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 |
|---|---|---|---|---|---|---|
The odd function $y=f(x)$ has a domain of $\mathbb{R}$, and when $x \geq 0$, $f(x) = 2x - x^2$. If the range of the function $y=f(x)$, where $x \in [a, b]$, is $[\frac{1}{b}, \frac{1}{a}]$, then the minimum value of $b$ is ______. | -1 | 0 | 7,978.5625 | -1 | 7,978.5625 | |
There is a uniformly growing grassland. If 20 cows are grazed, they will just finish eating all the grass in 60 days. If 30 cows are grazed, they will just finish eating all the grass in 35 days. Now, 6 cows are grazing on the grassland. After a month, 10 more cows are added. How many more days will it take for all the... | 84 | 0.25 | 6,567.625 | 4,824.25 | 7,148.75 | |
If the digits \( a_{i} (i=1,2, \cdots, 9) \) satisfy
$$
a_{9} < a_{8} < \cdots < a_{5} \text{ and } a_{5} > a_{4} > \cdots > a_{1} \text{, }
$$
then the nine-digit positive integer \(\bar{a}_{9} a_{8} \cdots a_{1}\) is called a “nine-digit peak number”, for example, 134698752. How many nine-digit peak numbers are ther... | 11875 | 0 | 7,642.9375 | -1 | 7,642.9375 | |
Given the function $f(x)=e^{ax}$, a line parallel to the $y$-axis is drawn through $A(a,0)$ and intersects the function $f(x)$ at point $P$. A tangent line to $f(x)$ at $P$ intersects the $x$-axis at point $B$. Find the minimum value of the area of $\triangle APB$. | \dfrac { \sqrt {2e}}{2} | 0 | 5,962.625 | -1 | 5,962.625 | |
From 51 consecutive odd numbers $1, 3, 5, \cdots, 101$, select $\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\mathrm{k}$? | 44 | 0 | 8,192 | -1 | 8,192 | |
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 50$ such that $i^x+i^y$ is a real number, and additionally, $x+y$ is divisible by $4$. | 288 | 0.0625 | 7,969.125 | 5,033 | 8,164.866667 | |
If the centroid of the inscribed triangle \( ABC \) of the curve \( y^{2}=4 \sqrt{2} x \) is its focus \( F \), then
\[ |FA|^{2} + |FB|^{2} + |FC|^{2} = \] | 27 | 0.6875 | 5,811.25 | 5,143.181818 | 7,281 | |
The height \( PO \) of the regular quadrilateral pyramid \( PABC D \) is 4, and the side of the base \( ABCD \) is 6. Points \( M \) and \( K \) are the midpoints of segments \( BC \) and \( CD \). Find the radius of the sphere inscribed in the pyramid \( PMKC \). | \frac{12}{13+\sqrt{41}} | 0 | 7,766.6875 | -1 | 7,766.6875 | |
What is the value of $2-(-2)^{-2}$? | \frac{7}{4} | 1. **Evaluate the exponentiation and inversion:**
The expression given is $2 - (-2)^{-2}$. We start by evaluating $(-2)^{-2}$. By the properties of exponents, $a^{-n} = \frac{1}{a^n}$, so:
\[
(-2)^{-2} = \frac{1}{(-2)^2}
\]
2. **Calculate $(-2)^2$:**
Since $(-2)^2 = (-2) \times (-2) = 4$, we substitute ... | 1 | 1,677.5 | 1,677.5 | -1 |
What is the value of the expression $\frac {x^2-x-6}{x-3}$ for $x=2$? Express your answer in simplest form. | 4 | 1 | 1,497 | 1,497 | -1 | |
A rectangle has dimensions 12 by 15, and a circle centered at one of its vertices has a radius of 15. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \( \pi \). | 180 + 168.75\pi | 0.0625 | 8,034.0625 | 5,665 | 8,192 | |
What is the largest six-digit number that can be obtained by removing nine digits from the number 778157260669103, without changing the order of its digits?
(a) 778152
(b) 781569
(c) 879103
(d) 986103
(e) 987776 | 879103 | 0 | 8,192 | -1 | 8,192 | |
The ratio of irises to roses in Nikki's garden is 2:5. She currently has 25 roses. She is planning to add 20 more roses and enough irises to keep the same ratio. How many irises will she have in total after this addition? | 18 | 0.6875 | 2,365.8125 | 2,891.272727 | 1,209.8 | |
In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.
Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round t... | 13/18 | 0 | 8,192 | -1 | 8,192 | |
Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$. | 744 | Note that if angle $BAC$ is obtuse, it would be impossible for the triangle to inscribed in a rectangle. This can easily be shown by drawing triangle ABC, where $A$ is obtuse. Therefore, angle A is acute. Let angle $CAS=n$ and angle $BAQ=m$. Then, $\overline{AS}=31\cos(n)$ and $\overline{AQ}=40\cos(m)$. Then the area o... | 0 | 8,192 | -1 | 8,192 |
Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[\{-8, -6, -4, -1, 1, 3, 5, 14\}.\] What is the minimum possible value of \[
(p+q+r+s)^2 + (t+u+v+w)^2
\] given that the sum $p+q+r+s$ is at least 5? | 26 | 0.9375 | 5,642.8125 | 5,472.866667 | 8,192 | |
In the Cartesian coordinate system $(xOy)$, let $l$ be a line with an angle of inclination $\alpha$ and parametric equations $\begin{cases} x = 3 + t\cos\alpha \\ y = t\sin\alpha \end{cases}$ (where $t$ is a parameter). The line $l$ intersects the curve $C$: $\begin{cases} x = \frac{1}{\cos\theta} \\ y = \tan\theta \en... | \frac{40}{3} | 0.3125 | 7,682.5 | 7,482.6 | 7,773.363636 | |
Given the function $f(x)=\sqrt{3}\sin x \cos x - \cos^2 x, (x \in \mathbb{R})$.
$(1)$ Find the intervals where $f(x)$ is monotonically increasing.
$(2)$ Find the maximum and minimum values of $f(x)$ on the interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$. | -\frac{3}{2} | 0.875 | 5,769.375 | 5,423.285714 | 8,192 | |
Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? | 8\pi - 16 | 0.75 | 6,168.6875 | 5,522.416667 | 8,107.5 | |
Given the function $f(x)=x^{3}-x^{2}+1$.
$(1)$ Find the equation of the tangent line to the function $f(x)$ at the point $(1,f(1))$;
$(2)$ Find the extreme values of the function $f(x)$. | \dfrac {23}{27} | 0.375 | 2,544.875 | 2,505.333333 | 2,568.6 | |
In an exam, there are 6 questions, and each question is solved by exactly 100 people. Each pair of examinees has at least one question that neither of them has solved. What is the minimum number of participants in the exam? | 200 | 0 | 7,548.125 | -1 | 7,548.125 | |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). If \(AD = 52\), \(BC = 65\), \(AB = 20\), and \(CD = 11\), find the area of the trapezoid. | 594 | 0.5625 | 7,499.4375 | 6,960.777778 | 8,192 | |
Suppose that \((x_{0}, y_{0})\) is a solution of the system:
\[
\begin{cases}
xy = 6 \\
x^2 y + xy^2 + x + y + c = 2
\end{cases}
\]
Find the value of \(d = x_{0}^{2} + y_{0}^{2}\). | 69 | 0 | 7,699.125 | -1 | 7,699.125 | |
The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values
when evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$? | 4/5 | 1. **Identify the Terms in the Expansion**:
The polynomial $(x+y)^9$ can be expanded using the binomial theorem, which states that:
\[
(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k
\]
For $n=9$, the second term (where $k=1$) is:
\[
\binom{9}{1} x^{9-1} y^1 = 9x^8y
\]
and the third term (wh... | 0.9375 | 2,902.0625 | 2,549.4 | 8,192 |
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airpor... | 210 | Let's denote the total distance from David's home to the airport as $d$ miles. According to the problem, David drives the first hour at 35 mph, covering 35 miles. If he continues at this speed, he would be 1 hour late. This means that the total time required to cover the distance $d$ at 35 mph should be $t+1$ hours, wh... | 0.8125 | 3,480.125 | 2,749.384615 | 6,646.666667 |
Find the number of pairs of integers \((a, b)\) with \(1 \leq a<b \leq 57\) such that \(a^{2}\) has a smaller remainder than \(b^{2}\) when divided by 57. | 738 | There are no such pairs when \(b=57\), so we may only consider pairs with \(1 \leq a<b \leq 56\). The key idea is that unless \(a^{2} \bmod 57=b^{2} \bmod 57,(a, b)\) can be paired with \((57-b, 57-a)\) and exactly one of them satisfies \(a^{2} \bmod 57<b^{2} \bmod 57\). Hence if \(X\) is the number of pairs \((a, b)\)... | 0 | 8,192 | -1 | 8,192 |
Let $S$ be a set of consecutive positive integers such that for any integer $n$ in $S$, the sum of the digits of $n$ is not a multiple of 11. Determine the largest possible number of elements of $S$. | 38 | We claim that the answer is 38. This can be achieved by taking the smallest integer in the set to be 999981. Then, our sums of digits of the integers in the set are $$45, \ldots, 53,45, \ldots, 54,1, \ldots, 10,2, \ldots, 10$$ none of which are divisible by 11.
Suppose now that we can find a larger set $S$: then we ca... | 0 | 8,192 | -1 | 8,192 |
A hall is organizing seats in rows for a lecture. Each complete row must contain $13$ chairs. Initially, the hall has $169$ chairs arranged. To maintain fully occupied rows with minimal empty seats, if $100$ students are expected to attend, how many chairs should be removed or added? | 65 | 0.625 | 5,368.5625 | 4,837 | 6,254.5 | |
What is the radius of the smallest sphere in which 4 spheres of radius 1 will fit? | \frac{2+\sqrt{6}}{2} | The centers of the smaller spheres lie on a tetrahedron. Let the points of the tetrahedron be $(1,1,1),(-1,-1,1),(-1,1,-1)$, and $(1,-1,-1)$. These points have distance $\sqrt{(} 3)$ from the center, and $\sqrt{(} 2)$ from each other, so the radius of the smallest sphere in which 4 spheres of radius $\sqrt{(2)}$ will f... | 0 | 5,768.125 | -1 | 5,768.125 |
What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \( d \) contained in this segment, there are no more than \( 1 + 1000 d^2 \) points? | 32 | 0.0625 | 8,091.0625 | 6,577 | 8,192 | |
In the AU tribe's language, there are two letters - "a" and "u". Certain sequences of these letters form words, where each word contains no fewer than one and no more than 13 letters. It is known that if any two words are written consecutively, the resulting sequence will not be a word. Find the maximum possible number... | 16256 | 0 | 8,137.875 | -1 | 8,137.875 | |
The truncated right circular cone has a large base radius 8 cm and a small base radius of 4 cm. The height of the truncated cone is 6 cm. How many $\text{cm}^3$ are in the volume of this solid? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
draw(ellipse((0,0),4,1)); draw(ellipse((0,3),2,1... | 224\pi | 1 | 2,736.9375 | 2,736.9375 | -1 | |
A printer received an annual order to print 10,000 posters each month. Each month's poster should have the month's name printed on it. Thus, he needs to print 10,000 posters with the word "JANUARY", 10,000 posters with the word "FEBRUARY", 10,000 posters with the word "MARCH", and so on. The typefaces used to print th... | 22 | 0 | 5,785.3125 | -1 | 5,785.3125 | |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer? | 215 | Let our answer be $n$. Write $n = 42a + b$, where $a, b$ are positive integers and $0 \leq b < 42$. Then note that $b, b + 42, ... , b + 42(a-1)$ are all primes.
If $b$ is $0\mod{5}$, then $b = 5$ because $5$ is the only prime divisible by $5$. We get $n = 215$ as our largest possibility in this case.
If $b$ is $1\mo... | 0 | 8,000.875 | -1 | 8,000.875 |
Given real numbers \( x \) and \( y \in (1, +\infty) \), and \( x y - 2 x - y + 1 = 0 \). Find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \). | 15 | 0.6875 | 6,962.4375 | 6,403.545455 | 8,192 | |
Given the point \( P(-2,5) \) lies on the circle \(\odot C: x^{2}+y^{2}-2x-2y-23=0\), and the line \( l: 3x+4y+8=0 \) intersects \(\odot C\) at points \( A \) and \( B \). Find \(\overrightarrow{AB} \cdot \overrightarrow{BC}\). | -32 | 0.5625 | 5,743.25 | 4,603.888889 | 7,208.142857 | |
There are three balls of the same size but different colors in a pocket. One ball is drawn each time, the color is recorded, and then it is put back. The drawing stops when all three colors of balls have been drawn. If it stops after exactly 5 draws, the number of different ways to draw is \_\_\_\_\_\_\_. | 42 | 0.25 | 7,010.6875 | 6,065 | 7,325.916667 | |
The distance between cities $A$ and $B$ is 435 km. A train left city $A$ at a speed of 45 km/h. After 40 minutes, another train left city $B$ heading towards the first train at a speed of 55 km/h. What distance will be between them one hour before they meet? | 100 | 0.4375 | 7,220.375 | 6,390.571429 | 7,865.777778 | |
Given $|m|=4$, $|n|=3$.
(1) When $m$ and $n$ have the same sign, find the value of $m-n$.
(2) When $m$ and $n$ have opposite signs, find the value of $m+n$. | -1 | 0.6875 | 6,257.75 | 6,130.545455 | 6,537.6 | |
There are 120 five-digit numbers formed by the digits 1, 2, 3, 4, 5, arranged in descending order. The 95th number is ______. | 21354 | 0.1875 | 7,763.75 | 6,892.666667 | 7,964.769231 | |
Will stands at a point \(P\) on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of \(n^{\circ}\) and \((n+1)^{\circ}\) with the tangent at \(P\), where \(n\) is a positive integer less than 90. The lasers reflect off of the walls, illuminating the ... | 28 | Note that we want the path drawn out by the lasers to come back to \(P\) in as few steps as possible. Observe that if a laser is fired with an angle of \(n\) degrees from the tangent, then the number of points it creates on the circle is \(\frac{180}{\operatorname{gcd}(180, n)}\). (Consider the regular polygon created ... | 0 | 8,192 | -1 | 8,192 |
Given that the product of the digits of a 3-digit positive integer equals 36, calculate the number of such integers. | 21 | 0.375 | 7,729.0625 | 6,957.5 | 8,192 | |
Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have? | 40 | 1. **Define Variables:**
Let $b$ represent the number of blueberry jelly beans Camilla originally had, and $c$ represent the number of cherry jelly beans she originally had.
2. **Set Up Initial Equations:**
According to the problem, Camilla had twice as many blueberry jelly beans as cherry jelly beans. This can ... | 1 | 1,424.9375 | 1,424.9375 | -1 |
How many positive integers \(N\) possess the property that exactly one of the numbers \(N\) and \((N+20)\) is a 4-digit number? | 40 | 0 | 5,056 | -1 | 5,056 | |
Given that the center of circle $M$ lies on the $y$-axis, the radius is $1$, and the chord intercepted by line $l: y = 2x + 2$ on circle $M$ has a length of $\frac{4\sqrt{5}}{5}$. Additionally, the circle center $M$ is located below line $l$.
(1) Find the equation of circle $M$;
(2) Let $A(t, 0), B(t + 5, 0) \, (-4 \... | \frac{125}{21} | 0.0625 | 8,192 | 8,192 | 8,192 | |
What are the rightmost three digits of $3^{1987}$? | 187 | 0 | 7,096.4375 | -1 | 7,096.4375 | |
Given a cone with a base radius of $1$ and a height of $\sqrt{3}$, both the apex of the cone and the base circle are on the surface of a sphere $O$, calculate the surface area of this sphere. | \frac{16\pi}{3} | 0.375 | 5,807.6875 | 5,529.166667 | 5,974.8 | |
In the set of four-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition, there are a total of numbers that are not divisible by 5. | 192 | 0.5 | 5,828.1875 | 3,891.625 | 7,764.75 | |
A stalker, to detect a gravitational anomaly (an area where the acceleration due to gravity changes sharply in magnitude), throws a small nut from the surface of the Earth at an angle \(\alpha = 30^\circ\) to the horizontal with a speed \(v_0 = 20 \, \text{m/s}\). The normal acceleration due to gravity is \(g = 10 \, \... | 40 | 0.4375 | 6,437.6875 | 5,094.428571 | 7,482.444444 | |
Let $f(x) = 3x-8$ and $g(f(x)) = 2x^2 + 5x - 3.$ Find $g(-5).$ | 4 | 1 | 2,946.9375 | 2,946.9375 | -1 | |
The lines $y=2$, $y=5$, $x=1$, and $x=a$ make a square. Find the product of the possible values for $a$. | -8 | 1 | 1,158.875 | 1,158.875 | -1 | |
\(\triangle ABC\) is equilateral with side length 4. \(D\) is a point on \(BC\) such that \(BD = 1\). If \(r\) and \(s\) are the radii of the inscribed circles of \(\triangle ADB\) and \(\triangle ADC\) respectively, find \(rs\). | 4 - \sqrt{13} | 0.75 | 6,810.875 | 6,361.75 | 8,158.25 | |
In a round glass, the axial cross-section of which is the graph of the function \( y = x^4 \), a cherry (a sphere with radius \( r \)) is placed. For which maximum \( r \) will the sphere touch the bottom point of the glass? (In other words, what is the maximum radius \( r \) of the circle lying in the region \( y \geq... | \frac{3 \cdot 2^{1/3}}{4} | 0 | 7,134.375 | -1 | 7,134.375 | |
If $\displaystyle\frac{q}{r} = 9$, $\displaystyle\frac{s}{r} = 6$, and $\displaystyle \frac{s}{t} = \frac{1}{2}$, then what is $\displaystyle\frac{t}{q}$? | \frac{4}{3} | 1 | 1,926.0625 | 1,926.0625 | -1 | |
What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43? | 19 | 0.1875 | 7,452.0625 | 4,245.666667 | 8,192 | |
For what value of $x$ does $3^{2x^{2}-5x+2} = 3^{2x^{2}+7x-4}$? Express your answer as a common fraction. | \frac{1}{2} | 1 | 1,569.5 | 1,569.5 | -1 | |
Given $α∈(\frac{π}{2},π)$, and $sin(α+\frac{π}{3})=\frac{12}{13}$, determine the value of $sin(\frac{π}{6}-α)+sin(\frac{2π}{3}-α)$. | \frac{7}{13} | 0.5625 | 6,555.1875 | 5,282.111111 | 8,192 | |
Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the sec... | \frac{9}{10} |
#### Step 1: Set up the equation
Let the volume of the first container be $A$ and the volume of the second container be $B$. According to the problem, $\frac{5}{6}$ of the first container's volume is equal to $\frac{3}{4}$ of the second container's volume when the water is transferred. Therefore, we can write the equa... | 1 | 1,654.5 | 1,654.5 | -1 |
In Mr. Johnson's class, 12 out of 20 students received an 'A' grade and the rest received a 'B' grade. Mrs. Smith, teaching a different class, observed that the proportion of students getting 'A' was the same. If Mrs. Smith has 30 students total, how many students received an 'A' grade? Moreover, if the same proportion... | 12 | 0.9375 | 559.875 | 563.133333 | 511 | |
In triangle \(ABC\), angle \(C\) is a right angle, and \(AC: AB = 4: 5\). A circle with its center on leg \(AC\) is tangent to the hypotenuse \(AB\) and intersects leg \(BC\) at point \(P\), such that \(BP: PC = 2: 3\). Find the ratio of the radius of the circle to leg \(BC\). | 13/20 | 0.3125 | 6,481.5625 | 4,643.8 | 7,316.909091 | |
How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$. | 4 | To solve this problem, we need to find all possible rearrangements of the string $abcd$ such that no two adjacent letters are also adjacent in the alphabet. The adjacent letters in the alphabet are $(a, b)$, $(b, c)$, and $(c, d)$. We must ensure that none of these pairs appear next to each other in any rearrangement.
... | 0 | 8,192 | -1 | 8,192 |
Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with side lengths as labeled. What is the area of the shaded $\text L$-shaped region? [asy]
/* AMC8 2000 #6 Problem */
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
draw((1,5)--(1,1)--(5,1));
draw((0,4)--(4,4)--(4,0));
fill((0,4)--(1,4)--(1,1)--(4... | 7 | 0.3125 | 7,580.0625 | 6,237.4 | 8,190.363636 | |
Given a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. One of its asymptotes is $x+\sqrt{2}y=0$. Point $M$ lies on the hyperbola, and $MF\_1 \perp x$-axis. If $F\_2$ is also a focus of the parabola $y^{2}=12x$, find the distance from $F\_1$... | \frac{6}{5} | 1 | 4,233.125 | 4,233.125 | -1 | |
Find the cross product of $\begin{pmatrix} 2 \\ 0 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 5 \\ -1 \\ 7 \end{pmatrix}.$ | \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix} | 0.9375 | 3,689.25 | 3,389.066667 | 8,192 | |
If $x = 101$ and $x^3y - 2x^2y + xy = 101000$, what is the value of $y$? | \frac{1}{10} | 0.75 | 4,108.6875 | 2,959.25 | 7,557 | |
Given the function $y=4^{x}-6\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained. | -1 | 0 | 2,526.4375 | -1 | 2,526.4375 | |
In a unit cube \( ABCD-A_1B_1C_1D_1 \), let \( O \) be the center of the square \( ABCD \). Points \( M \) and \( N \) are located on edges \( A_1D_1 \) and \( CC_1 \) respectively, with \( A_1M = \frac{1}{2} \) and \( CN = \frac{2}{3} \). Find the volume of the tetrahedron \( OMNB_1 \). | 11/72 | 0.625 | 6,738.6875 | 5,866.7 | 8,192 | |
A point $P$ is chosen in the interior of $\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the area of $\triangle ABC$.
[asy] size(200); pathp... | 144 | 0.8125 | 6,003 | 5,497.846154 | 8,192 | |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 6.\]Find the value of \[\frac{x + y}{x - y}.\] | -\sqrt{2} | 1 | 3,468.875 | 3,468.875 | -1 | |
Given the function $y=\sin x$ and $y=\sin (2x+ \frac {\pi}{3})$, determine the horizontal shift required to transform the graph of the function $y=\sin x$ into the graph of the function $y=\sin (2x+ \frac {\pi}{3})$. | \frac {\pi}{6} | 0.4375 | 5,878.6875 | 5,354.571429 | 6,286.333333 | |
You are in a place where 99% of the inhabitants are vampires and 1% are regular humans. On average, 90% of the vampires are correctly identified as vampires, and 90% of humans are correctly identified as humans. What is the probability that someone identified as a human is actually a human? | 1/12 | 0.6875 | 5,583.25 | 4,397.454545 | 8,192 | |
For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$. | 3 |
We are tasked with finding all positive integers \( n \) such that \( C(2^n + 1) = C(n) \), where \( C(k) \) denotes the sum of distinct prime divisors of the integer \( k \).
**Step 1: Understanding the function \( C(k) \)**
- The function \( C(k) \) evaluates to the sum of all distinct prime factors of \( k \).
- F... | 0.0625 | 8,164.3125 | 7,749 | 8,192 |
The parametric equation of curve $C_{1}$ is $\begin{cases} x=2+2\cos \alpha \\ y=2\sin \alpha \end{cases}$ ($\alpha$ is the parameter), with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The curve $C_{2}$: $\rho=2\cos \theta$ intersects with the polar ... | \frac{\pi}{3} | 0.0625 | 8,109 | 6,864 | 8,192 | |
A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle... | 75^\circ |
We are given a triangle \( ABC \) with a circle that touches the side \( AC \) at its midpoint \( M \), passes through the vertex \( B \), and intersects \( AB \) and \( BC \) at \( K \) and \( L \), respectively. The point \( N \) is located on the arc \( BL \) (not containing \( K \)) such that \( \angle LKN = \angl... | 0 | 8,192 | -1 | 8,192 |
Suppose there exists a convex $n$-gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of $n$. | 356 | We can't have $n=3$ since the sum of the angles must be $180^{\circ}$ but the sum of three odd numbers is odd. On the other hand, for $n=4$ we can take a quadrilateral with angle measures $83^{\circ}, 83^{\circ}, 97^{\circ}, 97^{\circ}$. The largest possible value of $n$ is 360. For larger $n$ we can't even have all an... | 0.1875 | 7,466.875 | 4,324.666667 | 8,192 |
A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Find the particle's position after $150$ moves. | (-5 \sqrt{2}, 5 + 5 \sqrt{2}) | 0 | 8,178.6875 | -1 | 8,178.6875 | |
Find the unique pair of positive integers $(a, b)$ with $a<b$ for which $$\frac{2020-a}{a} \cdot \frac{2020-b}{b}=2$$ | (505,1212) | If either $a$ or $b$ is larger than 2020, then both must be for the product to be positive. However, the resulting product would be less than 1, so this case is impossible. Now, we see that $\left(\frac{2020-a}{a}, \frac{2020-b}{b}\right)$ must be in the form $\left(\frac{x}{y}, \frac{2 y}{x}\right)$, in some order, fo... | 0 | 8,192 | -1 | 8,192 |
Let \(a, b, c, d\) be nonnegative real numbers such that \(a + b + c + d = 1\). Find the maximum value of
\[
\frac{ab}{a+b} + \frac{ac}{a+c} + \frac{ad}{a+d} + \frac{bc}{b+c} + \frac{bd}{b+d} + \frac{cd}{c+d}.
\] | \frac{1}{2} | 0 | 7,825.125 | -1 | 7,825.125 | |
Find the smallest positive integer \( n > 1 \) such that the arithmetic mean of \( 1^2, 2^2, 3^2, \cdots, n^2 \) is a perfect square. | 337 | 0.25 | 8,056.0625 | 7,648.25 | 8,192 | |
The line with equation $y = x$ is an axis of symmetry of the curve with equation
\[y = \frac{px + q}{rx + s},\]where $p,$ $q,$ $r,$ $s$ are all nonzero. Which of the following statements must hold?
(A) $p + q = 0$
(B) $p + r = 0$
(C) $p + s = 0$
(D) $q + r = 0$
(E) $q + s = 0$
(F) $r + s = 0$ | \text{(C)} | 0 | 6,515.375 | -1 | 6,515.375 | |
We have 10 points on a line A_{1}, A_{2} \cdots A_{10} in that order. Initially there are n chips on point A_{1}. Now we are allowed to perform two types of moves. Take two chips on A_{i}, remove them and place one chip on A_{i+1}, or take two chips on A_{i+1}, remove them, and place a chip on A_{i+2} and A_{i}. Find t... | 46 | We claim that n=46 is the minimum possible value of n. As having extra chips cannot hurt, it is always better to perform the second operation than the first operation, except on point A_{1}. Assign the value of a chip on point A_{i} to be i. Then the total value of the chips initially is n. Furthermore, both types of o... | 0 | 7,923.5625 | -1 | 7,923.5625 |
The inhabitants of the Isle of Concatenate use an extended alphabet of 25 letters (A through Y). Each word in their language has a maximum length of 5 letters, and every word must include the letter A at least once. How many such words are possible? | 1863701 | 0.8125 | 5,576.9375 | 4,973.461538 | 8,192 | |
A triangle has sides of lengths 6 cm and 8 cm that create a 45-degree angle between them. Calculate the length of the third side. | 5.67 | 0.25 | 3,901.5625 | 2,149 | 4,485.75 | |
What is the smallest positive value of $m$ so that the equation $18x^2 - mx + 252 = 0$ has integral solutions? | 162 | 0.0625 | 7,960.875 | 8,192 | 7,945.466667 | |
Suppose you have $6$ red shirts, $7$ green shirts, $9$ pairs of pants, $10$ blue hats, and $10$ red hats, all distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, if neither the hat nor the pants can match the shirt in color? | 1170 | 0.0625 | 8,023.9375 | 6,577 | 8,120.4 | |
Given a triangle $\triangle ABC$, where $2 \sqrt {2}(\sin ^{2}A-\sin ^{2}C)=(a-b)\sin B$, and the radius of the circumcircle is $\sqrt {2}$.
(1) Find $\angle C$;
(2) Find the maximum area of $\triangle ABC$. | \frac {3 \sqrt {3}}{2} | 0 | 7,574.5625 | -1 | 7,574.5625 | |
A triangle has three sides that are three consecutive natural numbers, and the largest angle is twice the smallest angle. The perimeter of this triangle is __________. | 15 | 0.8125 | 5,610.875 | 5,015.230769 | 8,192 | |
In an opaque bag, there are 2 red balls and 5 black balls, all identical in size and material. Balls are drawn one by one without replacement until all red balls are drawn. Calculate the expected number of draws. | \dfrac{16}{3} | 0.0625 | 8,022.125 | 5,474 | 8,192 | |
A rectangle has dimensions $8 \times 12$, and a circle centered at one of its corners has a radius of 10. Calculate the area of the union of the regions enclosed by the rectangle and the circle. | 96 + 75\pi | 0 | 8,192 | -1 | 8,192 | |
How many cubic feet are in three cubic yards? | 81 | 1 | 501.75 | 501.75 | -1 | |
Given a triangle \(ABC\) with an area of 2. Points \(P\), \(Q\), and \(R\) are taken on the medians \(AK\), \(BL\), and \(CN\) of the triangle \(ABC\) respectively, such that \(AP : PK = 1\), \(BQ : QL = 1:2\), and \(CR : RN = 5:4\). Find the area of the triangle \(PQR\). | 1/6 | 0.625 | 7,061.25 | 6,382.8 | 8,192 | |
The number of unordered pairs of edges of a given rectangular cuboid that determine a plane. | 66 | 0 | 7,274.375 | -1 | 7,274.375 | |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{4}} \left( x^{2} + 17.5 \right) \sin 2x \, dx
$$ | \frac{68 + \pi}{8} | 0.125 | 5,553.9375 | 7,617 | 5,259.214286 | |
Given $cos(\frac{π}{4}-α)=\frac{3}{5}$ and $sin(\frac{5π}{4}+β)=-\frac{12}{13}$, where $α∈(\frac{π}{4},\frac{3π}{4})$ and $β∈(0,\frac{π}{4})$, find $\frac{tanα}{tanβ}$. | -17 | 0.375 | 7,250.3125 | 5,903.166667 | 8,058.6 | |
If Kai will celebrate his 25th birthday in March 2020, in what year was Kai born? | 1995 | Kai was born 25 years before 2020 and so was born in the year $2020 - 25 = 1995$. | 0.8125 | 235.5625 | 233.846154 | 243 |
Given a right triangle $PQR$ with $\angle PQR = 90^\circ$, suppose $\cos Q = 0.6$ and $PQ = 15$. What is the length of $QR$? | 25 | 0.0625 | 5,095.4375 | 6,057 | 5,031.333333 | |
Let $z$ be a complex number such that $|z| = 13.$ Find $z \times \overline{z}.$ | 169 | 1 | 1,559.5 | 1,559.5 | -1 | |
(In the 15th Jiangsu Grade 7 First Trial) On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The building numbered $k$ has exactly $k$ (where $k=1, 2, 3, 4, 5$) workers from Factory A. The distance between two adjacent buildings is 50 meters. Factory A plans to build a station on ... | 150 | 0.25 | 7,838.625 | 6,778.5 | 8,192 | |
What is the area of the circle defined by \(x^2 - 8x + y^2 - 16y + 48 = 0\) that lies above the line \(y = 4\)? | 24\pi | 0 | 7,357 | -1 | 7,357 | |
Part of the graph of $f(x) = ax^3 + bx^2 + cx + d$ is shown. What is $b$?
[asy]
unitsize(1.5 cm);
real func(real x) {
return((x + 1)*(x - 1)*(x - 2));
}
draw(graph(func,-1.1,1.5));
draw((-1.5,0)--(1.5,0),Arrows(6));
draw((0,-1)--(0,2.5),Arrows(6));
label("$x$", (1.5,0), E);
label("$f(x)$", (0,2.5), N);
dot("$(-1... | -2 | 1 | 2,498.3125 | 2,498.3125 | -1 |
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