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 |
|---|---|---|---|---|---|---|
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6 and with their centers 13 units apart. If a plane \(\alpha\) is tangent to both spheres and intersects the cylindrical surface forming an ellipse, what is the length of the major axis of this ellipse? | 13 | 0.0625 | 8,077.5 | 6,360 | 8,192 | |
Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$ | 0 \text{ and } 1 |
To solve this problem, we need to find all real numbers \( a \) for which there exists a non-constant function \( f : \mathbb{R} \to \mathbb{R} \) satisfying the two conditions:
1. \( f(ax) = a^2f(x) \) for all \( x \in \mathbb{R} \).
2. \( f(f(x)) = a f(x) \) for all \( x \in \mathbb{R} \).
We will analyze these eq... | 0 | 7,497.375 | -1 | 7,497.375 |
For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? | 6 | To determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions, we analyze the discriminants of these quadratic equations.
1. **Analyze the discriminants:**
- For $x^2 + bx + c = 0$, the discriminant must be $\Delta_1 = b^2 - 4c... | 0.8125 | 6,661.25 | 6,308 | 8,192 |
The ratio of a geometric sequence is an integer. We know that there is a term in the sequence which is equal to the sum of some other terms of the sequence. What can the ratio of the sequence be? | -1 | 0 | 8,169.3125 | -1 | 8,169.3125 | |
David and Evan are playing a game. Evan thinks of a positive integer $N$ between 1 and 59, inclusive, and David tries to guess it. Each time David makes a guess, Evan will tell him whether the guess is greater than, equal to, or less than $N$. David wants to devise a strategy that will guarantee that he knows $N$ in fi... | 36440 | We can represent each strategy as a binary tree labeled with the integers from 1 to 59, where David starts at the root and moves to the right child if he is too low and to the left child if he is too high. Our tree must have at most 6 layers as David must guess at most 5 times. Once David has been told that he guessed ... | 0 | 7,119.1875 | -1 | 7,119.1875 |
A triangle with vertices \(A = (4, 3)\), \(B = (6, -2)\), and \(C = (7, 1)\) is reflected about the line \(x = 6\) to create a second triangle. Determine the area of the union of the two triangles. | 10 | 0 | 8,192 | -1 | 8,192 | |
The shortest distances between an interior diagonal of a rectangular parallelepiped, $P$, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$. | 750 | In the above diagram, we focus on the line that appears closest and is parallel to $BC$. All the blue lines are perpendicular lines to $BC$ and their other points are on $AB$, the main diagonal. The green lines are projections of the blue lines onto the bottom face; all of the green lines originate in the corner and re... | 0.1875 | 7,835.375 | 6,290 | 8,192 |
How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not? | 64 | 0.3125 | 6,762.375 | 3,617.2 | 8,192 | |
In a trapezoid with bases 3 and 4, find the length of the segment parallel to the bases that divides the area of the trapezoid in the ratio $5:2$, counting from the shorter base. | \sqrt{14} | 0.8125 | 6,283.4375 | 5,843 | 8,192 | |
Given a geometric sequence $\{a_n\}$, where the sum of the first $n$ terms is denoted as $S_n$, and $S_n = a\left(\frac{1}{4}\right)^{n-1} + 6$, find the value of $a$. | -\frac{3}{2} | 0.0625 | 8,124.3125 | 7,109 | 8,192 | |
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthl... | 346 | The arrangements must follow the pattern MVEMVE....... where each MVE consists of some Martians followed by some Venusians followed by some Earthlings, for $1, 2, 3, 4,$ or $5$ MVE's. If there are $k$ MVE's, then by stars and bars, there are ${4 \choose k-1}$ choices for the Martians in each block, and the same goes fo... | 0 | 8,192 | -1 | 8,192 |
A sphere is cut into four congruent wedges. The circumference of the sphere is $12\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\pi$.
Note: To measure the circumference, take the largest circle on the surface of the sphere. | 72\pi | 0.9375 | 1,421 | 969.6 | 8,192 | |
How many positive $3$-digit numbers are multiples of $20$, but not of $55$? | 41 | 0.875 | 5,207.0625 | 4,780.642857 | 8,192 | |
The price of the jacket was increased and then decreased by a certain percent, and then a 10% discount was applied, resulting in a final price that is 75% of the original price. Determine the percent by which the price was increased and then decreased. | 40.82 | 0.1875 | 5,217.5 | 4,444.333333 | 5,395.923077 | |
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number. | 505 | 0 | 8,192 | -1 | 8,192 | |
Find the product of the roots of the equation \[(3x^4 + 2x^3 - 9x + 30)(4x^3 - 20x^2 + 24x - 36) = 0.\] | -90 | 0 | 5,149.125 | -1 | 5,149.125 | |
Let \( a_{n} \) represent the closest positive integer to \( \sqrt{n} \) for \( n \in \mathbf{N}^{*} \). Suppose \( S=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{2000}} \). Determine the value of \( [S] \). | 88 | 0.4375 | 7,374 | 6,716.857143 | 7,885.111111 | |
A national team needs to select 4 out of 6 sprinters to participate in the 4×100m relay at the Asian Games. If one of them, A, cannot run the first leg, and another, B, cannot run the fourth leg, how many different methods are there to select the team? | 252 | 0.5 | 7,289 | 6,580.875 | 7,997.125 | |
A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | 758 | 0.4375 | 6,894.0625 | 6,040.428571 | 7,558 | |
What is the 7th term of an arithmetic sequence of 15 terms where the first term is 3 and the last term is 72? | 33 | 0 | 4,585.625 | -1 | 4,585.625 | |
Consider the system of equations:
\begin{align*}
8x - 5y &= a, \\
10y - 15x &= b.
\end{align*}
If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, calculate \(\frac{a}{b}\), assuming \(b\) is nonzero. | \frac{8}{15} | 0 | 8,056.25 | -1 | 8,056.25 | |
A heptagonal prism has ____ faces and ____ vertices. | 14 | 0.4375 | 424.125 | 434.142857 | 416.333333 | |
Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle. | \frac{2}{3} | 0.25 | 7,718.875 | 6,461 | 8,138.166667 | |
In three sugar bowls, there is an equal number of sugar cubes, and the cups are empty. If each cup receives $\frac{1}{18}$ of the contents of each sugar bowl, then each sugar bowl will have 12 more sugar cubes than each cup.
How many sugar cubes were originally in each sugar bowl? | 36 | 0 | 2,742.375 | -1 | 2,742.375 | |
With the same amount of a monoatomic ideal gas, two cyclic processes $1-2-3-4-1$ and $1-3-4-1$ are carried out. Find the ratio of their efficiencies. | 18/13 | 0 | 7,726.625 | -1 | 7,726.625 | |
Richard starts with the string HHMMMMTT. A move consists of replacing an instance of HM with MH , replacing an instance of MT with TM, or replacing an instance of TH with HT. Compute the number of possible strings he can end up with after performing zero or more moves. | 70 | The key claim is that the positions of the Ms fully determines the end configuration. Indeed, since all Hs are initially left of all Ts, the only successful swaps that can occur will involve Ms. So, picking $\binom{8}{4}=70$ spots for Ms and then filling in the remaining 4 spots with Hs first and then Ts gives all poss... | 0 | 8,101.875 | -1 | 8,101.875 |
How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | 60 | 0.5625 | 6,916.3125 | 5,924.111111 | 8,192 | |
Given positive real numbers \(a\) and \(b\) that satisfy \(ab(a+b) = 4\), find the minimum value of \(2a + b\). | 2\sqrt{3} | 0.4375 | 7,482.5625 | 6,570.428571 | 8,192 | |
Simplify and evaluate
(Ⅰ) Evaluate \\( \dfrac{ \sqrt{3}\sin (- \dfrac{20}{3}\pi)}{\tan \dfrac{11}{3}\pi}-\cos \dfrac{13}{4}\pi\cdot\tan (- \dfrac{35}{4}\pi) \).
(Ⅱ) Evaluate: \\( \dfrac{\sqrt{1-2\sin {10}^{\circ }\cos {10}^{\circ }}}{\cos {10}^{\circ }-\sqrt{1-{\cos }^{2}{170}^{\circ }}} \)
(Ⅲ) If \\( \sin \theta, \... | - \dfrac{ \sqrt{7}}{2} | 0 | 7,133.1875 | -1 | 7,133.1875 | |
Compute \((1+i^{-100}) + (2+i^{-99}) + (3+i^{-98}) + \cdots + (101+i^0) + (102+i^1) + \cdots + (201+i^{100})\). | 20302 | 0.25 | 6,738.75 | 5,147.25 | 7,269.25 | |
Find the center of the circle with equation $x^2 - 2x + y^2 - 4y - 28 = 0$. | (1, 2) | 1 | 1,887.8125 | 1,887.8125 | -1 | |
Given that $x$ and $y$ satisfy the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, find the minimum value of $x^2 + y^2$. | 14 - 2\sqrt{13} | 0.8125 | 5,933.1875 | 5,411.923077 | 8,192 | |
If $\Diamond4_7=\Diamond1_{8}$ and $\Diamond$ represents a digit, solve for $\Diamond$. | 3 | 1 | 1,422.625 | 1,422.625 | -1 | |
Let \( z \) be a complex number with a modulus of 1. Then the maximum value of \(\left|\frac{z+\mathrm{i}}{z+2}\right|\) is \(\ \ \ \ \ \ \). | \frac{2\sqrt{5}}{3} | 0 | 8,000.25 | -1 | 8,000.25 | |
For each natural number from 1 to 999, Damir subtracted the last digit from the first digit and wrote all the resulting 1000 differences on the board. For example, for the number 7, Damir wrote 0; for the number 105, he wrote (-4); for the number 61, he wrote 5.
What is the sum of all the numbers on the board? | 495 | 0 | 8,157.0625 | -1 | 8,157.0625 | |
Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after ... | 1641 | 0.0625 | 7,960.75 | 6,479 | 8,059.533333 | |
Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$. | 25 | 1 | 2,268.75 | 2,268.75 | -1 | |
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. | 31 | 0.375 | 7,453.25 | 6,222 | 8,192 | |
The year 2009 has the property that rearranging its digits never results in a smaller four-digit number (numbers do not start with zero). In which year will this property first repeat? | 2022 | 0.125 | 7,680.375 | 6,585.5 | 7,836.785714 | |
A \(3\times 5\) rectangle and a \(4\times 6\) rectangle need to be contained within a square without any overlapping at their interior points, and the square's sides are parallel to the sides of the given rectangles. Determine the smallest possible area of this square. | 81 | 0 | 8,192 | -1 | 8,192 | |
The horse walks a distance that is half of the previous day's distance each day, and after walking for 7 days, it has covered a total distance of 700 li. Calculate the total distance it will cover from the 8th day to the 14th day. | \frac {175}{32} | 0.5 | 7,372.625 | 7,425.875 | 7,319.375 | |
Cara is sitting at a circular table with her five friends as shown below. How many different possible pairs of people could Cara be sitting between?
[asy]
draw(circle((0,0),1));
label("$\_$",1.5dir(0));
label("$\_$",1.5dir(60));
label("Cara",1.5dir(120));
label("$\_$",1.5dir(180));
label("$\_$",1.5dir(240));
label("$\... | 10 | 0.1875 | 1,095.125 | 4,162.333333 | 387.307692 | |
In right triangle $BCD$ with $\angle D = 90^\circ$, we have $BC = 9$ and $BD = 4$. Find $\sin B$. | \frac{\sqrt{65}}{9} | 0 | 1,934.3125 | -1 | 1,934.3125 | |
A horse 24 feet from the center of a merry-go-round makes 32 revolutions. In order to travel the same distance, how many revolutions would a horse 8 feet from the center have to make? | 96 | 1 | 1,732 | 1,732 | -1 | |
How many different lines pass through at least two points in this 4-by-4 grid of lattice points? | 20 | 0 | 8,192 | -1 | 8,192 | |
One day, a group of young people came to the Platonic Academy located in the outskirts of Athens. The academy's gate was closed, and above the gate a sign read: "No one ignorant of geometry may enter!" Next to the sign was a diagram with four small rectangles of areas $20, 40, 48, \text{and } 42$ forming a larger recta... | 150 | 0.0625 | 8,031.9375 | 5,631 | 8,192 | |
If $\frac{x^2}{2^2} + \frac{y^2}{\sqrt{2}^2} = 1$, what is the largest possible value of $|x| + |y|$? | 2\sqrt{3} | 0 | 4,327.3125 | -1 | 4,327.3125 | |
Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 10000$, where the signs change after each perfect square. | 1000000 | 0 | 8,017.75 | -1 | 8,017.75 | |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is chosen at random from the region defined by the parallelogram, what is the probability that the point lies below or on the line $y = -1$? Express your answer as a common fraction. | \frac{1}{6} | 0.375 | 7,316.3125 | 5,856.833333 | 8,192 | |
Let $M$ denote the number of $8$-digit positive integers where the digits are in non-decreasing order. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed, and the digit zero can now be used.) | 310 | 0 | 6,202.75 | -1 | 6,202.75 | |
Sets $A, B$ , and $C$ satisfy $|A| = 92$ , $|B| = 35$ , $|C| = 63$ , $|A\cap B| = 16$ , $|A\cap C| = 51$ , $|B\cap C| = 19$ . Compute the number of possible values of $ |A \cap B \cap C|$ . | 10 | 0.5 | 6,305.4375 | 5,416.5 | 7,194.375 | |
Consider a parabola with vertex V and a focus F. There exists a point B on the parabola such that BF = 25 and BV = 24. Determine the sum of all possible values of the length FV. | \frac{50}{3} | 0 | 7,998.0625 | -1 | 7,998.0625 | |
Let $x, y, z$ be positive numbers satisfying the following system of equations:
$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$
Find the value of the expression $xy + yz + xz$. | 12 | 0.4375 | 7,526.375 | 6,670.571429 | 8,192 | |
Cara is sitting at a circular table with her six friends. How many different sets of two friends can Cara be directly sitting between? | 15 | 0.125 | 4,945.875 | 2,743.5 | 5,260.5 | |
In triangle \( \triangle ABC \), given that
$$
\angle A = 30^{\circ}, \quad 2 \overrightarrow{AB} \cdot \overrightarrow{AC} = 3 \overrightarrow{BC}^2,
$$
find the cosine of the largest angle of \( \triangle ABC \). | -\frac{1}{2} | 0.875 | 5,860.5625 | 5,527.5 | 8,192 | |
Given the function $f(x)=\sin \omega x (\omega > 0)$, translate the graph of this function to the left by $\dfrac{\pi}{4\omega}$ units to obtain the graph of the function $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\omega$ and is monotonically increasing in the interval $(-\omega,\omega)$, determine t... | \dfrac{\sqrt{\pi}}{2} | 0 | 6,161.0625 | -1 | 6,161.0625 | |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$ ? | 16 | 1 | 3,437.1875 | 3,437.1875 | -1 | |
A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance be... | 100 |
Let us consider the problem where Amy and Ben take turns placing stones on a 20x20 grid consisting of sites \((x, y)\) where \(x\) and \(y\) are integers between 1 and 20 inclusive. Amy's condition for placing a red stone is that the distance between any two red stones is not equal to \(\sqrt{5}\). This occurs specifi... | 0.375 | 7,519.5 | 6,490.166667 | 8,137.1 |
In the diagram, the square has a perimeter of $48$ and the triangle has a height of $48.$ If the square and the triangle have the same area, what is the value of $x?$ [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((3,0)--(6,0)--(6,5)--cycle);
draw((5.8,0)--(5.8,.2)--(6,.2));
label("$x$",(4.5,0),S);
label("48",(6,2... | 6 | 0.9375 | 3,900.1875 | 3,867.466667 | 4,391 | |
Given that the base area of a cone is $\pi$, and the lateral area is twice the base area, determine the surface area of the circumscribed sphere of the cone. | \frac{16\pi}{3} | 0.3125 | 5,583.875 | 4,085 | 6,265.181818 | |
Suppose that $a, b$ and $c$ are integers with $(x-a)(x-6)+3=(x+b)(x+c)$ for all real numbers $x$. What is the sum of all possible values of $b$? | -24 | We are told that $(x-a)(x-6)+3=(x+b)(x+c)$ for all real numbers $x$. In particular, this equation holds when $x=6$. Substituting $x=6$ gives $(6-a)(6-6)+3=(6+b)(6+c)$ or $3=(6+b)(6+c)$. Since $b$ and $c$ are integers, then $6+b$ and $6+c$ are integers, which means that $6+b$ is a divisor of 3. Therefore, the possible v... | 0.75 | 4,804.4375 | 4,291.083333 | 6,344.5 |
Given a sequence $\{a_n\}$ satisfying $a_1=1$ and $a_{n+1}= \frac {a_n}{a_n+2}$ $(n\in\mathbb{N}^*)$, find the value of $a_{10}$. | \frac {1}{1023} | 1 | 2,821.1875 | 2,821.1875 | -1 | |
$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$ , and $C$ . What is the magnitude of $\angle B AC$ in degrees? | 60 | 0.3125 | 2,987.4375 | 2,378.2 | 3,264.363636 | |
If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is: | $10\frac{2}{3}$ | 1. **Identify the Proportions**: The problem states that $64$ is divided into three parts proportional to $2$, $4$, and $6$. We can simplify these ratios by dividing each by the smallest number, which is $2$. This gives us the simplified ratio $1:2:3$.
2. **Set Up the Equations**: Let the three parts be $x$, $2x$, and... | 0 | 2,433.5625 | -1 | 2,433.5625 |
Football tickets are normally priced at $15 each. After buying 5 tickets, any additional tickets are sold at a discounted price of $12 each. If Jane has $150, what is the maximum number of tickets she can buy? | 11 | 1 | 931.75 | 931.75 | -1 | |
On a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exactly halfway between them. After some time, w... | 90 | 0 | 8,192 | -1 | 8,192 | |
Betty has a $3 \times 4$ grid of dots. She colors each dot either red or maroon. Compute the number of ways Betty can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. | 408 | First suppose no 3 by 1 row is all red or all blue. Then each row is either two red and one blue, or two blue and one red. There are 6 possible configurations of such a row, and as long as no row is repeated, there's no monochromatic rectangle This gives $6 \cdot 5 \cdot 4 \cdot 3=360$ possibilities. Now suppose we hav... | 0 | 8,056.375 | -1 | 8,056.375 |
Given $n \in \mathbb{N}^*$, in the expansion of $(x+2)^n$, the coefficient of the second term is $\frac{1}{5}$ of the coefficient of the third term.
(1) Find the value of $n$;
(2) Find the term with the maximum binomial coefficient in the expansion;
(3) If $(x+2)^n=a_0+a_1(x+1)+a_2(x+1)^2+\ldots+a_n(x+1)^n$, find... | 64 | 0.8125 | 4,600 | 4,299.461538 | 5,902.333333 | |
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, with $g(x) \neq 0$, $f(x)g'(x) > f'(x)g(x)$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \neq 1$), $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$. For the finite sequence $\frac{f(n)}{g(n)} = (n = 1, 2, \ldots, 0)$, find the probability that the su... | \frac{3}{5} | 0 | 6,489 | -1 | 6,489 | |
Given a regular tetrahedron A-BCD with an edge length of 1, and $\overrightarrow{AE} = 2\overrightarrow{EB}$, $\overrightarrow{AF} = 2\overrightarrow{FD}$, calculate $\overrightarrow{EF} \cdot \overrightarrow{DC}$. | -\frac{1}{3} | 0.875 | 5,237.375 | 4,815.285714 | 8,192 | |
If the coefficient of $x^6$ in the expansion of $(x^2-a)(x+\frac{1}{x})^{10}$ is 30, then find the value of $\int_{0}^{a} (3x^2+1) \, dx$. | 10 | 1 | 2,648.875 | 2,648.875 | -1 | |
A department needs to arrange a duty schedule for the National Day holiday (a total of 8 days) for four people: A, B, C, and D. It is known that:
- A and B each need to be on duty for 4 days.
- A cannot be on duty on the first day, and A and B cannot be on duty on the same day.
- C needs to be on duty for 3 days and c... | 700 | 0 | 8,192 | -1 | 8,192 | |
If the seven-digit number $854n526$ is divisible by $11$, what is $n$? | 5 | 1 | 2,058.625 | 2,058.625 | -1 | |
Solve the system
$$
\left\{\begin{array}{l}
x^{3}+3 y^{3}=11 \\
x^{2} y+x y^{2}=6
\end{array}\right.
$$
Calculate the values of the expression $\frac{x_{k}}{y_{k}}$ for each solution $\left(x_{k}, y_{k}\right)$ of the system and find the smallest among them. If necessary, round your answer to two decimal places. | -1.31 | 0.375 | 6,897.875 | 5,663.333333 | 7,638.6 | |
If the functions \( f(x) \) and \( g(x) \) are defined on \( \mathbf{R} \), and \( f(x-y)=f(x)g(y)-g(x)f(y) \), with \( f(-2)=f(1) \neq 0 \), what is \( g(1) + g(-1) \)? | -1 | 0.8125 | 5,397.125 | 4,752.153846 | 8,192 | |
In acute triangle $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (2, c)$ and $\overrightarrow{n} = (\frac{b}{2}\cos C - \sin A, \cos B)$, with $b = \sqrt{3}$ and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Fin... | \sqrt{3} | 0.8125 | 5,837.125 | 5,475.153846 | 7,405.666667 | |
A paper equilateral triangle of side length 2 on a table has vertices labeled \(A\), \(B\), and \(C\). Let \(M\) be the point on the sheet of paper halfway between \(A\) and \(C\). Over time, point \(M\) is lifted upwards, folding the triangle along segment \(BM\), while \(A\), \(B\), and \(C\) remain on the table. Thi... | \frac{\sqrt{3}}{6} | 0 | 8,192 | -1 | 8,192 | |
$f(x)$ is a monic polynomial such that $f(0)=4$ and $f(1)=10$. If $f(x)$ has degree $2$, what is $f(x)$? Express your answer in the form $ax^2+bx+c$, where $a$, $b$, and $c$ are real numbers. | x^2+5x+4 | 1 | 1,702.1875 | 1,702.1875 | -1 | |
The base 5 number $34x1_5$ is divisible by 31. What is the digit $x$? | 4 | 1 | 2,695.625 | 2,695.625 | -1 | |
A triangular region is bounded by the two coordinate axes and the line given by the equation $3x + y = 9$. Check if the point (1,1) lies inside this triangular region and find the region's area in square units. | \frac{27}{2} | 0.0625 | 1,843.1875 | 1,753 | 1,849.2 | |
Given the height of a cylinder is $1$, and the circumferences of its two bases are on the surface of the same sphere with a diameter of $2$, calculate the volume of the cylinder. | \dfrac{3\pi}{4} | 0.75 | 4,546.25 | 3,712.333333 | 7,048 | |
In a round-robin tournament among $8$ chess players (each pair plays one match), the scoring rules are: the winner of a match earns $2$ points, a draw results in $1$ point for each player, and the loser scores $0$ points. The final scores of the players are all different, and the score of the player in second place equ... | 12 | 0 | 8,192 | -1 | 8,192 | |
Elena drives 45 miles in the first hour, but realizes that she will be 45 minutes late if she continues at the same speed. She increases her speed by 20 miles per hour for the rest of the journey and arrives 15 minutes early. Determine the total distance from Elena's home to the convention center. | 191.25 | 0.625 | 5,973.625 | 4,853.2 | 7,841 | |
Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \). | 8/3 | 0 | 6,684.8125 | -1 | 6,684.8125 | |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the internal angles $A$, $B$, and $C$, respectively, and $\sin \left(A-B\right)\cos C=\cos B\sin \left(A-C\right)$. <br/>$(1)$ Determine the shape of triangle $\triangle ABC$; <br/>$(2)$ If triangle $\triangle ABC$ is an acute tr... | \frac{25}{16} | 0.375 | 6,999.875 | 6,152.5 | 7,508.3 | |
Point $P$ is a moving point on the parabola $y^{2}=4x$. The minimum value of the sum of the distances from point $P$ to point $A(0,-1)$ and from point $P$ to the line $x=-1$ is ______. | \sqrt{2} | 0.125 | 8,081.125 | 7,305 | 8,192 | |
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$? | 98 | 0 | 8,163.375 | -1 | 8,163.375 | |
Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three number... | 108 | Let $b$ be the finish number of Betty's peanuts. Then \[3b = 444-(5 + 9 + 25) = 405 = 3 \cdot 135 \implies b = 135, b+ 9 = 144.\]
Let $k > 1$ be the common ratio. Then \[\frac{144}{k} + 144 + k \cdot 144 = 444 \implies \frac{144}{k} + k \cdot 144 = 300\implies \frac{12}{k} + k \cdot 12 = 25\implies k = \frac{4}{3} \im... | 0.9375 | 3,483.375 | 3,169.466667 | 8,192 |
The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? | 34 | 1. **Calculate the total age of people in Room A**:
Since the average age of the 6 people in Room A is 40, the total age for Room A is calculated by multiplying the average age by the number of people:
\[
\text{Total age in Room A} = 6 \times 40 = 240.
\]
2. **Calculate the total age of people in Room B*... | 1 | 1,521.0625 | 1,521.0625 | -1 |
Triangle $A B C$ has incircle $\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\omega$ that is closer to $A$, and define $B_{3}$... | 14/65 | Notice that $A_{2}$ is the point of tangency of the excircle opposite $A$ to $B C$. Therefore, by considering the homothety centered at $A$ taking the excircle to the incircle, we notice that $A_{3}$ is the intersection of $\omega$ and the tangent line parallel to $B C$. It follows that $A_{1} B_{1} C_{1}$ is congruent... | 0 | 8,192 | -1 | 8,192 |
A $150\times 324\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes? | 768 | Consider a point travelling across the internal diagonal, and let the internal diagonal have a length of $d$. The point enters a new unit cube in the $x,y,z$ dimensions at multiples of $\frac{d}{150}, \frac{d}{324}, \frac{d}{375}$ respectively. We proceed by using PIE.
The point enters a new cube in the $x$ dimension ... | 0.75 | 4,220 | 3,780.916667 | 5,537.25 |
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds
$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ | f(x) = 2x \text{ and } f(x) = 0 |
To solve the functional equation problem, we need to identify all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the given functional equation for any real numbers \( x, y \):
\[
f(xf(y) + 2y) = f(xy) + xf(y) + f(f(y)).
\]
We will explore potential solutions by substituting specific values for \( x \) and... | 0.0625 | 8,034.1875 | 6,727 | 8,121.333333 |
A rectangle with integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area, assuming at least one of the dimensions is a prime number? | 116 | 0.375 | 5,841.1875 | 5,175.833333 | 6,240.4 | |
A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon inside it. Determine the fraction of the area of the larger octagon $ABCDEFGH$ that is enclosed by this smaller octagon. | \frac{1}{2} | 0 | 7,686.5625 | -1 | 7,686.5625 | |
A number \( a \) is randomly chosen from \( 1, 2, 3, \cdots, 10 \), and a number \( b \) is randomly chosen from \( -1, -2, -3, \cdots, -10 \). What is the probability that \( a^{2} + b \) is divisible by 3? | 37/100 | 0.625 | 7,429.75 | 7,012.8 | 8,124.666667 | |
A vessel with a capacity of 100 liters is filled with a brine solution containing 10 kg of dissolved salt. Every minute, 3 liters of water flows into it, and the same amount of the resulting mixture is pumped into another vessel of the same capacity, initially filled with water, from which the excess liquid overflows. ... | 333.33 | 0 | 7,295.625 | -1 | 7,295.625 | |
In a particular sequence, the first term is $a_1 = 1009$ and the second term is $a_2 = 1010$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \ge 1$. Determine $a_{1000}$. | 1675 | 0.125 | 8,120.375 | 7,619 | 8,192 | |
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle? | 65 | 0.75 | 5,201.75 | 4,458.583333 | 7,431.25 | |
In a square, points A and B are midpoints of two adjacent sides. A line segment is drawn from point A to the opposite vertex of the side that does not contain B, forming a triangle. What fraction of the interior of the square is shaded, if the triangle is left unshaded?
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle... | \frac{3}{4} | 0.5625 | 6,730.25 | 6,089.444444 | 7,554.142857 | |
Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.) | 550 | Recall that the arithmetic mean of all the $n$ digit palindromes is just the average of the largest and smallest $n$ digit palindromes, and in this case the $2$ palindromes are $101$ and $999$ and $\frac{101+999}{2}=\boxed{550},$ which is the final answer.
~ math31415926535 | 0.9375 | 3,697.6875 | 3,500.6 | 6,654 |
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