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 the ellipse $\frac {x^{2}}{9} + \frac {y^{2}}{4} = 1$, and the line $L: x + 2y - 10 = 0$.
(1) Does there exist a point $M$ on the ellipse for which the distance to line $L$ is minimal? If so, find the coordinates of point $M$ and the minimum distance.
(2) Does there exist a point $P$ on the ellipse for which th... | 3\sqrt {5} | 0 | 5,935.3125 | -1 | 5,935.3125 | |
For any positive integer \( n \), let \( f(n) = 70 + n^2 \) and let \( g(n) \) be the greatest common divisor (GCD) of \( f(n) \) and \( f(n+1) \). Find the greatest possible value of \( g(n) \). | 281 | 0.6875 | 5,571.125 | 4,379.818182 | 8,192 | |
Simplify and then evaluate the expression:
$$( \frac {x}{x-1}- \frac {x}{x^{2}-1})÷ \frac {x^{2}-x}{x^{2}-2x+1}$$
where $$x= \sqrt {2}-1$$ | 1- \frac { \sqrt {2}}{2} | 0 | 4,738.125 | -1 | 4,738.125 | |
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$. | \frac{1}{6} | To solve the problem, we need to find the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$. This condition implies that $\lfloor\log_{10}4x\rfloor = \lfloor\log_{10}x\rfloor$.
1. **Understanding the Floor Function and Logarithm:**
- The expression $\lfloor\log_{10}4x\rfloor = \lfloor\log_... | 0.4375 | 7,728.5 | 7,132.571429 | 8,192 |
Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored? | 402 | 0.0625 | 8,144.75 | 8,035 | 8,152.066667 | |
Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of fruit per day for a week, and must consume at least one apple before any orange, how many different orders can Cory eat these fruits? The fruits within each category are indistinguishable. | 105 | 0 | 7,370.5625 | -1 | 7,370.5625 | |
Find the minimum value of the function \( f(x)=\cos 4x + 6\cos 3x + 17\cos 2x + 30\cos x \) for \( x \in \mathbb{R} \). | -18 | 0.75 | 6,460.3125 | 5,948.5 | 7,995.75 | |
Let \(p\) be a prime number and \(\mathbb{F}_{p}\) be the field of residues modulo \(p\). Let \(W\) be the smallest set of polynomials with coefficients in \(\mathbb{F}_{p}\) such that the polynomials \(x+1\) and \(x^{p-2}+x^{p-3}+\cdots+x^{2}+2x+1\) are in \(W\), and for any polynomials \(h_{1}(x)\) and \(h_{2}(x)\) i... | p! | Note that both of our polynomials are bijective functions on \(\mathbb{F}_{p}\): \(f_{1}(x)=x+1\) is the cycle \(0 \rightarrow 1 \rightarrow 2 \rightarrow \cdots \rightarrow(p-1) \rightarrow 0\) and \(f_{2}(x)=x^{p-2}+x^{p-3}+\cdots+x^{2}+2x+1\) is the transposition \(0 \leftrightarrow 1\) (this follows from the formul... | 0 | 7,936.0625 | -1 | 7,936.0625 |
We roll five dice, each a different color. In how many ways can the sum of the rolls be 11? | 205 | 0.125 | 7,943.8125 | 6,537.5 | 8,144.714286 | |
For how many four-digit whole numbers does the sum of the digits equal $30$? | 20 | 0 | 7,965.125 | -1 | 7,965.125 | |
Let \\(f(x)=a(x-5)^{2}+6\ln x\\), where \\(a\in\mathbb{R}\\), the tangent line of the curve \\(y=f(x)\\) at point \\((1,f(1))\\) intersects the \\(y\\)-axis at point \\((0,6)\\).
\\((1)\\) Determine the value of \\(a\\);
\\((2)\\) Find the intervals of monotonicity and the extremum of the function \\(f(x)\\). | 2+6\ln 3 | 0.0625 | 5,244.375 | 4,551 | 5,290.6 | |
The altitude \(AH\) and the angle bisector \(CL\) of triangle \(ABC\) intersect at point \(O\). Find the angle \(BAC\) if it is known that the difference between the angle \(COH\) and half of the angle \(ABC\) is \(46^\circ\). | 92 | 0.0625 | 8,016.1875 | 5,379 | 8,192 | |
A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.
[asy] import three; import solids; size(5cm); currentprojection=o... | 384 | We can use the same method as in Solution 2 to find the side length of the equilateral triangle, which is $4\sqrt3$. From here, its area is \[\dfrac{\bigl(4\sqrt3\bigr)^2\sqrt3}4=12\sqrt3.\] The leg of the isosceles right triangle is $\dfrac{4\sqrt3}{\sqrt2}=2\sqrt6$, and the horizontal distance from the vertex to the ... | 0 | 8,172.625 | -1 | 8,172.625 |
In a triangle \( \triangle ABC \), \(a\), \(b\), and \(c\) are the sides opposite to angles \(A\), \(B\), and \(C\) respectively, with \(B= \dfrac {2\pi}{3}\). If \(a^{2}+c^{2}=4ac\), then find the value of \( \dfrac {\sin (A+C)}{\sin A\sin C} \). | \dfrac{10\sqrt{3}}{3} | 0 | 7,124.8125 | -1 | 7,124.8125 | |
One of the mascots for the 2012 Olympic Games is called 'Wenlock' because the town of Wenlock in Shropshire first held the Wenlock Olympian Games in 1850. How many years ago was that?
A) 62
B) 152
C) 158
D) 162
E) 172 | 162 | 0.375 | 278.125 | 248 | 296.2 | |
Suppose that \(\begin{array}{c} a \\ b \\ c \end{array}\) means $a+b-c$.
For example, \(\begin{array}{c} 5 \\ 4 \\ 6 \end{array}\) is $5+4-6 = 3$.
Then the sum \(\begin{array}{c} 3 \\ 2 \\ 5 \end{array}\) + \(\begin{array}{c} 4 \\ 1 \\ 6 \end{array}\) is | 1 | 1. **Interpret the given symbol**: According to the problem, the symbol
\[
\triangle
\]
represents the operation $a + b - c$ where $a$, $b$, and $c$ are the numbers at the vertices of the triangle.
2. **Apply the operation to the first triangle**: The first triangle has numbers 1, 3, and 4. Using the ope... | 0 | 1,791.125 | -1 | 1,791.125 |
When a student multiplied the number $66$ by the repeating decimal, \(1.\overline{ab}\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-digit number $ab$? | 75 | 1. **Understanding the Problem:**
We are given a repeating decimal $1.\overline{ab}$, where $a$ and $b$ are digits, and the student mistakenly multiplied $66$ by $1.ab$ instead of $1.\overline{ab}$. The error in the calculation resulted in an answer that was $0.5$ less than the correct answer. We need to find the tw... | 0.9375 | 3,422.25 | 3,104.266667 | 8,192 |
The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the len... | \frac{1}{2} | 0.0625 | 8,103.125 | 6,770 | 8,192 | |
What is the sum of all four-digit integers from 1000 to 5000, where each number increases by 4 from the previous one? | 3003000 | 0.875 | 4,072.625 | 3,484.142857 | 8,192 | |
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v$, $w$, $x$, $y$, and $z$. Find $y+z$.
[asy]
path a=(0,0)--(1,0)--(1,1)--(0,1)--cycle;
for (int i=0; i<3; ++i) {
for (int j=0; j<3; ++j) {
draw(shift((i,j))*a);
};}
label("25",(... | 46 | 0.875 | 3,571.4375 | 2,911.357143 | 8,192 | |
Find $x$ such that $\log_x 81=\log_2 16$. | 3 | 1 | 1,664.3125 | 1,664.3125 | -1 | |
A belt is placed without slack around two non-crossing circular pulleys which have radii of $15$ inches and $5$ inches respectively. The distance between the points where the belt contacts the pulleys is $30$ inches. Determine the distance between the centers of the two pulleys.
A) $20$ inches
B) $10\sqrt{10}$ inches
C... | 10\sqrt{10} | 0 | 2,488.1875 | -1 | 2,488.1875 | |
The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$? | \sqrt{3} | 0.3125 | 7,936.375 | 7,374 | 8,192 | |
A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$ , and $L$ . What is the smallest possible integer value for $L$ ? | 21 | 0.8125 | 5,927.1875 | 5,404.538462 | 8,192 | |
Given a positive integer \( n \) (\( n \geqslant 6 \)), let set \( A \) be composed of sums of any 5 consecutive positive integers not greater than \( n \), and let set \( B \) be composed of sums of any 6 consecutive positive integers not greater than \( n \). If the number of elements in the intersection of sets \( A... | 12106 | 0.0625 | 7,283.9375 | 4,768 | 7,451.666667 | |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and the altitude on side $BC$ is $\frac{a}{2}$. Determine the maximum value of $\frac{c}{b}$. | \sqrt{2} + 1 | 0 | 7,772.5625 | -1 | 7,772.5625 | |
Convert the binary number $111011001001_{(2)}$ to its corresponding decimal number. | 3785 | 0.6875 | 7,003.6875 | 6,463.545455 | 8,192 | |
A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the third prize is more than that for the second p... | 2700 | 0.125 | 8,021.625 | 6,829 | 8,192 | |
If four distinct positive integers $a$, $b$, $c$, $d$ satisfy $\left(6-a\right)\left(6-b\right)\left(6-c\right)\left(6-d\right)=9$, then $a+b+c+d$ is ______. | 24 | 0.9375 | 4,372.25 | 4,117.6 | 8,192 | |
The number of minutes in a week is closest to: | 10000 | 0.125 | 6,890.5625 | 3,507.5 | 7,373.857143 | |
Given that a rectangle with length $3x$ inches and width $x + 5$ inches has the property that its area and perimeter have equal values, what is $x$? | 1 | 1 | 1,976.5 | 1,976.5 | -1 | |
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ( $0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,... | 64 | 0 | 8,142.8125 | -1 | 8,142.8125 | |
What is the least positive integer with exactly $12$ positive factors? | 96 | 0 | 4,010.3125 | -1 | 4,010.3125 | |
A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median m of the list is a member of the list. If the list member m were replaced by m+10, the mean and median of the new list would be 24 and m+10, respectively. If m were instead replaced by m-8, the median of the new list wou... | 20 | 1. **Understanding the problem and setting up equations:**
Let the number of integers in the list be $n$. The list has a mean of $22$, so the sum of all integers in the list is $22n$.
2. **Analyzing the effect of replacing $m$ with $m+10$:**
When $m$ is replaced by $m+10$, the sum of the integers becomes $22n + ... | 0.0625 | 8,032.25 | 5,636 | 8,192 |
A projection takes $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ to $\begin{pmatrix} \frac{3}{2} \\ -\frac{3}{2} \end{pmatrix}.$ Which vector does the projection take $\begin{pmatrix} -4 \\ 1 \end{pmatrix}$ to? | \begin{pmatrix} -5/2 \\ 5/2 \end{pmatrix} | 0 | 5,114.25 | -1 | 5,114.25 | |
Find the largest \( n \) such that the sum of the fourth powers of any \( n \) prime numbers greater than 10 is divisible by \( n \). | 240 | 0.0625 | 8,009.75 | 7,770 | 8,025.733333 | |
A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at ... | 23 | 0 | 6,008.5625 | -1 | 6,008.5625 | |
The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\circ$ and the common difference being $5^\circ$. Determine the number of sides $n$ for the polygon. | n = 9 | 0.9375 | 3,552.4375 | 3,243.133333 | 8,192 | |
One digit of the decimal representation of $\frac{5}{7}$ is randomly selected. What is the probability that the digit is a 4? Express your answer as a common fraction. | \frac{1}{6} | 1 | 1,784.3125 | 1,784.3125 | -1 | |
In each cell of a $15 \times 15$ table, the number $-1, 0,$ or $+1$ is written such that the sum of the numbers in any row is nonpositive and the sum of the numbers in any column is nonnegative. What is the minimum number of zeros that can be written in the cells of the table? | 15 | 0.125 | 8,086.9375 | 7,554 | 8,163.071429 | |
When two fair 8-sided dice (labeled from 1 to 8) are tossed, the numbers \(a\) and \(b\) are obtained. What is the probability that the two-digit number \(ab\) (where \(a\) and \(b\) are digits) and both \(a\) and \(b\) are divisible by 4? | \frac{1}{16} | 1 | 3,227.375 | 3,227.375 | -1 | |
Evaluate \[\frac{3}{\log_8{5000^4}} + \frac{2}{\log_9{5000^4}},\] giving your answer as a fraction in lowest terms. | \frac{1}{4} | 0 | 8,192 | -1 | 8,192 | |
What is the sum of all the integers from 100 to 2000? | 1996050 | 0.875 | 3,562.0625 | 2,900.642857 | 8,192 | |
Evaluate: $6 - 5\left[7 - (\sqrt{16} + 2)^2\right] \cdot 3.$ | -429 | 0 | 2,093.4375 | -1 | 2,093.4375 | |
The numbers \(a, b, c, d\) belong to the interval \([-11.5, 11.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 552 | 0.25 | 7,889.375 | 6,981.5 | 8,192 | |
Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:
$$
\left|x^3 + y^3 + z^3 + 1\right| \leqslant C \left|x^5 + y^5 + z^5 + 1\right|.
$$ | \frac{9}{10} | 0 | 8,192 | -1 | 8,192 | |
$(100^2-99^2) + (98^2-97^2) + \ldots + (2^2-1^2) = \ $ | 5050 | 0.875 | 5,161.3125 | 5,109.428571 | 5,524.5 | |
For a positive integer \( n \), define \( s(n) \) as the smallest positive integer \( t \) such that \( n \) is a factor of \( t! \). Compute the number of positive integers \( n \) for which \( s(n) = 13 \). | 792 | 0.25 | 7,527.8125 | 5,535.25 | 8,192 | |
How many $3$-digit positive integers have digits whose product equals $30$? | 12 | 0.75 | 6,763.0625 | 6,286.75 | 8,192 | |
Among all pairs of real numbers $(x, y)$ such that $\cos \sin x = \cos \sin y$ with $-\frac{15\pi}{2} \le x, y \le \frac{15\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{4} | 0.0625 | 8,019.625 | 8,192 | 8,008.133333 | |
In the numbers from $1$ to $2002$, the number of positive integers that contain exactly one digit $0$ is: | 414 | 0.0625 | 8,042.3125 | 5,797 | 8,192 | |
Let $S(x)$ denote the sum of the digits of a positive integer $x$. Find the maximum possible value of $S(x+2019)-S(x)$. | 12 | We note that $S(a+b) \leq S(a)+S(b)$ for all positive $a$ and $b$, since carrying over will only decrease the sum of digits. (A bit more rigorously, one can show that $S\left(x+a \cdot 10^{b}\right)-S(x) \leq a$ for $0 \leq a \leq 9$.) Hence we have $S(x+2019)-S(x) \leq S(2019)=12$, and equality can be achieved with $x... | 0 | 8,192 | -1 | 8,192 |
Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$. | 27 | 0.5 | 6,944.5 | 5,782.5 | 8,106.5 | |
When the repeating decimal $0.\overline{12}$ is expressed as a common fraction in lowest terms, what is the sum of its numerator and denominator? | 37 | 1 | 1,348.25 | 1,348.25 | -1 | |
In a zoo, oranges, bananas, and coconuts were brought to feed three monkeys, with an equal number of each type of fruit. The first monkey was fed only oranges and bananas, with the number of bananas being 40% more than the number of oranges. The second monkey was fed only bananas and coconuts, with the number of coconu... | 1/2 | 0.375 | 5,393 | 4,267.833333 | 6,068.1 | |
The Dyslexian alphabet consists of consonants and vowels. It so happens that a finite sequence of letters is a word in Dyslexian precisely if it alternates between consonants and vowels (it may begin with either). There are 4800 five-letter words in Dyslexian. How many letters are in the alphabet? | 12 | 12 Suppose there are $c$ consonants, $v$ vowels. Then there are $c \cdot v \cdot c \cdot v \cdot c+$ $v \cdot c \cdot v \cdot c \cdot v=(c v)^{2}(c+v)$ five-letter words. Thus, $c+v=4800 /(c v)^{2}=3 \cdot(40 / c v)^{2}$, so $c v$ is a divisor of 40. If $c v \leq 10$, we have $c+v \geq 48$, impossible for $c, v$ intege... | 0.625 | 6,152 | 5,399.8 | 7,405.666667 |
A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches? | 384 | 1 | 1,033.1875 | 1,033.1875 | -1 | |
Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcasting methods. | 36 | 0 | 7,138.5625 | -1 | 7,138.5625 | |
In triangle \(ABC\), \(\angle A = 60^\circ\). Points \(M\) and \(N\) are on sides \(AB\) and \(AC\) respectively, such that the circumcenter of triangle \(ABC\) bisects segment \(MN\). Find the ratio \(AN:MB\). | 2:1 | 0.3125 | 7,428.9375 | 6,961 | 7,641.636364 | |
Which number is closest to \(-3.4\) on a number line? | -3 | On a number line, \(-3.4\) is between \(-4\) and \(-3\). This means that \(-3.4\) is closer to \(-3\) than to \(-4\), and so the answer is \(-3\). | 0.625 | 462 | 423.4 | 526.333333 |
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move... | $(6,2,1)$ | To solve this problem, we use the concept of nim-values from combinatorial game theory. The nim-value of a game configuration determines whether a position is winning or losing. A position with a nim-value of $0$ is losing (if both players play optimally), and any other nim-value is winning.
#### Step 1: Calculate nim... | 0 | 8,055.25 | -1 | 8,055.25 |
Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\$2.53$. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the c... | \$1.65 | 0.9375 | 2,806.6875 | 2,556.466667 | 6,560 | |
Let positive numbers $x$ and $y$ satisfy: $x > y$, $x+2y=3$. Find the minimum value of $\frac{1}{x-y} + \frac{9}{x+5y}$. | \frac{8}{3} | 0.8125 | 5,692.6875 | 5,115.923077 | 8,192 | |
On each cell of a $200 \times 200$ grid, we place a car, which faces in one of the four cardinal directions. In a move, one chooses a car that does not have a car immediately in front of it, and slides it one cell forward. If a move would cause a car to exit the grid, the car is removed instead. The cars are placed so ... | 6014950 | Let $n=100$. The answer is $\frac{1}{2} n\left(12 n^{2}+3 n-1\right)=6014950$. A construction for an $8 \times 8$ grid instead (so $n=4$ ):  Label the rows and columns from 1 to $2 n$, and let ... | 0 | 7,881.25 | -1 | 7,881.25 |
Tom, Sara, and Jim split $\$1200$ among themselves to be used in investments. Each starts with a different amount. At the end of one year, they have a total of $\$1800$. Sara triples her money by the end of the year, Jim doubles his money, whereas Tom loses $\$200$. What was Tom's original amount of money? | 400 | 0 | 7,832.875 | -1 | 7,832.875 | |
We can view these conditions as a geometry diagram as seen below. So, we know that $\frac{e}{f}=\frac{3}{4}$ (since $e=a-b=\frac{3}{4} c-\frac{3}{4} d=\frac{3}{4} f$ and we know that $\sqrt{e^{2}+f^{2}}=15$ (since this is $\left.\sqrt{a^{2}+c^{2}}-\sqrt{b^{2}+d^{2}}\right)$. Also, note that $a c+b d-a d-b c=(a-b)(c-d)=... | 108 | The value of $ef$ is 108. | 0.9375 | 4,272.4375 | 4,011.133333 | 8,192 |
Given points $F_{1}$, $F_{2}$ are the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and point $M(x_{0},y_{0})$ ($x_{0} > 0$, $y_{0} > 0$) lies on the asymptote of the hyperbola such that $MF_{1} \perp MF_{2}$. If there is a parabola with focus $F_{2}$ described by $y^2 = 2px$ ($p > 0... | 2 + \sqrt{5} | 0.6875 | 5,486.375 | 4,563.636364 | 7,516.4 | |
The Hangzhou Asian Games are underway, and table tennis, known as China's "national sport," is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other side serving two balls, ... | \frac{3}{4} | 0 | 7,762.25 | -1 | 7,762.25 | |
What is the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$? | 28 | To find the largest difference that can be formed by subtracting two numbers chosen from the set $\{ -16,-4,0,2,4,12 \}$, we need to consider the subtraction $a - b$, where $a$ and $b$ are elements of the set. The goal is to maximize this expression.
1. **Maximize $a$:** The maximum value of $a$ is the largest number ... | 1 | 2,716.4375 | 2,716.4375 | -1 |
Define $n!!$ as in the original problem. Evaluate $\sum_{i=1}^{5} \frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms. | \frac{437}{256} | 0.875 | 4,935.9375 | 4,470.785714 | 8,192 | |
Given four non-coplanar points \(A, B, C, D\) in space where the distances between any two points are distinct, consider a plane \(\alpha\) that satisfies the following properties: The distances from three of the points \(A, B, C, D\) to \(\alpha\) are equal, while the distance from the fourth point to \(\alpha\) is tw... | 32 | 0 | 7,936 | -1 | 7,936 | |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$? | 50 | 0.0625 | 8,185.625 | 8,090 | 8,192 | |
Calculate the area of an isosceles triangle if the length of its altitude, drawn to one of the equal sides, is $12 \mathrm{~cm}$, and the length of the base is $15 \mathrm{~cm}$. | 90 | 0 | 6,544.875 | -1 | 6,544.875 | |
Given the set $M = \{1,2,3,4\}$, let $A$ be a subset of $M$. The product of all elements in set $A$ is called the "cumulative value" of set $A$. It is stipulated that if set $A$ has only one element, its cumulative value is the value of that element, and the cumulative value of the empty set is 0. Find the number of su... | 13 | 0.0625 | 6,249.4375 | 4,837 | 6,343.6 | |
In the diagram, \(PQRS\) is a square with side length 8. Points \(T\) and \(U\) are on \(PS\) and \(QR\) respectively with \(QU = TS = 1\). The length of \(TU\) is closest to | 10 | 0.75 | 7,272.25 | 6,965.666667 | 8,192 | |
Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer. | 15015 | 0 | 8,192 | -1 | 8,192 | |
In the Cartesian coordinate plane \( xOy \), the coordinates of point \( F \) are \((1,0)\), and points \( A \) and \( B \) lie on the parabola \( y^2 = 4x \). It is given that \( \overrightarrow{OA} \cdot \overrightarrow{OB} = -4 \) and \( |\overrightarrow{FA}| - |\overrightarrow{FB}| = 4\sqrt{3} \). Find the value of... | -11 | 0.875 | 5,102.125 | 4,769.285714 | 7,432 | |
The proportion of top-quality products at this enterprise is $31 \%$. What is the most likely number of top-quality products in a randomly selected batch of 75 products? | 23 | 0.9375 | 4,290.6875 | 4,030.6 | 8,192 | |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 0.1875 | 7,694.9375 | 5,541 | 8,192 | |
Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$. What is the smallest possible value of $n$? | 10 | 1. **Understanding the Problem:**
We need to find the smallest integer $n$ such that when $x$, $y$, and $z$ are chosen independently from the interval $[0, n]$, the probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac{1}{2}$.
2. **Setting Up the Problem:**
The total... | 0.125 | 8,063.9375 | 7,167.5 | 8,192 |
Given that $\overrightarrow{a}=(2,3)$, $\overrightarrow{b}=(-4,7)$, and $\overrightarrow{a}+\overrightarrow{c}=\overrightarrow{0}$, find the projection of $\overrightarrow{c}$ on the direction of $\overrightarrow{b}$. | -\frac{\sqrt{65}}{5} | 0 | 5,028.625 | -1 | 5,028.625 | |
My school's Physics Club has 22 members. It needs to select 3 officers: chairman, vice-chairman, and sergeant-at-arms. Each person can hold at most one office. Two of the members, Penelope and Quentin, will only be officers if the other one is also an officer. (In other words, either both Penelope and Quentin are o... | 6960 | 0.625 | 5,912 | 4,544 | 8,192 | |
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $ \frac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive inte... | 330 | 0.125 | 7,539.0625 | 4,182 | 8,018.642857 | |
500 × 3986 × 0.3986 × 5 = ? | 0.25 \times 3986^2 | 0 | 440.8125 | -1 | 440.8125 | |
Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest triangle is 16 centimeters. What is the length ... | \frac{6\sqrt{6}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given a cone whose vertex and the circumference of its base are both on the same sphere, if the radius of the sphere is $1$, then when the volume of the cone is maximized, the height of the cone is ______. | \dfrac{4}{3} | 0.5625 | 6,586 | 5,706.333333 | 7,717 | |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | 1+3\sqrt{2} | 1 | 4,245.5 | 4,245.5 | -1 | |
Given that $\cos \alpha =\dfrac{4}{5}$ and $\cos (\alpha +\beta )=\dfrac{5}{13}$, where $\alpha$ and $\beta$ are acute angles.
1. Find the value of $\sin 2\alpha$.
2. Find the value of $\sin \beta$. | \dfrac{33}{65} | 0.625 | 6,728.75 | 6,009.8 | 7,927 | |
The expression $\frac{4k+8}{4}$ simplifies to an expression of the form $ak+b$ where $a$ and $b$ are integers. Find $\frac{a}{b}$ . | \frac{1}{2} | 1 | 1,277.5625 | 1,277.5625 | -1 | |
Find the smallest possible value of the expression $$\frac{(a+b)^2+(b-c)^2+(c-a)^2}{b^2},$$where $b > c > a$ are real numbers, and $b \neq 0.$ | \frac{4}{3} | 0.5 | 6,614 | 5,627 | 7,601 | |
Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous path following the stripes that goes from ... | \frac{3}{64} | 0 | 8,083.5 | -1 | 8,083.5 | |
Determine the number of solutions to the equation
\[\tan (7 \pi \cos \theta) = \cot (3 \pi \sin \theta)\] where $\theta \in (0, 3\pi).$ | 90 | 0 | 8,192 | -1 | 8,192 | |
Is there a number in which every digit from 1 to 9 is used exactly once, and for which the number formed by the first $k$ digits is divisible by $k$ for every $k$ from 1 to 9? | 381654729 | 0.0625 | 7,960.6875 | 4,491 | 8,192 | |
Given that $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2015} + b^{2014}$. | -1 | 0.25 | 7,339.8125 | 7,153 | 7,402.083333 | |
Teacher Zhang led the students of class 6 (1) to plant trees. The students can be divided into 5 equal groups. It is known that each teacher and student plants the same number of trees, with a total of 527 trees planted. How many students are there in class 6 (1)? | 30 | 0.1875 | 1,230.6875 | 1,613.666667 | 1,142.307692 | |
Restore the digits. On the board, the product of three consecutive even numbers was written. During the break, Vasya erased some of the digits. As a result, the board shows $87*****8$. Help Petya find the missing digits in the product. | 87526608 | 0 | 7,663.8125 | -1 | 7,663.8125 | |
A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he enc... | 342 | List all the numbers from $1$ through $1024$, then do the process yourself!!! It will take about 25 minutes (if you don't start to see the pattern), but that's okay, eventually, you will get $\boxed{342}$.
(Note: If you try to do this, first look through all the problems! -Guy) | 0 | 7,945 | -1 | 7,945 |
Evaluate $\lceil-2.4\rceil$. | -2 | 1 | 1,645.8125 | 1,645.8125 | -1 | |
How many positive three-digit integers less than 500 have at least two digits that are the same? | 112 | 0.625 | 5,581.625 | 4,015.4 | 8,192 | |
A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their... | 3/4 | 0.1875 | 7,816.625 | 6,403.333333 | 8,142.769231 |
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