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 function $f(x)=\sin 2x+2\cos ^{2}x-1$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$. | \frac{\pi}{2} | 0.9375 | 4,590.375 | 4,350.266667 | 8,192 | |
In a right triangle JKL, where $\angle J$ is $90^\circ$, side JL is known to be 12 units, and the hypotenuse KL is 13 units. Calculate $\tan K$ and $\cos L$. | \frac{5}{13} | 0 | 1,943.4375 | -1 | 1,943.4375 | |
If $x \%$ of 60 is 12, what is $15 \%$ of $x$? | 3 | Since $x \%$ of 60 is 12, then $\frac{x}{100} \cdot 60=12$ or $x=\frac{12 \cdot 100}{60}=20$. Therefore, $15 \%$ of $x$ is $15 \%$ of 20, or $0.15 \cdot 20=3$. | 1 | 413.75 | 413.75 | -1 |
In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is: | \frac{\sqrt{5}-1}{2} | 1. **Identify the sequence properties**: We are given a geometric progression with positive terms, where any term is equal to the sum of the next two terms. Let the first term be $a$ and the common ratio be $r$.
2. **Set up the equation**: For any term $a_n$ in the geometric progression, we have:
\[
a_n = a_{n+... | 0 | 2,048.75 | -1 | 2,048.75 |
A standard deck of 52 cards is arranged randomly. What is the probability that the top three cards alternate in color, starting with a red card, then a black card, followed by another red card? | \frac{13}{102} | 0.5625 | 5,382.75 | 5,145 | 5,688.428571 | |
For any real value of $x$ the maximum value of $8x - 3x^2$ is: | \frac{16}{3} | 1. **Identify the function and its type**:
Let $f(x) = 8x - 3x^2$. This is a quadratic function.
2. **Determine the nature of the quadratic**:
Since the coefficient of $x^2$ is negative ($-3$), the parabola opens downwards. This means the vertex of the parabola will give the maximum value of the function.
3. *... | 1 | 2,729.9375 | 2,729.9375 | -1 |
In a circle with center $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$, a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is: | \frac{r^2\sqrt{3}}{32} | 1. **Identify the Geometry of Triangle $AOB$:**
Since $AB = r$ and $AO = OB = r$ (radii of the circle), triangle $AOB$ is isosceles with $AO = OB = AB$. Additionally, since all sides are equal, $\triangle AOB$ is an equilateral triangle. Therefore, each angle in $\triangle AOB$ is $60^\circ$.
2. **Determine the Len... | 0 | 7,003.375 | -1 | 7,003.375 |
Given $A(-1,\cos \theta)$, $B(\sin \theta,1)$, if $|\overrightarrow{OA}+\overrightarrow{OB}|=|\overrightarrow{OA}-\overrightarrow{OB}|$, then find the value of the acute angle $\theta$. | \frac{\pi}{4} | 0.9375 | 3,187.625 | 3,200 | 3,002 | |
How many factors are there in the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ if we know that it ends with 1981 zeros? | 7935 | 0.1875 | 8,040.3125 | 8,060.333333 | 8,035.692308 | |
Given a geometric progression $\{a_n\}$ with the first term $a_1=2$ and the sum of the first $n$ terms as $S_n$, and the equation $S_5 + 4S_3 = 5S_4$ holds, find the maximum term of the sequence $\left\{ \frac{2\log_{2}a_n + 1}{\log_{2}a_n - 6} \right\}$. | 15 | 0.9375 | 5,528.125 | 5,350.533333 | 8,192 | |
Voldemort bought a book for $\$5$. It was one-tenth of its original price. What was the original price in dollars? | \$50 | 1 | 803.8125 | 803.8125 | -1 | |
A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many boys are on the chess team? | 6 | 1 | 1,848.6875 | 1,848.6875 | -1 | |
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\text{th}}$-graders to $6^{\text{th}}$-graders is $5:3$, and the the ratio of $8^{\text{th}}$-graders to $7^{\text{th}}$-graders is $8:5$. What is the smallest number of students that could be participatin... | 89 |
#### Step 1: Establish the ratios
We are given two ratios:
- The ratio of $8^\text{th}$-graders to $6^\text{th}$-graders is $5:3$.
- The ratio of $8^\text{th}$-graders to $7^\text{th}$-graders is $8:5$.
#### Step 2: Equalize the number of $8^\text{th}$-graders
To combine these ratios into a single ratio involving $8^... | 1 | 2,361.5625 | 2,361.5625 | -1 |
Let $F_1 = \left( -3, 1 - \frac{\sqrt{5}}{4} \right)$ and $F_ 2= \left( -3, 1 + \frac{\sqrt{5}}{4} \right).$ Then the set of points $P$ such that
\[|PF_1 - PF_2| = 1\]form a hyperbola. The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1,\]where $a, b > 0.$ Find $h + k ... | -\frac{5}{4} | 0.9375 | 4,309.6875 | 4,268.466667 | 4,928 | |
There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)? | 337 | 0.4375 | 6,994.375 | 6,241.857143 | 7,579.666667 | |
What is the units digit of $3^{2004}$? | 1 | 1 | 1,715.4375 | 1,715.4375 | -1 | |
A point $(x,y)$ is randomly selected such that $0 \le x \le 3$ and $0 \le y \le 6$. What is the probability that $x+y \le 4$? Express your answer as a common fraction. | \frac{5}{12} | 0.8125 | 6,432.625 | 6,026.615385 | 8,192 | |
In square $ABCD$ , $\overline{AC}$ and $\overline{BD}$ meet at point $E$ .
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$ .
If $\overline{AF}$ meets $\overline{ED}$ at point $G$ , and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$ . | 48 | 0.6875 | 5,668 | 4,608 | 8,000 | |
Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids. | 134 | 0.0625 | 6,469.3125 | 8,192 | 6,354.466667 | |
Andrea notices that the 40-foot tree next to her is casting a 10-foot shadow. How tall, in inches, is Andrea if she is casting a 15-inch shadow at the same time? | 60 | 0.9375 | 1,735.375 | 1,814.133333 | 554 | |
Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \dfrac{2\sqrt{3}}{3})$, with the left and right foci being $F_1$ and $F_2$ respectively. The circle $x^2 + y^2 = 2$ intersects with the line $x + y + b = 0$ at a chord of length $2$.
(Ⅰ) Find the standard equatio... | \dfrac{2\sqrt{3}}{3} | 0 | 7,064.25 | -1 | 7,064.25 | |
Given the numbers 1, 2, 3, 4, 5, there are $5!$ permutations $a_1, a_2, a_3, a_4, a_5$. Find the number of distinct permutations where $a_k \geq k - 2$ for all $k = 1, 2, 3, 4, 5$. | 54 | 0.3125 | 7,111.125 | 4,733.2 | 8,192 | |
Find the smallest solution to the equation \[\frac{1}{x-2} + \frac{1}{x-4} = \frac{3}{x-3}.\] | 3 - \sqrt3 | 0.9375 | 3,873.75 | 3,585.866667 | 8,192 | |
There are five positive integers that are divisors of each number in the list $$48, 144, 24, 192, 216, 120.$$ Find the sum of these positive integers. | 16 | 0 | 7,059.75 | -1 | 7,059.75 | |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}{cccccc}&A&B&B&C&B\ +&B&C&A&D&A\ \hline &D&B&D&D&D\end{array}$ | 7 | 1. **Analyze the first column**: The sum $A + B$ must yield a single digit, $D$, and no carry-over since the sum of the digits in the first column is a single digit. This implies $A + B < 10$.
2. **Analyze the fourth column**: The sum $C + D$ must yield the digit $D$. This implies $C + D = D$ or $C = 0$ (since adding ... | 0.0625 | 8,192 | 8,192 | 8,192 |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{8}$ be the 8 zeros of the polynomial $z^{8} - 16^8$. For each $j$, let $w_j$ be either $z_j$, $-z_j$, or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{8} w_j.\] | 32 + 32 \sqrt{2} | 0.0625 | 7,976.1875 | 6,362 | 8,083.8 | |
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon ... | 7 | 1. **Assign Variables:**
Let the side length of the octagon be $x$. Since $AP = BQ < 4$ and $PQ = 8$, we have $AP = BQ = \frac{8-x}{2}$.
2. **Use the Pythagorean Theorem:**
Since $ABCDEFGH$ is equilateral, all sides are equal, and $BQ = CQ = x$. Therefore, $CQ = \frac{6-x}{2}$ because $QR = 6$ and $C$ and $D$ ar... | 0.125 | 7,990.1875 | 7,357 | 8,080.642857 |
The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;
for(int i = -4; i <= 5; ++i) {
draw((i,-1)--(i,6), dashed+med... | 2 | 0.0625 | 8,004.5625 | 6,710 | 8,090.866667 | |
Screws are sold in packs of $10$ and $12$ . Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$ ? | 60 | 0.25 | 7,403.125 | 5,875.5 | 7,912.333333 | |
Find any quadruple of positive integers $(a, b, c, d)$ satisfying $a^{3}+b^{4}+c^{5}=d^{11}$ and $a b c<10^{5}$. | (128,32,16,4) \text{ or } (160,16,8,4) | It's easy to guess that there are solutions such that $a, b, c, d$ are in the form of $n^{x}$, where $n$ is a rather small number. After a few attempts, we can see that we obtain simple equations when $n=2$ or $n=3$ : for $n=2$, the equation becomes in the form of $2^{t}+2^{t}+2^{t+1}=2^{t+2}$ for some non-negative int... | 0 | 8,192 | -1 | 8,192 |
Let $A = (-1,1,2),$ $B = (1,2,3),$ and $C = (t,1,1),$ where $t$ is a real number. Find the smallest possible area of triangle $ABC.$ | \frac{\sqrt{3}}{2} | 0 | 5,479.4375 | -1 | 5,479.4375 | |
Calculate $\lim_{n \to \infty} \frac{C_n^2}{n^2+1}$. | \frac{1}{2} | 0.4375 | 5,256.4375 | 2,438.571429 | 7,448.111111 | |
Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$? | 1011100 | 0.625 | 4,675.3125 | 3,634 | 6,410.833333 | |
Let $S$ be the set of points in the Cartesian plane that satisfy
$\Big|\big||x|-2\big|-1\Big|+\Big|\big||y|-2\big|-1\Big|=1.$
If a model of $S$ were built from wire of negligible thickness, then the total length of wire required would be $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by ... | 66 | Let $f(x) = \Big|\big||x|-2\big|-1\Big|$, $f(x) \ge 0$. Then $f(x) + f(y) = 1 \Longrightarrow f(x), f(y) \le 1 \Longrightarrow x, y \le 4$. We only have a $4\times 4$ area, so guessing points and graphing won't be too bad of an idea. Since $f(x) = f(-x)$, there's a symmetry about all four quadrants, so just consider th... | 0 | 8,192 | -1 | 8,192 |
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$. | 106 | Let $b$, $n$ denote birch tree and not-birch tree, respectively. Notice that we only need $4$ $n$s to separate the $5$ $b$s. Specifically, \[b,n,b,n,b,n,b,n,b\] Since we have $7$ $n$s, we are placing the extra $3$ $n$s into the $6$ intervals beside the $b$s.
Now doing simple casework.
If all $3$ $n$s are in the same ... | 0.875 | 5,619.3125 | 5,251.785714 | 8,192 |
Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240? | 17 | 0.4375 | 7,595 | 6,827.428571 | 8,192 | |
Determine the number of integers $D$ such that whenever $a$ and $b$ are both real numbers with $-1 / 4<a, b<1 / 4$, then $\left|a^{2}-D b^{2}\right|<1$. | 32 | We have $$-1<a^{2}-D b^{2}<1 \Rightarrow \frac{a^{2}-1}{b^{2}}<D<\frac{a^{2}+1}{b^{2}}$$ We have $\frac{a^{2}-1}{b^{2}}$ is maximal at $-15=\frac{.25^{2}-1}{.25^{2}}$ and $\frac{a^{2}+1}{b^{2}}$ is minimal at $\frac{0^{2}+1}{.25^{2}}=16$. However, since we cannot have $a, b= \pm .25$, checking border cases of -15 and 1... | 0 | 6,972.6875 | -1 | 6,972.6875 |
The altitude of an equilateral triangle is $\sqrt{12}$ units. What is the area and the perimeter of the triangle, expressed in simplest radical form? | 12 | 1 | 1,765.375 | 1,765.375 | -1 | |
Kate has saved up $4444_8$ dollars for a trip to France. A round-trip airline ticket costs $1000_{10}$ dollars. In base ten, how many dollars will she have left for lodging and food? | 1340 | 1 | 2,123.125 | 2,123.125 | -1 | |
In how many ways can a tetromino in the shape of the letter $Z$ be placed on a chessboard (size $8 \times 8$ squares) so that it is located exactly on the cells of the board and within the board's boundaries? The tetromino can be rotated and flipped. Justify your answer. | 168 | 0 | 7,411.75 | -1 | 7,411.75 | |
Translate the function $y=\sqrt{3}\cos x+\sin x$ $(x\in\mathbb{R})$ to the left by $m$ $(m > 0)$ units, and the resulting graph is symmetric about the $y$-axis. Find the minimum value of $m$. | \frac{\pi}{6} | 0.5625 | 6,243.0625 | 5,286.777778 | 7,472.571429 | |
Let $f$ be a polynomial such that, for all real number $x$ , $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$ .
Compute $f(2018)$ . | -2019 | 1 | 4,904 | 4,904 | -1 | |
We consider an $n \times n$ table, with $n\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable? | \frac{n(n+1)}{2} |
Given an \( n \times n \) table, where \( n \geq 1 \), the task is to determine the maximum number of cells \( k \) that can be colored such that there is a unique way to place \( n \) tokens on the colored cells. Importantly, no two tokens should be in the same row or column.
To approach this problem, consider the c... | 0 | 8,140.5 | -1 | 8,140.5 |
A set of marbles can be divided in equal shares among $2$, $3$, $4$, $5$, or $6$ children with no marbles left over. What is the least number of marbles that the set could have? | 60 | 1 | 2,721.375 | 2,721.375 | -1 | |
A square pyramid has a base edge of 32 inches and an altitude of 1 foot. A square pyramid whose altitude is one-fourth of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the original pyramid? | \frac{63}{64} | 0.9375 | 2,892.1875 | 2,538.866667 | 8,192 | |
Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:
(i) \( f(1)=0 \),
(ii) \( f(p)=1 \) for all prime numbers \( p \),
(iii) \( f(xy)=y f(x)+x f(y) \) for all \( x, y \in \mathbb{Z}_{>0} \).
Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n)=n \).
(... | 3125 | 0.0625 | 8,106.375 | 6,822 | 8,192 | |
Suppose \( g(x) \) is a rational function such that \( 4g\left(\frac{1}{x}\right) + \frac{3g(x)}{x} = x^3 \) for \( x \neq 0 \). Find \( g(-3) \). | -\frac{6565}{189} | 0.1875 | 7,928.4375 | 6,786.333333 | 8,192 | |
Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally? | 50 | 1 | 1,500.75 | 1,500.75 | -1 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $C= \dfrac {\pi}{3}$, $b=8$. The area of $\triangle ABC$ is $10 \sqrt {3}$.
(I) Find the value of $c$;
(II) Find the value of $\cos (B-C)$. | \dfrac {13}{14} | 0.9375 | 3,858.875 | 3,885.533333 | 3,459 | |
What is the smallest positive integer $n$ such that $\frac{n}{n+53}$ is equal to a terminating decimal? | 11 | 0.6875 | 5,417.25 | 4,527.272727 | 7,375.2 | |
If $f(n)=\tfrac{1}{3} n(n+1)(n+2)$, then $f(r)-f(r-1)$ equals: | r(r+1) | To solve for $f(r) - f(r-1)$, we first need to express each function in terms of $r$.
1. **Calculate $f(r)$:**
\[
f(r) = \frac{1}{3} r(r+1)(r+2)
\]
2. **Calculate $f(r-1)$:**
\[
f(r-1) = \frac{1}{3} (r-1)r(r+1)
\]
3. **Subtract $f(r-1)$ from $f(r)$:**
\[
f(r) - f(r-1) = \frac{1}{3} r(r+1)(r+2... | 0.6875 | 4,466.375 | 3,701 | 6,150.2 |
The sum of two numbers is $10$; their product is $20$. The sum of their reciprocals is: | \frac{1}{2} | Given the equations:
1. \(x + y = 10\)
2. \(xy = 20\)
We are asked to find the sum of the reciprocals of \(x\) and \(y\), which is \(\frac{1}{x} + \frac{1}{y}\).
Using the identity for the sum of reciprocals:
\[
\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}
\]
Substituting the values from the given equations:
\[
\fra... | 1 | 1,972.6875 | 1,972.6875 | -1 |
A proposal will make years that end in double zeroes a leap year only if the year leaves a remainder of 200 or 600 when divided by 900. Under this proposal, how many leap years will there be that end in double zeroes between 1996 and 4096? | 5 | 0.5625 | 7,030.4375 | 6,219.222222 | 8,073.428571 | |
If 2035 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 50 | 0.0625 | 7,601.125 | 8,192 | 7,561.733333 | |
The average of seven numbers in a list is 62. The average of the first four numbers is 55. What is the average of the last three numbers? | 71.\overline{3} | 0 | 1,959.5 | -1 | 1,959.5 | |
Given the functions $f(x)=2\sin \left(\pi x+\frac{\pi }{3}\right)$ and $g(x)=2\cos \left(\pi x+\frac{\pi }{3}\right)$, find the area of the triangle formed by the intersection points of the two functions within the interval $\left[-\frac{4}{3},\frac{7}{6}\right]$. | 2\sqrt{2} | 0.75 | 6,335.4375 | 5,885.166667 | 7,686.25 | |
Given the system of equations
\begin{align*}
4x+2y &= c, \\
6y - 12x &= d,
\end{align*}
where \(d \neq 0\), find the value of \(\frac{c}{d}\). | -\frac{1}{3} | 0.0625 | 7,902.4375 | 6,544 | 7,993 | |
The side of rhombus \(ABCD\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.
Find the distance between the points where this circle touches the sides \(AB\) and \(BC\), if the diagonal \(AC\) is less than the diagonal \(BD\). | 3.84 | 0 | 7,613.0625 | -1 | 7,613.0625 | |
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$ , and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$ . | 10 | 0.4375 | 7,062.1875 | 5,894.714286 | 7,970.222222 | |
In the expression $\frac{x + 1}{x - 1}$ each $x$ is replaced by $\frac{x + 1}{x - 1}$. The resulting expression, evaluated for $x = \frac{1}{2}$, equals: | 3 | 1. **Substitute $x$ with $\frac{x+1}{x-1}$ in the expression:**
\[
\frac{\left(\frac{x+1}{x-1}\right) + 1}{\left(\frac{x+1}{x-1}\right) - 1}
\]
2. **Simplify the expression:**
- For the numerator:
\[
\frac{x+1}{x-1} + 1 = \frac{x+1}{x-1} + \frac{x-1}{x-1} = \frac{x+1 + x-1}{x-1} = \frac{2x}{x-1}
... | 0 | 3,888.4375 | -1 | 3,888.4375 |
Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, the sum of the first $n$ terms is $S_n$, and $S_3 = 7$. Also, $a_1, a_2, a_3 - 1$ form an arithmetic sequence. For the sequence $\{b_n\}$, the sum of the first $n$ terms is $T_n$, and $6T_n = (3n+1)b_n + 2$, where $n \in \mathbb{N}^+$.
(Ⅰ) Find the gen... | 3318 | 0.25 | 6,210.8125 | 5,671.75 | 6,390.5 | |
Given the graph of $y=\cos 2x$, obtain the graph of $y=\sin 2x$. | \frac{\pi}{4} | 0.3125 | 5,674 | 5,601.6 | 5,706.909091 | |
Let \(x_1, x_2, \ldots, x_n\) be real numbers in arithmetic sequence, which satisfy \(|x_i| < 1\) for \(i = 1, 2, \dots, n,\) and \[|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of \(n\)? | 26 | 0.125 | 7,800.5 | 5,298.5 | 8,157.928571 | |
Compute the integer $k > 2$ for which
\[\log_{10} (k - 2)! + \log_{10} (k - 1)! + 2 = 2 \log_{10} k!.\] | 5 | 1 | 2,863.8125 | 2,863.8125 | -1 | |
29 boys and 15 girls came to the ball. Some of the boys danced with some of the girls (at most once with each person in the pair). After the ball, each individual told their parents how many times they danced. What is the maximum number of different numbers that the children could mention? | 29 | 0 | 8,127.125 | -1 | 8,127.125 | |
Four fair coins are to be flipped. What is the probability that all four will be heads or all four will be tails? Express your answer as a common fraction. | \frac{1}{8} | 1 | 1,308.3125 | 1,308.3125 | -1 | |
If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for
$\begin{array}{ccc} X & X & X \ & Y & X \ + & & X \ \hline \end{array}$
has the form | $YYZ$ | 1. **Identify the structure of the sum**: The problem involves summing three numbers where the digits are represented by $X$ and $Y$. The numbers are $XXX$, $YX$, and $X$. We can rewrite these numbers in terms of their decimal values:
- $XXX = 100X + 10X + X = 111X$
- $YX = 10Y + X$
- $X = X$
2. **Combine the... | 0 | 7,943.875 | -1 | 7,943.875 |
Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2500$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$? | 625 | 0.3125 | 7,867.6875 | 7,154.2 | 8,192 | |
Find the units digit of $9^{8^7}$. | 1 | 1 | 2,680.9375 | 2,680.9375 | -1 | |
A shop specializing in small refrigerators sells an average of 50 refrigerators per month. The cost of each refrigerator is $1200. Since shipping is included in the sale, the shop needs to pay a $20 shipping fee for each refrigerator. In addition, the shop has to pay a "storefront fee" of $10,000 to the JD website each... | 1824 | 0.25 | 5,365.3125 | 4,165 | 5,765.416667 | |
The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$? | 54 | We are given the equation \(x^2+y^2 = 3|x-y| + 3|x+y|\) and need to find the area of the region it describes. We will consider different cases based on the absolute values \(|x-y|\) and \(|x+y|\).
#### Case 1: \(|x-y|=x-y, |x+y|=x+y\)
Substituting these into the equation, we get:
\[
x^2+y^2 = 3(x-y) + 3(x+y) = 6x
\]
R... | 0.0625 | 8,080.6875 | 6,411 | 8,192 |
The number 5.6 may be expressed uniquely (ignoring order) as a product $\underline{a} \cdot \underline{b} \times \underline{c} . \underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\underline{a} \cdot \underline{b}+\underline{c} . \underline{d}$. | 5.1 | We want $\overline{a b} \times \overline{c d}=560=2^{4} \times 5 \times 7$. To avoid a zero digit, we need to group the 5 with the 7 to get 3.5 and 1.6 , and our answer is $3.5+1.6=5.1$. | 0 | 8,178.125 | -1 | 8,178.125 |
A car license plate contains three letters and three digits, for example, A123BE. The allowable letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Katya considers a plate number lucky if the second letter is a consonant, the first digit is odd, and the thi... | 288000 | 0.5 | 4,389.75 | 3,168.25 | 5,611.25 | |
How many distinct divisors does the number a) 800; b) 126000 have? | 120 | 0.8125 | 2,413 | 2,450.692308 | 2,249.666667 | |
Determine the slope of line $\overline{CD}$ where circles given by $x^2 + y^2 - 6x + 4y - 5 = 0$ and $x^2 + y^2 - 10x + 16y + 24 = 0$ intersect at points $C$ and $D$. | \frac{1}{3} | 0.9375 | 4,612.5 | 4,373.866667 | 8,192 | |
Given the set \( M = \{1, 3, 5, 7, 9\} \), find the non-empty set \( A \) such that:
1. Adding 4 to each element in \( A \) results in a subset of \( M \).
2. Subtracting 4 from each element in \( A \) also results in a subset of \( M \).
Determine the set \( A \). | {5} | 0 | 5,847.4375 | -1 | 5,847.4375 | |
In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B, \angle C, \angle E,$ and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1).$ Find $AB$. | 46 | Let the side length be called $x$, so $x=AB=BC=CD=DE=EF=AF$.
The diagonal $BF=\sqrt{AB^2+AF^2}=\sqrt{x^2+x^2}=x\sqrt{2}$. Then the areas of the triangles AFB and CDE in total are $\frac{x^2}{2}\cdot 2$, and the area of the rectangle BCEF equals $x\cdot x\sqrt{2}=x^2\sqrt{2}$
Then we have to solve the equation
$2116(\... | 0.0625 | 8,171.8125 | 7,869 | 8,192 |
Find the number of functions of the form \( f(x) = ax^3 + bx^2 + cx + d \) such that
\[ f(x)f(-x) = f(x^3). \] | 16 | 0 | 5,742.3125 | -1 | 5,742.3125 | |
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy... | 157 | Assume without loss of generality that the first card laid out is red. Then the arrangements that satisfy Kathy’s requirements are RRRRR, RRRRG, RRRGG, RRGGG, and RGGGG. The probability that Kathy will lay out one of these arrangements is \[\frac49\cdot\frac38\cdot\frac27\cdot\frac16\] \[\frac49\cdot\frac38\cdot\frac27... | 0 | 8,104.3125 | -1 | 8,104.3125 |
Of the numbers 1, 2, 3, ..., 15, which number has the greatest number of divisors (the dots mean that we are including all the integers between 1 and 15)? | 12 | 1 | 2,058.75 | 2,058.75 | -1 | |
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ Find the prime factorization of the minimum possible value of $x$ and determine the sum of the exponents and the prime factors. | 31 | 0.75 | 4,074 | 3,735.083333 | 5,090.75 | |
Grandma has two balls of yarn: one large and one small. From the large ball, she can either knit a sweater and three socks, or five identical hats. From the small ball, she can either knit half a sweater or two hats. (In both cases, all the yarn will be used up.) What is the maximum number of socks Grandma can knit usi... | 21 | 0 | 7,937.5 | -1 | 7,937.5 | |
The letters of the word 'GAUSS' and the digits in the number '1998' are each cycled separately. If the pattern continues in this way, what number will appear in front of 'GAUSS 1998'? | 20 | 0.8125 | 4,230.3125 | 3,316.076923 | 8,192 | |
A line is described by the equation $y-4=4(x-8)$. What is the sum of its $x$-intercept and $y$-intercept? | -21 | 1 | 1,994.0625 | 1,994.0625 | -1 | |
The Tianfu Greenway is a popular check-in spot for the people of Chengdu. According to statistics, there is a linear relationship between the number of tourists on the Tianfu Greenway, denoted as $x$ (in units of 10,000 people), and the economic income of the surrounding businesses, denoted as $y$ (in units of 10,000 y... | 88 | 0.5625 | 5,143.4375 | 3,312.777778 | 7,497.142857 | |
A right circular cone is inscribed in a right prism as shown. What is the ratio of the volume of the cone to the volume of the prism? Express your answer as a common fraction in terms of $\pi$. [asy]
import three; import graph3;
defaultpen(linewidth(0.8));
size(200);
draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle);
dra... | \frac{\pi}{12} | 0.9375 | 4,334.125 | 4,076.933333 | 8,192 | |
Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 1 | 2,050.25 | 2,050.25 | -1 | |
In regular pentagon $ABCDE$, diagonal $AC$ is drawn, as shown. Given that each interior angle of a regular pentagon measures 108 degrees, what is the measure of angle $CAB$?
[asy]
size(4cm,4cm);
defaultpen(linewidth(1pt)+fontsize(10pt));
pair A,B,C,D,E;
A = (0,0);
B = dir(108);
C = B+dir(39);
D = C+dir(-39);
E = (1,0... | 36 | 0.875 | 5,067.8125 | 4,621.5 | 8,192 | |
Let $S=\{1,2, \ldots 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1)=1$, where $f^{(i)}(x)=f\left(f^{(i-1)}(x)\right)$. What is the expected value of $n$? | \frac{2017}{2} | Say that $n=k$. Then $1, f(1), f^{2}(1), \ldots, f^{(k-1)}(1)$ are all distinct, which means there are 2015. $2014 \cdots(2016-k+1)$ ways to assign these values. There is 1 possible value of $f^{k}(1)$, and $(2016-k)$ ! ways to assign the image of the $2016-k$ remaining values. Thus the probability that $n=k$ is $\frac... | 0.0625 | 7,919.5 | 3,832 | 8,192 |
How many different routes are there from point $A$ to point $B$ if you can only move to the right or down along the drawn segments?
[asy]
unitsize(0.09inch);
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((5,0)--(5,10));
draw((0,5)--(10,5));
dot((0,10));
dot((10,0));
label("$A$",(0,10),NW);
label("$B$",(10,0),SE);
... | 6 | 0 | 5,104.375 | -1 | 5,104.375 | |
In a right-angled triangle $ABC$, where $AB = AC = 1$, an ellipse is constructed with point $C$ as one of its foci. The other focus of the ellipse lies on side $AB$, and the ellipse passes through points $A$ and $B$. Determine the focal length of the ellipse. | \frac{\sqrt{5}}{2} | 0 | 3,840.5625 | -1 | 3,840.5625 | |
Calculate the sum of $213_4 + 132_4 + 321_4$ and express your answer in base $4$. | 1332_4 | 0.75 | 4,618.25 | 3,427 | 8,192 | |
In the diagram, $\triangle ABF$, $\triangle BCF$, and $\triangle CDF$ are right-angled, with $\angle ABF=\angle BCF = 90^\circ$ and $\angle CDF = 45^\circ$, and $AF=36$. Find the length of $CF$.
[asy]
pair A, B, C, D, F;
A=(0,25);
B=(0,0);
C=(0,-12);
D=(12, -12);
F=(24,0);
draw(A--B--C--D--F--A);
draw(B--F);
draw(C--F)... | 36 | 0 | 7,953 | -1 | 7,953 | |
Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$ | 128 | 0.1875 | 7,979 | 8,192 | 7,929.846154 | |
Given the function g defined on the set of positive rational numbers by g(x \cdot y) = g(x) + g(y) for all positive rational numbers x and y, and g(n) = n^2 for every prime number n, calculate g(x) for x = \frac{25}{21}. | -8 | 1 | 4,363.125 | 4,363.125 | -1 | |
Given a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its asymptotes intersect the parabola $y^2 = 4x$ at two points A and B, distinct from the origin O. Let F be the focus of the parabola $y^2 = 4x$. If $\angle AFB = \frac{2\pi}{3}$, then the eccentricity of the hyperbola is ________. | \frac{\sqrt{21}}{3} | 0 | 7,809.8125 | -1 | 7,809.8125 | |
Define a function $g(z) = (3 + i)z^2 + \alpha z + \gamma$ for all complex $z$, where $\alpha$ and $\gamma$ are complex numbers. Assume that $g(1)$ and $g(i)$ both yield real numbers. Determine the smallest possible value of $|\alpha| + |\gamma|$. | \sqrt{2} | 0.4375 | 7,337.75 | 6,979.857143 | 7,616.111111 | |
Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $ , $ f (n) $ is an integer greater than or equal to $ 0 $ . $\bullet$ $f (n) = 2010 $ , if $ n $ ends in $ 7 $ . For example, $ f (137) = 2010 $ . $\bullet$ If $ a $ is a divisor of $ b $ , then: $ ... | 2010 | 0 | 7,797.75 | -1 | 7,797.75 | |
Find the period of the repetend of the fraction $\frac{39}{1428}$ by using *binary* numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\c... | 24 | 0.75 | 5,033.5 | 4,431.583333 | 6,839.25 | |
Solve the equation \(3 \cdot 4^{\log_{x} 2} - 46 \cdot 2^{\log_{x} 2 - 1} = 8\). | \sqrt[3]{2} | 1 | 2,842.125 | 2,842.125 | -1 |
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