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 |
|---|---|---|---|---|---|---|
Given \( f(x) = \sum_{k=1}^{2017} \frac{\cos k x}{\cos^k x} \), find \( f\left(\frac{\pi}{2018}\right) \). | -1 | 0 | 8,192 | -1 | 8,192 | |
How many positive integers $N$ less than $1000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$? (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.) | 412 | 0.3125 | 7,493.3125 | 6,385 | 7,997.090909 | |
How many even divisors does \(8!\) have, and how many of those are also multiples of both 2 and 3? | 84 | 0 | 4,561.1875 | -1 | 4,561.1875 | |
Alexis imagines a $2008\times 2008$ grid of integers arranged sequentially in the following way:
\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-... | 1004 | 0.8125 | 6,508.8125 | 6,120.384615 | 8,192 | |
Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum. | 105 | 0.0625 | 7,872.375 | 7,679 | 7,885.266667 | |
If the radius of a circle is increased $100\%$, the area is increased: | 300\% | 1. **Understanding the percentage increase in radius**: The problem states that the radius of a circle is increased by $100\%$. This means the new radius is $100\%$ more than the original radius, effectively doubling it. If the original radius is $r$, then the new radius becomes $2r$.
2. **Calculating the new area**: ... | 1 | 1,350.875 | 1,350.875 | -1 |
Given the equation \(x + 11y + 11z = n\) where \(n \in \mathbf{Z}_{+}\), there are 16,653 sets of positive integer solutions \((x, y, z)\). Find the minimum value of \(n\). | 2014 | 0.5625 | 5,825.4375 | 4,321 | 7,759.714286 | |
A company decides to increase the price of a product by 20%. If they aim to increase their total income by 10% despite the decrease in demand, by what proportion must the demand decrease to meet this new income goal? | \frac{1}{12} | 0.75 | 2,639.6875 | 2,360.833333 | 3,476.25 | |
A mathematical demonstration showed that there were distinct positive integers such that $97^4 + 84^4 + 27^4 + 3^4 = m^4$. Calculate the value of $m$. | 108 | 0 | 8,192 | -1 | 8,192 | |
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\). | 86 | 0.1875 | 6,271.875 | 7,354 | 6,022.153846 | |
Let $P(n)=\left(n-1^{3}\right)\left(n-2^{3}\right) \ldots\left(n-40^{3}\right)$ for positive integers $n$. Suppose that $d$ is the largest positive integer that divides $P(n)$ for every integer $n>2023$. If $d$ is a product of $m$ (not necessarily distinct) prime numbers, compute $m$. | 48 | We first investigate what primes divide $d$. Notice that a prime $p$ divides $P(n)$ for all $n \geq 2024$ if and only if $\left\{1^{3}, 2^{3}, \ldots, 40^{3}\right\}$ contains all residues in modulo $p$. Hence, $p \leq 40$. Moreover, $x^{3} \equiv 1$ must not have other solution in modulo $p$ than 1, so $p \not \equiv ... | 0 | 8,127.75 | -1 | 8,127.75 |
A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box each time. Winning is defined as drawing tw... | \frac{6}{7} | 0.0625 | 6,860.75 | 5,430 | 6,956.133333 | |
Let $f(x)$ and $g(x)$ be nonzero polynomials such that
\[f(g(x)) = f(x) g(x).\]If $g(2) = 37,$ find $g(x).$ | x^2 + 33x - 33 | 0.25 | 7,494.9375 | 5,548.5 | 8,143.75 | |
A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476? | 1989 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given that each of the smaller rectangles has a shorter side of 7 feet and a longer side with a length three times that of the shorter side, and that two smaller rectangles are placed adjacent to each other by their longer sides and the third rectangle is placed adjacent by its long side to one of the other two rectang... | 588 | 0.0625 | 7,493.25 | 7,583 | 7,487.266667 | |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and $\sin A$, $\sin B$, and $\sin C$ form a geometric sequence. When $B$ takes the maximum value, the maximum value of $\sin A + \sin C$ is _____. | \sqrt{3} | 0.125 | 8,048.1875 | 7,041.5 | 8,192 | |
Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered... | 12 | 0 | 8,192 | -1 | 8,192 | |
In a similar mistake during multiplication, Tom reversed the digits of the two-digit number c and mistakenly multiplied the reversed number with d, resulting in the erroneous product of 143. Determine the correct product of c and d. | 341 | 0.75 | 4,897.75 | 3,987 | 7,630 | |
Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$. | 600 | By Vietas, the sum of the roots is $-a$ and the product is $b$. Therefore, both roots are nonpositive. For each value of $a$ from $1$ to $100$, the number of $b$ values is the number of ways to sum two numbers between $0$ and $a-1$ inclusive to $a$. This is just $1 + 2 + 2 + 3 + 3 +... 50 + 50 + 51 = 2600$. Thus, the a... | 0.0625 | 7,876.3125 | 7,973 | 7,869.866667 |
The natural number \( a \) is divisible by 21 and has 105 distinct divisors, including 1 and \( a \). Find the smallest such \( a \). | 254016 | 0 | 8,046 | -1 | 8,046 | |
Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed into a higher-numbered bin than the orang... | \frac{7}{16} | 0 | 8,192 | -1 | 8,192 | |
A particular integer is the smallest multiple of 72, each of whose digits is either 0 or 1. How many digits does this integer have? | 12 | 0.375 | 6,932.1875 | 4,832.5 | 8,192 | |
A positive integer $n$ is $inverosimil$ if there exists $n$ integers not necessarily distinct such that the sum and the product of this integers are equal to $n$. How many positive integers less than or equal to $2022$ are $inverosimils$? | 1010 |
We are tasked with determining how many positive integers \( n \leq 2022 \) are $inversosimil$, which means \( n \) can be expressed with \( n \) integers such that both the sum and the product of these integers equal \( n \).
To solve this problem, let's first consider the sequence of integers that can satisfy the c... | 0 | 8,177.125 | -1 | 8,177.125 |
For each permutation $a_1,a_2,a_3,\cdots,a_{10}$ of the integers $1,2,3,\cdots,10$, form the sum
\[|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
| 58 | 0.8125 | 4,621.875 | 4,202 | 6,441.333333 | |
Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$. | 506 | 0 | 8,192 | -1 | 8,192 | |
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,y+z^2+x^4&=0,z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$ , find the modulo $2007$ residue of $m$ . | 2004 | 0 | 8,112.5625 | -1 | 8,112.5625 | |
Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers \( x \). | 2\sqrt{5} | 0.5 | 7,351.625 | 6,570.625 | 8,132.625 | |
Find all real numbers $x$ so that the product $(x + i)((x + 1) + i)((x + 2) + i)$ is pure imaginary. Enter all the solutions, separated by commas. | -3,-1,1 | 0 | 3,920.8125 | -1 | 3,920.8125 | |
Let \( A \), \( B \), and \( C \) be pairwise independent events with equal probabilities, and \( A \cap B \cap C = \varnothing \). Find the maximum possible value for the probability \( \mathrm{P}(A) \). | 1/2 | 0.25 | 7,842 | 6,792 | 8,192 | |
Given a power function $y = f(x)$ whose graph passes through the point $(4, 2)$, find $f\left( \frac{1}{2} \right)$. | \frac{\sqrt{2}}{2} | 0 | 7,031 | -1 | 7,031 | |
A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$ . | 25 | 0.5 | 6,079.875 | 4,852.875 | 7,306.875 | |
Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $b.$
[asy]import TrigMacros;
size(400);
real f(real x)
{
return 2*sin(3*x + pi) + 1;
}
draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-4,4,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);... | 3 | 1 | 4,765 | 4,765 | -1 | |
In a right cone with a base radius of \(15\) cm and a height of \(30\) cm, a sphere is inscribed. The radius of the sphere can be expressed as \(b\sqrt{d} - b\) cm. Find the value of \(b + d\). | 12.5 | 0 | 8,174.625 | -1 | 8,174.625 | |
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls. | 15 | 0.625 | 5,311.5 | 4,810.1 | 6,147.166667 | |
What percent of the square $EFGH$ is shaded? All angles in the diagram are right angles, and the side length of the square is 8 units. In this square:
- A smaller square in one corner measuring 2 units per side is shaded.
- A larger square region, excluding a central square of side 3 units, occupying from corners (2,2)... | 17.1875\% | 0.1875 | 4,314.25 | 3,285.666667 | 4,551.615385 | |
Given that $0 < α < \frac{π}{2}$ and $\frac{π}{2} < β < π$, with $\cos(α + \frac{π}{4}) = \frac{1}{3}$ and $\cos(\frac{π}{4} - \frac{β}{2}) = \frac{\sqrt{3}}{3}$,
1. Find the value of $\cos β$;
2. Find the value of $\cos(2α + β)$. | -1 | 0 | 7,656.375 | -1 | 7,656.375 | |
Find all the roots of $\left(x^{2}+3 x+2\right)\left(x^{2}-7 x+12\right)\left(x^{2}-2 x-1\right)+24=0$. | 0, 2, 1 \pm \sqrt{6}, 1 \pm 2 \sqrt{2} | We re-factor as $(x+1)(x-3)(x+2)(x-4)\left(x^{2}-2 x-1\right)+24$, or $\left(x^{2}-2 x-3\right)\left(x^{2}-2 x-8\right)\left(x^{2}-2 x-1\right)+24$, and this becomes $(y-4)(y-9)(y-2)+24$ where $y=(x-1)^{2}$. Now, $(y-4)(y-9)(y-2)+24=(y-8)(y-6)(y-1)$, so $y$ is 1, 6, or 8. Thus the roots of the original polynomial are $... | 0 | 8,026.875 | -1 | 8,026.875 |
In the diagram, \(ABCD\) is a square with a side length of \(8 \, \text{cm}\). Point \(E\) is on \(AB\) and point \(F\) is on \(DC\) so that \(\triangle AEF\) is right-angled at \(E\). If the area of \(\triangle AEF\) is \(30\%\) of the area of \(ABCD\), what is the length of \(AE\)? | 4.8 | 1 | 4,075.375 | 4,075.375 | -1 | |
What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\in\mathbb{N}$ can take? | 3 |
We need to find the smallest value of the sum of the digits of the expression \( 3n^2 + n + 1 \) for \( n \in \mathbb{N} \).
To determine this, we will explore values of \( n \) and calculate the resulting expression \( 3n^2 + n + 1 \), then find the sum of its digits. Our goal is to find the smallest possible sum.
... | 0.375 | 7,462.75 | 6,247.333333 | 8,192 |
$\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\\ ... | s^2 \le 8r^2 | 0 | 4,706.6875 | -1 | 4,706.6875 | |
The minimum value of the function $y=|x-1|$ is 0, the minimum value of the function $y=|x-1|+|x-2|$ is 1, and the minimum value of the function $y=|x-1|+|x-2|+|x-3|$ is 2. What is the minimum value of the function $y=|x-1|+|x-2|+\ldots+|x-10|$? | 25 | 0.875 | 4,783.75 | 4,296.857143 | 8,192 | |
I have two 12-sided dice, each with 3 maroon sides, 4 teal sides, 4 cyan sides, and one sparkly side. If I roll both dice simultaneously, what is the probability that they will display the same color? | \frac{7}{24} | 1 | 2,828.25 | 2,828.25 | -1 | |
Compute $\arctan (\tan 65^{\circ}-2 \tan 40^{\circ})$. (Express your answer in degrees as an angle between $0^{\circ}$ and $180^{\circ}$.) | 25^{\circ} | First Solution: We have $\tan 65^{\circ}-2 \tan 40^{\circ}=\cot 25^{\circ}-2 \cot 50^{\circ}=\cot 25^{\circ}-\frac{\cot ^{2} 25^{\circ}-1}{\cot 25^{\circ}}=\frac{1}{\cot 25^{\circ}}=\tan 25^{\circ}$. Therefore, the answer is $25^{\circ}$. Second Solution: We have $\tan 65^{\circ}-2 \tan 40^{\circ}=\frac{1+\tan 20^{\cir... | 0.875 | 5,382.5 | 4,981.142857 | 8,192 |
The second hand on a clock is 8 cm long. How far in centimeters does the tip of the second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$. | 720\pi | 1 | 1,540.5 | 1,540.5 | -1 | |
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? | 8 | 1. **Calculate the area of the smaller circle (red area):**
The diameter of the smaller circle is $1$ inch, so its radius is $\frac{1}{2}$ inch. The area of a circle is given by the formula $A = \pi r^2$. Therefore, the area of the smaller circle is:
\[
A_{\text{red}} = \pi \left(\frac{1}{2}\right)^2 = \frac{\... | 0.5 | 3,041.0625 | 2,071.875 | 4,010.25 |
Consider a modified sequence whose $n$th term is defined by $a_n = (-1)^n \cdot \lfloor \frac{3n+1}{2} \rfloor$. What is the average of the first $150$ terms of this sequence? | -\frac{1}{6} | 0 | 5,140.875 | -1 | 5,140.875 | |
Horizontal parallel segments $A B=10$ and $C D=15$ are the bases of trapezoid $A B C D$. Circle $\gamma$ of radius 6 has center within the trapezoid and is tangent to sides $A B, B C$, and $D A$. If side $C D$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $A B C D$. | \frac{225}{2} | Suppose that the center of the circle is $O$ and the circle intersects $C D$ at $X$ and $Y$. Since $\angle X O Y=120^{\circ}$ and triangle $X O Y$ is isosceles, the distance from $O$ to $X Y$ is $6 \cdot \sin \left(30^{\circ}\right)=3$. On the other hand, the distance from $O$ to $A B$ is 6 as the circle is tangent to ... | 0 | 8,173 | -1 | 8,173 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\begin{cases}x=2\cos \alpha \\ y=\sin \alpha \\ \end{cases}$ (where $\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.
1. Find the polar coordin... | \frac{4}{5} | 0.4375 | 7,193.9375 | 5,910.714286 | 8,192 | |
Given that the left focus of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ is $F$, the eccentricity is $\frac{\sqrt{2}}{2}$, and the distance between the left intersection point of the ellipse with the $x$-axis and point $F$ is $\sqrt{2} - 1$.
(I) Find the equation of the ellipse;
(II) The lin... | \frac{3}{2} | 0.8125 | 6,337.4375 | 6,007.230769 | 7,768.333333 | |
Four people, A, B, C, and D, stand in a line from left to right and are numbered 1, 2, 3, and 4 respectively. They have the following conversation:
A: Both people to my left and my right are taller than me.
B: Both people to my left and my right are shorter than me.
C: I am the tallest.
D: There is no one to my right.... | 2314 | 0 | 7,513.9375 | -1 | 7,513.9375 | |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ with distinct terms, given that $a_3a_5=3a_7$, and $S_3=9$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Let $T_n$ be the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$, find the m... | \frac{1}{16} | 0.875 | 6,047.75 | 5,741.428571 | 8,192 | |
Convert $\rm{A}03_{16}$ to a base 10 integer, where the 'digits' A through F represent the values 10, 11, 12, 13, 14, and 15 in order. | 2563 | 0.8125 | 3,586.625 | 2,523.846154 | 8,192 | |
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\frac{12}5\sqrt2$. What is the volume of the tetrahedron? | \frac{24}{5} | 1. **Identify the Midpoint and Lengths**:
Let the midpoint of $CD$ be $E$. Since $CD = \frac{12}{5}\sqrt{2}$, we have $CE = \frac{1}{2}CD = \frac{6}{5}\sqrt{2}$.
2. **Use the Pythagorean Theorem in $\triangle ACE$ and $\triangle BCE$**:
- For $\triangle ACE$, we have $AC = 3$ and $CE = \frac{6}{5}\sqrt{2}$. By t... | 0.5625 | 6,987.375 | 6,050.444444 | 8,192 |
When the vectors $\begin{pmatrix} -5 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}.$ | \begin{pmatrix} -34/53 \\ 119/53 \end{pmatrix} | 0 | 5,763.375 | -1 | 5,763.375 | |
Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this? | 16 | 0 | 8,192 | -1 | 8,192 | |
Find all the roots of the equation
\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]
(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))
In the answer, specify the sum of the found roots. | 21 | 0.125 | 7,348.75 | 5,460.5 | 7,618.5 | |
The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\omega$ is needed to rotate the vessel around its axis so that the water overflows from the top of the vessel? | 9.9 | 0 | 7,886.875 | -1 | 7,886.875 | |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) with its left focus $F$ and right vertex $A$, upper vertex $B$. If the distance from point $F$ to line $AB$ is $\frac{5\sqrt{14}}{14}b$, find the eccentricity of the ellipse. | \frac{2}{3} | 0.875 | 4,776.3125 | 4,288.357143 | 8,192 | |
Find all roots of the polynomial $x^3-5x^2+3x+9$. Enter your answer as a list of numbers separated by commas. If a root occurs more than once, enter it as many times as its multiplicity. | -1,3,3 | 0.625 | 1,787.4375 | 1,896.2 | 1,606.166667 | |
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician l... | 12 |
Given the problem, let's denote the three boxes as \( R \) (red), \( W \) (white), and \( B \) (blue). Each box must contain at least one card, and the numbers on the cards range from 1 to 100. The magician must be able to determine the box from which no card has been drawn using only the sum of the numbers on the two... | 0 | 7,947.0625 | -1 | 7,947.0625 |
How many 5-digit numbers have at least one zero? | 30,951 | 0 | 7,166.0625 | -1 | 7,166.0625 | |
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$ . We have such strange rules:
(1) Define $P(n)$ : the number of prime numbers that are smaller than $n$ . Only when $P(i)\equiv j\pmod7$ , wolf $i$ may eat sheep $j$ (he can also choose not to eat the s... | 288 | 0 | 8,121.5625 | -1 | 8,121.5625 | |
The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face? | 16 | 0.125 | 7,970.6875 | 6,421.5 | 8,192 | |
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the poin... | 25 | Suppose that there are $n$ players participating in the tournament. We break this up into a group of the weakest ten, and the other $n-10$ people. Note that the $10$ players who played each other generated a total of $\dbinom{10}{2} = 45$ points playing each other. Thus, they earned $45$ playing the $n-10$ other people... | 0.0625 | 7,998.3125 | 5,093 | 8,192 |
A recent report about the amount of plastic created in the last 65 years stated that the 8.3 billion tonnes produced is as heavy as 25000 Empire State Buildings in New York or a billion elephants. On that basis, how many elephants have the same total weight as the Empire State Building? | 40000 | 0 | 529.9375 | -1 | 529.9375 | |
The integers $a$ , $b$ , $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$ . Determine the largest possible value of $d$ , | 2016 | 0.0625 | 8,159.625 | 7,674 | 8,192 | |
Given that the line $y=x-2$ intersects the hyperbola $C: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a>0, b>0)$ at the intersection points $A$ and $B (not coincident)$. The perpendicular bisector of line segment $AB$ passes through the point $(4,0)$. The eccentricity of the hyperbola $C$ is ______. | \frac{2\sqrt{3}}{3} | 0 | 3,826.8125 | -1 | 3,826.8125 | |
Let $A B C D$ be a rectangle with $A B=3$ and $B C=7$. Let $W$ be a point on segment $A B$ such that $A W=1$. Let $X, Y, Z$ be points on segments $B C, C D, D A$, respectively, so that quadrilateral $W X Y Z$ is a rectangle, and $B X<X C$. Determine the length of segment $B X$. | $\frac{7-\sqrt{41}}{2}$ | We note that $$\angle Y X C=90-\angle W X B=\angle X W B=90-\angle A W Z=\angle A Z W$$ gives us that $X Y C \cong Z W A$ and $X Y Z \sim W X B$. Consequently, we get that $Y C=A W=1$. From $X Y Z \sim W X B$, we get that $$\frac{B X}{B W}=\frac{C Y}{C X} \Rightarrow \frac{B X}{2}=\frac{1}{7-B X}$$ from which we get $$... | 0 | 7,442.75 | -1 | 7,442.75 |
12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn? | 1320 | 1 | 1,653.0625 | 1,653.0625 | -1 | |
In $\triangle ABC$, it is known that $\cos C+(\cos A- \sqrt {3}\sin A)\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $\sin (A- \frac {π}{3})= \frac {3}{5}$, find $\sin 2C$. | \frac {24+7 \sqrt {3}}{50} | 0 | 6,113.4375 | -1 | 6,113.4375 | |
On Monday, Mukesh travelled \(x \mathrm{~km}\) at a constant speed of \(90 \mathrm{~km} / \mathrm{h}\). On Tuesday, he travelled on the same route at a constant speed of \(120 \mathrm{~km} / \mathrm{h}\). His trip on Tuesday took 16 minutes less than his trip on Monday. What is the value of \(x\)? | 96 | We recall that time \(=\frac{\text { distance }}{\text { speed }}\). Travelling \(x \mathrm{~km}\) at \(90 \mathrm{~km} / \mathrm{h}\) takes \(\frac{x}{90}\) hours. Travelling \(x \mathrm{~km}\) at \(120 \mathrm{~km} / \mathrm{h}\) takes \(\frac{x}{120}\) hours. We are told that the difference between these lengths of ... | 1 | 1,981.6875 | 1,981.6875 | -1 |
Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's? | \frac{1}{80} | To solve this problem, we need to count the number of valid assignments of numbers to Al, Bill, and Cal such that Al's number is a multiple of Bill's number, and Bill's number is a multiple of Cal's number. We then divide this count by the total number of ways to assign the numbers.
1. **Total number of ways to assign... | 0.5625 | 7,257.125 | 6,530 | 8,192 |
The probability that Asha will lose a game is $\frac{4}{9}$. If it is impossible to tie, what is the probability that Asha will win the game? Express your answer as a common fraction. | \frac{5}{9} | 1 | 988.125 | 988.125 | -1 | |
In the triangle shown, what is the positive difference between the greatest and least possible integral values of $x$?
[asy]
defaultpen(linewidth(0.7));
pair a,b,c;
b = (1,2); c = (3,0);
draw(a--b--c--a);
label("$x$",a--b,dir(135)); label("5",b--c,dir(45)); label("6",c--a,S);
[/asy] | 8 | 1 | 2,147.5625 | 2,147.5625 | -1 | |
In triangle $BCD$, $\angle C = 90^\circ$, $CD = 3$, and $BD = \sqrt{13}$. What is $\tan B$? | \frac{3}{2} | 1 | 2,267.9375 | 2,267.9375 | -1 | |
A solid is formed by rotating a triangle with sides of lengths 3, 4, and 5 around the line containing its shortest side. Find the surface area of this solid. | 36\pi | 0.3125 | 6,028.3125 | 5,625.4 | 6,211.454545 | |
The quadratic equations \(x^{2} + px + q\) and \(x^{2} + ax + b\) each have one root. Among the numbers \(p, q, a, b\) there are 16, 64, and 1024. What can the fourth number be?
If there are multiple possible answers, input the larger one into the system, and specify all of them in the written solution. | 262144 | 0.0625 | 7,799.6875 | 4,840 | 7,997 | |
Suppose that \(x_1+1=x_2+2=x_3+3=\cdots=x_{2010}+2010=x_1+x_2+x_3+\cdots+x_{2010}+2011\). Find the value of \(\left\lfloor|T|\right\rfloor\), where \(T=\sum_{n=1}^{2010}x_n\). | 1005 | 0 | 5,062.9375 | -1 | 5,062.9375 | |
Given: $$\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\in\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}$
Find the value of $a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n}a_{n}$. | 81 | 0.9375 | 2,358.1875 | 2,334.466667 | 2,714 | |
A merchant placed on display some dresses, each with a marked price. He then posted a sign "$1/3$ off on these dresses." The cost of the dresses was $3/4$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was: | \frac{1}{2} | 1. **Define Variables:**
Let $x$ be the marked price of the dresses.
2. **Calculate the Selling Price:**
The dresses are sold at $\frac{1}{3}$ off the marked price. Therefore, the selling price is:
\[
\text{Selling Price} = x - \frac{1}{3}x = \frac{2}{3}x
\]
3. **Calculate the Cost Price:**
The cost... | 0.8125 | 3,106.8125 | 2,465.846154 | 5,884.333333 |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, let $P$ be a point on the ellipse such that the projection of $P$ onto the $x$-axis is the left focus $F_1$. Let $A$ be the intersection point of the ellipse with the positive semi-major axis, and let $B$ be the intersection point of the ellipse with... | \frac{\sqrt{2}}{2} | 0 | 3,787.8125 | -1 | 3,787.8125 | |
A sphere with volume $V$ is inside a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$. Find the maximum value of $V$. | \frac{9\pi}{2} | 0 | 7,145.25 | -1 | 7,145.25 | |
Given that real numbers \( x \) and \( y \) satisfy the equation \( 1 + \cos^{2}(x + y - 1) = \frac{x^{2} + y^{2} + 2(x + 1)(1 - y)}{x - y + 1} \), find the minimum value of \( xy \). | \frac{1}{4} | 0.6875 | 6,769.6875 | 6,123.181818 | 8,192 | |
What is the smallest five-digit number divisible by 4 that can be formed with the digits 1, 2, 3, 4, and 9? | 13492 | 0.3125 | 7,784.125 | 6,886.8 | 8,192 | |
A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is $12$ m. What is the area (in $m^2$ ) of the room?
 $\frac{1}{2}$
B) $\... | \frac{35}{66} | 0 | 6,729.1875 | -1 | 6,729.1875 | |
In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$? | 20 | We work from right to left as we would if doing this calculation by hand. In the units column, we have $L-1$ giving 1. Thus, $L=2$. (There is no borrowing required.) In the tens column, we have $3-N$ giving 5. Since 5 is larger than 3, we must borrow from the hundreds column. Thus, $13-N$ gives 5, which means $N=8$. In... | 0 | 8,185.75 | -1 | 8,185.75 |
Given the function $f(x)= e^{-2x}+1$, find the area of the triangle formed by the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ and the lines $y=0$ and $y=x$. | \frac{1}{3} | 0.8125 | 5,734 | 5,166.769231 | 8,192 | |
By partial fractions,
\[\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4}\]for some constants $A,$ $B,$ $C,$ $D,$ and $E.$ Find $A + B + C + D + E.$ | 0 | 0.5625 | 6,022.25 | 4,334.666667 | 8,192 | |
Find the smallest natural number that consists of identical digits and is divisible by 18. | 666 | 0.625 | 6,447.5625 | 5,400.9 | 8,192 | |
Triangle $ABC$ has a right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$.
[asy]
unitsize(0.2 cm);
pair A, B, C, P;
A = (0,14);
B = (0,0);
C = (21*sqrt(3),0);
P = intersectionpoint(arc(B,6,0,180),arc(C,33,0,180));
draw(A--B--C--cycle);
dra... | 33 | 0.8125 | 5,639.0625 | 5,049.923077 | 8,192 | |
What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect square? | 11 | 1 | 2,625.6875 | 2,625.6875 | -1 | |
Find the sum of the first eight prime numbers that have a units digit of 3. | 394 | 0.8125 | 3,128.4375 | 2,663.692308 | 5,142.333333 | |
Find the largest prime factor of $-x^{10}-x^{8}-x^{6}-x^{4}-x^{2}-1$, where $x=2 i$, $i=\sqrt{-1}$. | 13 | 13. | 0.9375 | 3,838.5 | 3,548.266667 | 8,192 |
According to the standard convention for exponentiation,
\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible? | 1 | To solve this problem, we need to evaluate the expression $2^{2^{2^{2}}}$ with different parenthesizations and determine how many distinct values can be obtained.
The expression $2^{2^{2^{2}}}$ can be parenthesized in the following ways:
1. $2^{(2^{(2^2)})}$
2. $2^{((2^2)^2)}$
3. $((2^2)^2)^2$
4. $(2^{(2^2)})^2$
5. $(... | 0.625 | 5,905.375 | 4,533.4 | 8,192 |
Given $-\frac{\pi}{2} < x < 0$, $\sin x + \cos x = \frac{1}{5}$.
(1) Find the value of $\sin x - \cos x$.
(2) Find the value of $\tan x$. | -\frac{3}{4} | 1 | 2,407.625 | 2,407.625 | -1 | |
What is $\dbinom{n}{1}$ for any positive integer $n$? | n | 1 | 1,619.5 | 1,619.5 | -1 | |
Given that $a$ and $b$ are positive real numbers satisfying $9a^{2}+b^{2}=1$, find the maximum value of $\frac{ab}{3a+b}$. | \frac{\sqrt{2}}{12} | 0 | 7,942.875 | -1 | 7,942.875 |
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