$ and let $S$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{RP}$ that is not on the same side of $\overline{... | 14 | 0 | 8,157.375 | -1 | 8,157.375 | |
Determine the value of
\[2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right).\] | 4002 | 0.125 | 7,197.9375 | 3,144.5 | 7,777 | |
Three Triangles: Within triangle \(ABC\), a random point \(M\) is chosen. What is the probability that the area of one of the triangles \(ABM, BCM,\) and \(CAM\) will be greater than the sum of the areas of the other two? | 0.75 | 0 | 7,900.25 | -1 | 7,900.25 | |
The sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n + 1} - a_n)} - 1 = \frac {1}{n + \frac {2}{3}}$ for $n \geq 1$. Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$. | 41 | We notice that by multiplying the equation from an arbitrary $a_n$ all the way to $a_1$, we get: \[5^{a_n-a_1}=\dfrac{n+\tfrac23}{1+\tfrac23}\] This simplifies to \[5^{a_n}=3n+2.\] We can now test powers of $5$.
$5$ - that gives us $n=1$, which is useless.
$25$ - that gives a non-integer $n$.
$125$ - that gives $n=\box... | 0.875 | 6,394.3125 | 6,137.5 | 8,192 |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in\mathbb{R}$ \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\][i] | f(x) = x^2 + 1 |
To determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the given functional equation for all \( x, y \in \mathbb{R} \):
\[
f(x)f(y) - f(x-1) - f(y+1) = f(xy) + 2x - 2y - 4,
\]
we proceed as follows.
### Step 1: Substitute Special Values
1. **Substitute \( x = 0 \) and \( y = 0 \):**
\[
f... | 0 | 8,192 | -1 | 8,192 |
Let \( A \) be a 4-digit integer. When both the first digit (left-most) and the third digit are increased by \( n \), and the second digit and the fourth digit are decreased by \( n \), the new number is \( n \) times \( A \). Find the value of \( A \). | 1818 | 0.5625 | 6,048.4375 | 4,381.222222 | 8,192 | |
If $x$ and $y$ are positive integers with $x+y=31$, what is the largest possible value of $x y$? | 240 | First, we note that the values of $x$ and $y$ cannot be equal since they are integers and $x+y$ is odd. Next, we look at the case when $x>y$. We list the fifteen possible pairs of values for $x$ and $y$ and the corresponding values of $x y$. Therefore, the largest possible value for $x y$ is 240. Note that the largest ... | 1 | 808.4375 | 808.4375 | -1 |
Compute the expressions
\[ C = 3 \times 4 + 5 \times 6 + 7 \times 8 + \cdots + 43 \times 44 + 45 \]
and
\[ D = 3 + 4 \times 5 + 6 \times 7 + \cdots + 42 \times 43 + 44 \times 45 \]
and find the positive difference between integers $C$ and $D$. | 882 | 0 | 6,113.75 | -1 | 6,113.75 | |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( AC \), \( D \) is a point on side \( BC \) such that \( AD \) is the angle bisector of \( \angle BAC \), and \( P \) is the point of intersection of \( AD \) and \( BM \). Given that \( AB = 10 \, \text{cm} \), \( AC = 30 \, \text{cm} \), and the area ... | 20 | 0.625 | 6,882.875 | 6,122.4 | 8,150.333333 | |
In this 5x5 square array of 25 dots, five dots are to be chosen at random. What is the probability that the five dots will be collinear? Express your answer as a common fraction.
[asy]
size(59);
for(int i = 0; i < 5; ++i)
for(int j = 0; j < 5; ++j)
dot((i, j), linewidth(7));
[/asy] | \frac{12}{53130} | 0 | 8,142.8125 | -1 | 8,142.8125 | |
Given the function $y=x^{2}+bx+3$ (where $b$ is a real number), the range of $y$ is $\left[0,+\infty \right)$. Find the value of the real number $c$ if the solution set of the inequality $x^{2}+bx+3 \lt c$ is $m-8 \lt x \lt m$. | 16 | 0.875 | 4,392.5625 | 3,849.785714 | 8,192 | |
Given vectors $\overrightarrow{a}=(\cos x, \sin x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x, 2\cos x-\sqrt{3}\sin x)$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.
$(2)$ If the maximum value of the function $g(x)=f(x-\frac{\pi}{6})+af(\frac{... | 2\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 227.052227052227 | 0 | 7,861.75 | -1 | 7,861.75 | |
In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\overline{AB}$ and $\overline{BC}$. No point of circle $Q$ lies outside of $\triangle ABC$. The radius of circle $Q$ can be expr... | 254 | 0.5625 | 6,870.1875 | 6,271.333333 | 7,640.142857 | |
Tio Mané has two boxes, one with seven distinct balls numbered from 1 to 7 and another with eight distinct balls numbered with all prime numbers less than 20. He draws one ball from each box.
Calculate the probability that the product is odd. What is the probability that the product of the numbers on the drawn balls i... | \frac{1}{2} | 0.75 | 4,939.3125 | 4,119.833333 | 7,397.75 | |
Given vectors $\overrightarrow{a}=(2\cos \alpha,\sin ^{2}\alpha)$, $\overrightarrow{b}=(2\sin \alpha,t)$, where $\alpha\in(0, \frac {\pi}{2})$, and $t$ is a real number.
$(1)$ If $\overrightarrow{a}- \overrightarrow{b}=( \frac {2}{5},0)$, find the value of $t$;
$(2)$ If $t=1$, and $\overrightarrow{a}\cdot \overrigh... | \frac {23}{7} | 0.875 | 4,699.375 | 4,529.071429 | 5,891.5 | |
What is the area of the region defined by the equation $x^2 + y^2 - 10 = 4y - 10x + 4$? | 43\pi | 0.875 | 2,221.125 | 2,308.785714 | 1,607.5 | |
A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not exceed $20$, then $q(x)=\dfrac{1260}{x+1}$;... | 240000 | 0 | 5,292.75 | -1 | 5,292.75 | |
In a paper, a $4 \times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.
Observe that the diagonal $AB$ intersects the grid at 9 points.
If the grid were of size $12 \times 17$, at how many points would the diagonal $AB$ intersect the grid? | 29 | 0.25 | 6,534.5 | 5,266.25 | 6,957.25 | |
The coefficients of the polynomial
\[x^4 + bx^3 + cx^2 + dx + e = 0\]are all integers. Let $n$ be the exact number of integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x + 3)^2 (x^2 + 4x + 11) = 0$ has two integer roots counting multiplicity, because the root $-3$ is counted twice.... | 0, 1, 2, 4 | 0.0625 | 6,591.875 | 6,642 | 6,588.533333 | |
Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test? $\textbf{(A)} 92 \qquad\textbf{(B)} 94 \qquad ... | 100 | 0 | 8,192 | -1 | 8,192 | |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | 16.67\% | 0.25 | 3,567.9375 | 2,345 | 3,975.583333 | |
Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at 5:00 p.m. Calculate the total number of mi... | 105 | 0.0625 | 654.5 | 639 | 655.533333 | |
Quadrilateral $ABCD$ has right angles at $B$ and $D$, and $AC=3$. If $ABCD$ has two sides with distinct integer lengths, then what is the area of $ABCD$? Express your answer in simplest radical form. | \sqrt 2+\sqrt 5 | 0 | 8,142.5 | -1 | 8,142.5 | |
Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$ . Define $\{x \} = x - \lfloor x \rfloor$ . For example, $\{ 3 \} = 3-3 = 0$ , $\{ \pi \} = \pi - 3$ , and $\{ - \pi \} = 4-\pi$ . If $\{n\} + \{ 3n\} = 1.4$ , then find the sum of all possible values of $100\{n\}$ .
*Proposed by Isab... | 145 | 1 | 3,990.8125 | 3,990.8125 | -1 | |
In a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)? | 35 | 0 | 8,192 | -1 | 8,192 | |
Given that 10 is the arithmetic mean of the set $\{6, 13, 18, 4, x\}$, what is the value of $x$? | 9 | 1 | 1,229.75 | 1,229.75 | -1 | |
How many positive integers $n$ less than 150 have a corresponding integer $m$ not divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | 50 | 0.0625 | 6,640.8125 | 6,040 | 6,680.866667 | |
How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not.
(a) 10
(b) 13
(c) 18
(d) 22
(e) 25 | 22 | 0 | 7,712.1875 | -1 | 7,712.1875 | |
In the provided polygon, each side is perpendicular to its adjacent sides, and all 36 of the sides are congruent. The perimeter of the polygon is 72. Inside, the polygon is divided into rectangles instead of squares. Find the area of the polygon. | 72 | 0.0625 | 6,582.75 | 6,975 | 6,556.6 | |
Solve
\[\arcsin x + \arcsin 2x = \frac{\pi}{3}.\] | \frac{\sqrt{21}}{14} | 0 | 6,953.8125 | -1 | 6,953.8125 | |
A club is creating a special series of membership cards that includes a code consisting of five characters selected from the letters B, E, T, A and the digits in 2023. No character should appear in a code more often than it appears in "BETA" or "2023" (B, E, T, A appear once each; 2 is twice; 0 and 3 appear once each).... | 312 | 0 | 8,059.125 | -1 | 8,059.125 | |
The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude. | 12.8470588235 | 0 | 5,062.5625 | -1 | 5,062.5625 | |
Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d\), and \(e\) respectively. We know that:
1. \(c = b \cdot e\)
2. \(a + b = d\)
3. \(e - d = a\)
Frankie picks up the card which has the largest integer on its re... | 103 | 0 | 8,192 | -1 | 8,192 | |
A function $f$ is defined for all real numbers and satisfies the conditions $f(3+x) = f(3-x)$ and $f(8+x) = f(8-x)$ for all $x$. If $f(0) = 0$, determine the minimum number of roots that $f(x) = 0$ must have in the interval $-500 \leq x \leq 500$. | 201 | 0.0625 | 8,081.125 | 6,418 | 8,192 | |
If $M(\frac{p}{2}$,$p)$ is a point on the parabola $y^{2}=2px\left(p \gt 0\right)$, and the distance from $M$ to the point $\left(1,0\right)$ is 1 greater than the distance to the $y$-axis, find:<br/>
$(1)$ The equation of the parabola.<br/>
$(2)$ Let line $l$ intersect the parabola at points $A$ and $B$. The circle wi... | 2\sqrt{5} | 0 | 8,192 | -1 | 8,192 | |
Find the greatest constant $M,$ so that
\[\frac{a^2 + b^2}{c^2} > M\]whenever $a,$ $b,$ and $c$ are the sides of a triangle. | \frac{1}{2} | 0.8125 | 6,557.0625 | 6,179.769231 | 8,192 | |
A quadrilateral pyramid \(SABCD\) is given, with a base that is a trapezoid \(ABCD\). The ratio of the bases \(AD\) and \(BC\) of this trapezoid is 2. Construct the cross-section of the pyramid with a plane passing through point \(D\) and the midpoints of the edges \(SA\) and \(SB\). In what ratio does this plane divid... | 2:1 | 0.5 | 6,517.25 | 5,452.875 | 7,581.625 | |
Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 1 | 2,302.125 | 2,302.125 | -1 | |
Points $A, B, C$ in the plane satisfy $\overline{A B}=2002, \overline{A C}=9999$. The circles with diameters $A B$ and $A C$ intersect at $A$ and $D$. If $\overline{A D}=37$, what is the shortest distance from point $A$ to line $B C$? | 37 | $\angle A D B=\angle A D C=\pi / 2$ since $D$ lies on the circles with $A B$ and $A C$ as diameters, so $D$ is the foot of the perpendicular from $A$ to line $B C$, and the answer is the given 37. | 0.0625 | 8,183.875 | 8,062 | 8,192 |
What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? | 48 | 1. **Understanding the Triangle Inequality Theorem**: The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides $a$, $b$, and $c$, this can be expressed as:
- $a + b > c$
- $a + c > b$
- $b + c ... | 0.9375 | 1,440.4375 | 1,360.2 | 2,644 |
A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\_1$ are $\begin{cases} x=\sqrt{2}\sin(\alpha + \frac{\pi}{4}) \\ y=\sin(2\alpha) + 1 \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with $... | \frac{\sqrt{7}}{2} - 1 | 0 | 8,073 | -1 | 8,073 | |
Find all real numbers $x$ such that \[3 \le \frac{x}{2x-5} < 8.\](Give your answer in interval notation.) | (\tfrac83, 3] | 0.375 | 7,003.8125 | 5,968 | 7,625.3 | |
If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? | -2 | 1 | 2,732.5625 | 2,732.5625 | -1 | |
Given the points M(2,0) and N(a,b) in the Cartesian coordinate system, with the Manhattan distance between M and N defined as d(M,N) = |x₁ - x₂| + |y₁ - y₂|, and d(M,N) = 2, find the sum of the minimum and maximum values of a² + b² - 4a. | -2 | 0.5 | 7,117.9375 | 6,578.125 | 7,657.75 | |
Given that a shop advertises everything as "half price in today's sale," and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total original price for both items that the customer... | 20\% | 0.0625 | 3,690.625 | 4,250 | 3,653.333333 | |
On a complex plane map of a fictional continent, city A is located at the origin $0$, city B is at $3900i$, and city C is at $1170 + 1560i$. Calculate the distance from city C to city A on this plane. | 1950 | 1 | 3,006.875 | 3,006.875 | -1 | |
A line passing through one focus F_1 of the ellipse 4x^{2}+2y^{2}=1 intersects the ellipse at points A and B. Then, points A, B, and the other focus F_2 of the ellipse form ∆ABF_2. Calculate the perimeter of ∆ABF_2. | 2 \sqrt {2} | 0 | 7,726.1875 | -1 | 7,726.1875 | |
If $g(x) = 2x^2 - 3$ and $h(x) = 4x^3 +1$, what is the value of $g(h(-1))$? | 15 | 1 | 1,686.0625 | 1,686.0625 | -1 | |
On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? | \frac{9}{16} | 1. **Total number of squares on the checkerboard**: The checkerboard is composed of $8 \times 8 = 64$ unit squares.
2. **Counting the squares on the perimeter**:
- The top and bottom rows each have 8 squares.
- The left and right columns each have 8 squares, but this count includes the corners twice (once for t... | 1 | 2,515.75 | 2,515.75 | -1 |
Football tickets cost $\$13.50$ each. What is the maximum number of tickets Jane can buy with $\$100.00$? | 7 | 1 | 2,783.875 | 2,783.875 | -1 | |
Given triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ lying on the line $x + 2y = 8$, find the area of triangle $ABC$. | 4.5 | 0 | 8,192 | -1 | 8,192 | |
Find the length of the parametric curve described by
\[(x,y) = (2 \sin t, 2 \cos t)\]from $t = 0$ to $t = \pi.$ | 2 \pi | 1 | 1,655.1875 | 1,655.1875 | -1 | |
Solve the application problem by setting up equations:<br/>A gift manufacturing factory receives an order for a batch of teddy bears and plans to produce them in a certain number of days. If they produce $20$ teddy bears per day, they will be $100$ short of the order. If they produce $23$ teddy bears per day, they will... | 40 | 0.8125 | 1,247.1875 | 1,346 | 819 | |
\( P \) is a moving point on the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\). The tangent line to the ellipse at point \( P \) intersects the circle \(\odot O\): \(x^{2}+y^{2}=12\) at points \( M \) and \( N \). The tangents to \(\odot O\) at \( M \) and \( N \) intersect at point \( Q \).
(1) Find the equation of th... | \frac{\sqrt{3}}{2} | 0 | 7,928.4375 | -1 | 7,928.4375 | |
In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$. | 693 | 0.0625 | 8,167.5625 | 7,801 | 8,192 | |
I am dining at a Mexican restaurant with a friend who is vegan and allergic to nuts. The restaurant menu lists 8 dishes that are vegan. These vegan options constitute one-fourth of the entire menu. However, 5 of these vegan dishes contain nuts. What fraction of the menu can my friend eat? | \frac{3}{32} | 0.625 | 2,068.875 | 1,695 | 2,692 | |
In the diagram, $\angle PQR=\angle PRQ$. If $QR=5$ and $PR=7$, what is the perimeter of $\triangle PQR$? [asy]
draw((0,0)--(2.5,7.43)--(5,0)--cycle);
label("5",(2.5,0),S);
label("$Q$",(0,0),SW);
label("$R$",(5,0),SE);
label("$P$",(2.5,7.43),N);
label("7",(4.2,3.7));
[/asy] | 19 | 1 | 2,498.4375 | 2,498.4375 | -1 | |
In $\triangle ABC$, $\angle A = 90^\circ$ and $\tan C = 3$. If $BC = 90$, what is the length of $AB$, and what is the perimeter of triangle ABC? | 36\sqrt{10} + 90 | 0 | 2,483.5 | -1 | 2,483.5 | |
In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? | 2-\sqrt{3} | 1. **Assign Variables:**
Let $BE = x$, $EK = a$, and $EJ = b$. Since $ABCD$ is a square with side length $1$, we have $AB = BC = CD = DA = 1$.
2. **Use Pythagorean Theorem in $\triangle BEK$:**
Since $\triangle BEK$ is a right triangle, by the Pythagorean theorem, we have:
\[ x^2 = a^2 + b^2. \]
3. **Congrue... | 0 | 8,143.375 | -1 | 8,143.375 |
If I have a $4\times 4$ chess board, in how many ways can I place four distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 576 | 1 | 4,975.9375 | 4,975.9375 | -1 | |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|= \sqrt {3}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}- \overrightarrow {b}) \perp \overrightarrow {a}$, find the projection of $\overrightarrow {a}$ on $\overrightarrow {b}$. | \frac {3}{2} | 1 | 3,016.5625 | 3,016.5625 | -1 | |
How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square? | 343 | 0.125 | 3,361.1875 | 2,261.5 | 3,518.285714 | |
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? | \frac{2}{5} | 1. **Define Variables:**
Let $m$ represent the total amount of money Brianna has. Let $c$ represent the cost of one CD, and let $n$ be the total number of CDs she wants to buy.
2. **Establish Relationships:**
According to the problem, Brianna uses one fifth of her money to buy one third of the CDs. This can be e... | 1 | 2,371.0625 | 2,371.0625 | -1 |
Which of the following quantities is the largest? (Write $A$, $B$, or $C$.) \[ A.\ \ \frac{2006}{2005}+\frac{2006}{2007} \qquad B.\ \ \frac{2006}{2007}+\frac{2008}{2007} \qquad C.\ \ \frac{2007}{2006}+\frac{2007}{2008} \] | \text{A} | 0 | 7,130.8125 | -1 | 7,130.8125 | |
Given circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 60^\circ$, and $\angle FOB = 90^\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle. | \frac{1}{12} | 1 | 4,168.875 | 4,168.875 | -1 | |
(1) Given $0 < x < \frac{1}{2}$, find the maximum value of $y= \frac{1}{2}x(1-2x)$;
(2) Given $x > 0$, find the maximum value of $y=2-x- \frac{4}{x}$;
(3) Given $x$, $y\in\mathbb{R}_{+}$, and $x+y=4$, find the minimum value of $\frac{1}{x}+ \frac{3}{y}$. | 1+ \frac{ \sqrt{3}}{2} | 0 | 5,078.25 | -1 | 5,078.25 | |
If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 5,$ then find
\[\begin{vmatrix} a - c & b - d \\ c & d \end{vmatrix}.\] | 5 | 1 | 2,847.9375 | 2,847.9375 | -1 | |
The sequence consists of 19 ones and 49 zeros arranged in a random order. A group is defined as the maximal subsequence of identical symbols. For example, in the sequence 110001001111, there are five groups: two ones, then three zeros, then one one, then two zeros, and finally four ones. Find the expected value of the ... | 2.83 | 0 | 8,133.625 | -1 | 8,133.625 | |
Let $f(x) = 2a^{x} - 2a^{-x}$ where $a > 0$ and $a \neq 1$. <br/> $(1)$ Discuss the monotonicity of the function $f(x)$; <br/> $(2)$ If $f(1) = 3$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \in [0,3]$, find the minimum value of $g(x)$. | -2 | 0.6875 | 6,285.4375 | 6,298.909091 | 6,255.8 | |
If $f^{-1}(g(x))=x^2-4$ and $g$ has an inverse, find $g^{-1}(f(10))$. | \sqrt{14} | 0.4375 | 7,360.4375 | 6,291.285714 | 8,192 | |
Let \(a\), \(b\), and \(c\) be angles such that:
\[
\sin a = \cot b, \quad \sin b = \cot c, \quad \sin c = \cot a.
\]
Find the largest possible value of \(\cos a\). | \sqrt{\frac{3 - \sqrt{5}}{2}} | 0 | 7,239.8125 | -1 | 7,239.8125 | |
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangeme... | 630 | 0 | 8,192 | -1 | 8,192 | |
For each of the possible outcomes of rolling three coins (TTT, THT, TTH, HHT, HTH, HHH), a fair die is rolled for each two heads that appear together. What is the probability that the sum of the die rolls is odd? | \frac{1}{4} | 0.25 | 6,118.1875 | 6,066.75 | 6,135.333333 | |
$A B C$ is a triangle with $A B=15, B C=14$, and $C A=13$. The altitude from $A$ to $B C$ is extended to meet the circumcircle of $A B C$ at $D$. Find $A D$. | \frac{63}{4} | Let the altitude from $A$ to $B C$ meet $B C$ at $E$. The altitude $A E$ has length 12 ; one way to see this is that it splits the triangle $A B C$ into a $9-12-15$ right triangle and a $5-12-13$ right triangle; from this, we also know that $B E=9$ and $C E=5$. Now, by Power of a Point, $A E \cdot D E=B E \cdot C E$, s... | 0.9375 | 4,842.0625 | 4,618.733333 | 8,192 |
A traffic light cycles repeatedly in the following order: green for 45 seconds, then yellow for 5 seconds, and red for 50 seconds. Cody picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching? | \frac{3}{20} | 0.3125 | 7,432.5625 | 5,987 | 8,089.636364 | |
Equilateral triangle $ABC$ has a side length of $\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\angle ABD_1 = 30^\circ$ and $\angle ABD_2 = 150^\circ$.... | 576 | 0 | 7,988.125 | -1 | 7,988.125 | |
Subtract $111.11$ from $333.33.$ Express the result as a decimal to the nearest hundredth. | 222.22 | 1 | 1,933.125 | 1,933.125 | -1 | |
The circle is divided by points \(A, B, C, D\) such that \(\sim AB: BC: CD: DA = 2: 3: 5: 6\). Chords \(AC\) and \(BD\) are drawn, intersecting at point \(M\).
Find the angle \(AMB\). | 78.75 | 0.625 | 6,104.3125 | 4,851.7 | 8,192 | |
The number of four-digit even numbers formed without repeating digits from the numbers $2$, $0$, $1$, $7$ is ______. | 10 | 0.75 | 4,523.5625 | 3,653.166667 | 7,134.75 | |
Given $\boldsymbol{a} = (\cos \alpha, \sin \alpha)$ and $\boldsymbol{b} = (\cos \beta, \sin \beta)$, the relationship between $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $|k \boldsymbol{a} + \boldsymbol{b}| - \sqrt{3}|\boldsymbol{a} - k \boldsymbol{b}|$, where $k > 0$. Find the minimum value of $\boldsymbol{a} \... | \frac{1}{2} | 0.5 | 6,988.25 | 5,784.5 | 8,192 | |
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? | 12 | 0.9375 | 3,347.5625 | 3,024.6 | 8,192 | |
On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red... | 13 | 0 | 7,802.625 | -1 | 7,802.625 | |
How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. | 248 | 0 | 8,192 | -1 | 8,192 | |
The sum of sides \( AB \) and \( BC \) of triangle \( ABC \) is 11, angle \( B \) is \( 60^\circ \), and the radius of the inscribed circle is \(\frac{2}{\sqrt{3}}\). It is also known that side \( AB \) is longer than side \( BC \). Find the height of the triangle dropped from vertex \( A \). | 4\sqrt{3} | 0.875 | 5,447.0625 | 5,054.928571 | 8,192 | |
Use all digits from 1 to 9 to form three three-digit numbers such that their product is:
a) the smallest;
b) the largest. | 941 \times 852 \times 763 | 0 | 8,192 | -1 | 8,192 | |
Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a\), \(b\), and \(c\) are positive real numbers. | \frac{3}{\sqrt[3]{162}} | 0 | 7,552.25 | -1 | 7,552.25 | |
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the dig... | 420 | [Clarification: You are supposed to find the number of all possible tuples of prices, $(A, B, C)$, that could have been on that day.]
Since we have three numbers, consider the number of ways we can put these three numbers together in a string of 7 digits. For example, if $A=113, B=13, C=31$, then the string is
\[11313... | 0 | 8,192 | -1 | 8,192 |
The Gropkas of Papua New Guinea have ten letters in their alphabet: A, E, G, I, K, O, R, U, and V. Suppose license plates of four letters use only the letters in the Gropka alphabet. How many possible license plates are there of four letters that begin with either A or E, end with V, cannot contain P, and have no lette... | 84 | 0.625 | 4,967.875 | 4,907.6 | 5,068.333333 | |
A high school is holding a speech contest with 10 participants. There are 3 students from Class 1, 2 students from Class 2, and 5 students from other classes. Using a draw to determine the speaking order, what is the probability that the 3 students from Class 1 are placed consecutively (in consecutive speaking slots) a... | $\frac{1}{20}$ | 0 | 6,289.5625 | -1 | 6,289.5625 | |
What is the largest digit $N$ for which $2345N$ is divisible by 6? | 4 | 1 | 3,282.0625 | 3,282.0625 | -1 | |
Determine the exact value of the series
\[
\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.
\] | \frac{1}{2} | 0.125 | 7,477.1875 | 5,243.5 | 7,796.285714 | |
The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? | 17.67 | 0 | 2,452.25 | -1 | 2,452.25 | |
Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza. | 65.31\% | 0.3125 | 3,918.9375 | 3,068.4 | 4,305.545455 | |
Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from more than half of the other 7 balls. | \frac{35}{128} | 0.375 | 6,505.4375 | 5,411.666667 | 7,161.7 | |
Find $\tan \frac{9 \pi}{4}.$ | 1 | 1 | 3,108.8125 | 3,108.8125 | -1 | |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number? | 1221 | 0.3125 | 7,161 | 5,531.2 | 7,901.818182 |
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