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 |
|---|---|---|---|---|---|---|
A math conference is presenting a lecture series with six different lecturers. If Dr. Smith's lecture depends on Dr. Jones's lecture, so that Dr. Smith must be scheduled at some time after Dr. Jones, in how many orders can the six lecturers be scheduled? | 360 | 1 | 2,201.625 | 2,201.625 | -1 | |
Let $a,$ $b,$ $c$ be the roots of $3x^3 - 3x^2 + 11x - 8 = 0.$ Find $ab + ac + bc.$ | \frac{11}{3} | 1 | 4,147.6875 | 4,147.6875 | -1 | |
A club has $5$ members from each of $3$ different schools, for a total of $15$ members. How many possible ways are there to arrange a presidency meeting under the following conditions:
i. The club must choose one of the $3$ schools at which to host the meeting, and
ii. The host school sends $2$ representatives to the... | 750 | 0.8125 | 3,616.5 | 2,718.153846 | 7,509.333333 | |
Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote?
A) 997
B) 999
C) 1111
D) 1113
E) 1115 | 1113 | 0 | 6,319 | -1 | 6,319 | |
If $x$ is a real number and $\lceil x \rceil = 9,$ how many possible values are there for $\lceil x^2 \rceil$? | 17 | 0.125 | 7,103.625 | 7,141 | 7,098.285714 | |
Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? | 4 | To solve this problem, we need to determine how many noncongruent triangles can be formed using any three of the six points $R$, $S$, $T$, $X$, $Y$, and $Z$, where $X$, $Y$, and $Z$ are the midpoints of the sides of equilateral triangle $\triangle RST$.
#### Step 1: Identify possible triangles
We start by noting that ... | 0.0625 | 8,187.1875 | 8,115 | 8,192 |
Three distinct integers $a, b,$ and $c$ satisfy the following three conditions: $abc=17955$, $a, b,$ and $c$ form an arithmetic sequence in that order, and $(3a+b), (3b+c),$ and $(3c+a)$ form a geometric sequence in that order. What is the value of $a+b+c$? | -63 | Since $a, b$ and $c$ form an arithmetic sequence in this order, then $a=b-d$ and $c=b+d$ for some real number $d$. We note that $d \neq 0$, since otherwise we would have $a=b=c$ and then $abc=17955$ would tell us that $b^{3}=17955$ or $b=\sqrt[3]{17955}$, which is not an integer. Writing the terms of the geometric sequ... | 0.875 | 4,800.375 | 4,315.857143 | 8,192 |
A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00. | 1400 | 0.8125 | 547.125 | 567.153846 | 460.333333 | |
Given that the equation $2kx+2m=6-2x+nk$ has a solution independent of $k$, the value of $4m+2n$ is ______. | 12 | 1 | 2,649.875 | 2,649.875 | -1 | |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to points $F_1(0, -\sqrt{3})$ and $F_2(0, \sqrt{3})$ is equal to 4. Let the trajectory of point $P$ be $C$.
(1) Find the equation of trajectory $C$;
(2) Let line $l: y=kx+1$ intersect curve $C$ at points $A$ and $B$. For what value of $k$ is ... | \frac{4\sqrt{65}}{17} | 0 | 6,205.5 | -1 | 6,205.5 | |
What expression is never a prime number when $p$ is a prime number? | $p^2+26$ | We need to determine which of the given expressions is never a prime number when $p$ is a prime number. To do this, we will analyze each expression modulo a small integer to see if it can be factored or if it always results in a composite number.
1. **Expression A: $p^2 + 16$**
- Consider $p^2 + 16 \pmod{3}$.
- ... | 0 | 5,188.8125 | -1 | 5,188.8125 |
Find the product of $1011_2 \cdot 101_2$. Express your answer in base 2. | 110111 | 0 | 5,932.875 | -1 | 5,932.875 | |
In $\triangle ABC$, it is given that $BD:DC = 3:2$ and $AE:EC = 3:4$. Point $M$ is the intersection of $AD$ and $BE$. If the area of $\triangle ABC$ is 1, what is the area of $\triangle BMD$? | $\frac{4}{15}$ | 0 | 7,663.1875 | -1 | 7,663.1875 | |
Determine all real numbers $x$ for which the function $$g(x) = \frac{1}{2+\frac{1}{1+\frac{1}{x-1}}}$$ is undefined, and find their sum. | \frac{4}{3} | 0.75 | 4,613.0625 | 4,385.25 | 5,296.5 | |
A palindrome is a positive integer that reads the same backwards as forwards, such as 82328. What is the smallest 5 -digit palindrome that is a multiple of 99 ? | 54945 | Write the number as $X Y Z Y X$. This is the same as $10000 X+1000 Y+100 Z+10 Y+X=$ $99(101 X+10 Y+Z)+20 Y+2 X+Z$. We thus want $20 Y+2 X+Z$ to be a multiple of 99 , with $X$ as small as possible. This expression cannot be larger than $20 \cdot 9+2 \cdot 9+9=207$, and it is greater than 0 (since $X \neq 0$ ), so for th... | 0 | 7,755.375 | -1 | 7,755.375 |
Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that... | 134 | By looking at the problem and drawing a few pictures, it quickly becomes obvious that one cannot draw a circle that covers $2$ disjoint areas of the $12$-gon without including all the vertices in between those areas. In other words, in order for a subset to be communal, all the vertices in the subset must be adjacent t... | 0 | 8,192 | -1 | 8,192 |
Given a square initially painted black, with $\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white. | \frac{729}{8192} | 0.875 | 4,436.9375 | 4,403.285714 | 4,672.5 | |
Max sold glasses of lemonade for 25 cents each. He sold 41 glasses on Saturday and 53 glasses on Sunday. What were his total sales for these two days? | $23.50 | 0 | 353 | -1 | 353 | |
Steven subtracts the units digit from the tens digit for each two-digit number. He then finds the sum of all his answers. What is the value of Steven's sum? | 45 | 0.75 | 5,946.9375 | 5,198.583333 | 8,192 | |
The numbers \( a, b, c, d \) belong to the interval \([-13.5, 13.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 756 | 0.25 | 7,963.6875 | 7,727.25 | 8,042.5 | |
A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$? | 19 | 0.3125 | 7,753.3125 | 6,788.2 | 8,192 | |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters with no restrictions? (The triplets are treated as distinguishable.) | 3003 | 0.9375 | 3,321.4375 | 2,996.733333 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. The radius of the circumcircle is $1$, and it is given that $\frac{\tan A}{\tan B} = \frac{2c-b}{b}$. Find the maximum value of the area of $\triangle ABC$. | \frac{\sqrt{3}}{2} | 0 | 7,478.375 | -1 | 7,478.375 | |
Let $x$ and $y$ be real numbers such that
\[4x^2 + 8xy + 5y^2 = 1.\]Let $m$ and $M$ be the minimum and maximum values of $2x^2 + 3xy + 2y^2,$ respectively. Find the product $mM.$ | \frac{7}{16} | 0.5625 | 6,804.6875 | 5,817.666667 | 8,073.714286 | |
Find all positive integers $n$ that satisfy the following inequalities: $$ -46 \leq \frac{2023}{46-n} \leq 46-n $$ | 90 | 0.3125 | 6,710.375 | 6,231.6 | 6,928 | |
Four people can mow a lawn in 6 hours. How many more people will be needed to mow the lawn in 4 hours, assuming each person mows at the same rate? | 2 | 1 | 2,218.0625 | 2,218.0625 | -1 | |
At Frank's Fruit Market, 6 bananas cost as much as 4 apples, and 5 apples cost as much as 3 oranges. In addition, 4 oranges cost as much as 7 pears. How many pears cost as much as 36 bananas? | 28 | 0 | 5,125.625 | -1 | 5,125.625 | |
The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{72}$. Find the greatest possible value of $k$. | 2\sqrt{26} | 0.4375 | 3,376.125 | 2,681.857143 | 3,916.111111 | |
The figure shows a square of side $y$ units divided into a square of side $x$ units and four congruent rectangles. What is the perimeter, in units, of one of the four congruent rectangles? Express your answer in terms of $y$. [asy]
size(4cm);
defaultpen(linewidth(1pt)+fontsize(12pt));
draw((0,0)--(0,4)--(4,4)--(4,0)--c... | 2y | 0.375 | 7,351.875 | 5,951.666667 | 8,192 | |
The positive integer divisors of 252, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 14? | 70 | 0 | 8,192 | -1 | 8,192 | |
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} | 0.9375 | 5,277.0625 | 5,082.733333 | 8,192 | |
Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\circ}$. Find the ratio of the radii of the circles. | 1/3 | 0 | 7,905.875 | -1 | 7,905.875 | |
Daniel works at an electronics store, and he claims that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost. If 12 customers buy a toaster that costs $\$500$, according to Daniel's theory, how many customers would buy a toaster that costs $\$750$? | 8 | 1 | 1,226.4375 | 1,226.4375 | -1 | |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{20} | 0 | 7,872.25 | -1 | 7,872.25 | |
Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \). | 85 | 0.25 | 6,607.8125 | 6,624 | 6,602.416667 | |
The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? | 10 | 1. **Identify the number to factorize**: We start with the number $16{,}383$. We note that $16{,}384 = 2^{14}$, so $16{,}383 = 2^{14} - 1$.
2. **Factorize $16{,}383$**: We use the difference of squares to factorize $16{,}383$:
\[
16{,}383 = 2^{14} - 1 = (2^7)^2 - 1^2 = (2^7 + 1)(2^7 - 1).
\]
Calculating th... | 1 | 2,403.375 | 2,403.375 | -1 |
Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative
integers, defined by the following conditions :
$ (i)$ $ f(0) \equal{} 0$; $ (ii)$ $ f(2n) \equal{} 2f(n)$ and $ (iii)$ $ f(2n \plus{} 1) \equal{} n \plus{} 2f(n)$ for all $ n\geq 0$.
$ (a)$ Determine the... | 2^k - 1 |
### Part (a)
We have the function \( f: \mathbb{N}_0 \rightarrow \mathbb{N}_0 \) defined by:
- \( f(0) = 0 \)
- \( f(2n) = 2f(n) \)
- \( f(2n + 1) = n + 2f(n) \)
We need to determine the sets:
- \( L = \{ n \mid f(n) < f(n + 1) \} \)
- \( E = \{ n \mid f(n) = f(n + 1) \} \)
- \( G = \{ n \mid f(n) > f(n + 1) \} \)
... | 0 | 8,192 | -1 | 8,192 |
A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine the fractional part of the final mixture tha... | \frac{243}{1024} | 0.875 | 5,256.1875 | 4,836.785714 | 8,192 | |
In triangle $ABC$ , $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$ . Let $D$ be a foot of perpendicular from point $A$ to side $BC$ , $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$ . Find $\angle DAE$ | 30 | 0 | 8,022.5 | -1 | 8,022.5 | |
Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area $12$ square units? | 8 | 1. **Given Information and Formula for Area**: We know that the area of a triangle can be expressed as \([PQR] = \frac{1}{2} \cdot PQ \cdot h_R\), where \(h_R\) is the perpendicular distance from point \(R\) to line \(PQ\). Given that \(PQ = 8\) and the area of triangle \(PQR\) is \(12\) square units, we can set up the... | 0.75 | 6,357.125 | 5,872.583333 | 7,810.75 |
Given $x^{2}-5x-2006=0$, evaluate the algebraic expression $\dfrac {(x-2)^{3}-(x-1)^{2}+1}{x-2}$. | 2010 | 0.9375 | 3,407.4375 | 3,088.466667 | 8,192 | |
Triangle $ABC$ is isosceles with angle $B$ congruent to angle $C$. The measure of angle $C$ is four times the measure of angle $A$. What is the number of degrees in the measure of angle $B$? | 80 | 1 | 1,575.4375 | 1,575.4375 | -1 | |
Two individuals, A and B, travel from point $A$ to point $B$. A departs 48 minutes before B. When A has traveled $\frac{2}{3}$ of the total distance, B catches up to A. If B immediately returns to point $A$ after arriving at point $B$ at the same speed, B meets A 6 minutes after leaving point $B$. How many more minutes... | 12 | 0 | 8,039.3125 | -1 | 8,039.3125 | |
For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^6$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y$? | 997 | \begin{eqnarray*} \frac{x+y}{2} &=& \sqrt{xy} + 2\\ x+y-4 &=& 2\sqrt{xy}\\ y - 2\sqrt{xy} + x &=& 4\\ \sqrt{y} - \sqrt{x} &=& \pm 2\end{eqnarray*}
Because $y > x$, we only consider $+2$.
For simplicity, we can count how many valid pairs of $(\sqrt{x},\sqrt{y})$ that satisfy our equation.
The maximum that $\sqrt{y}$ ... | 0.8125 | 5,957.8125 | 5,442.230769 | 8,192 |
A square has two diagonals, and a convex pentagon has five diagonals. How many diagonals does a convex decagon have? | 35 | 1 | 1,497.4375 | 1,497.4375 | -1 | |
What is the measure of the smaller angle between the hands of a 12-hour clock at 12:25 pm, in degrees? Express your answer as a decimal to the nearest tenth. | 137.5\text{ degrees} | 1 | 3,251 | 3,251 | -1 | |
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{3}{\pi} | 0.6875 | 5,193.4375 | 4,386.181818 | 6,969.4 | |
Given the function $f(x) = (2-a)(x-1) - 2\ln x$
(1) When $a=1$, find the intervals of monotonicity for $f(x)$.
(2) If the function $f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$. | 2 - 4\ln 2 | 0.5625 | 7,536.8125 | 7,027.222222 | 8,192 | |
What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} | 0.9375 | 2,337.75 | 2,163.4 | 4,953 | |
Recently, many cities in China have been intensifying efforts to develop the "night economy" to meet the diverse consumption needs of different groups and to boost employment, drive entrepreneurship, and enhance regional economic development vitality. A handicraft seller at a night market found through a survey of dail... | 441 | 0.625 | 5,643.4375 | 4,842.9 | 6,977.666667 | |
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her? | 54 | 1. **Identify the Periodicity**:
Alice, Beatrix, and Claire visit every 3, 4, and 5 days respectively. The least common multiple (LCM) of 3, 4, and 5 is:
\[
\text{LCM}(3, 4, 5) = 60
\]
This means every 60 days, all three friends visit Daphne together.
2. **Divide the 365-day period**:
Since the visi... | 0.5 | 6,961 | 5,730 | 8,192 |
What is the smallest positive integer with exactly 16 positive divisors? | 384 | 0 | 5,088.8125 | -1 | 5,088.8125 | |
Alice has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal rounded to the... | 0.171 | 0.625 | 6,886.3125 | 6,102.9 | 8,192 | |
Given the expression \[2 - (-3) - 4 - (-5) - 6 - (-7) \times 2,\] calculate its value. | -14 | 0 | 4,115.875 | -1 | 4,115.875 | |
(1) Use the Horner's method to calculate the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 79x^3 - 8x^2 + 35x + 12$ when $x = -4$, find the value of $v_3$.
(2) Convert the hexadecimal number $210_{(6)}$ into a decimal number. | 78 | 0.5625 | 3,866.5 | 3,369.444444 | 4,505.571429 | |
The probability that a car driving on this road does not stop at point A and the probability that it does not stop at point B and the probability that it does not stop at point C are $\left(1-\frac{25}{60}\right)$, $\left(1-\frac{35}{60}\right)$, and $\left(1-\frac{45}{60}\right)$, respectively. | \frac{35}{192} | 0.0625 | 4,944.875 | 7,301 | 4,787.8 | |
A point is marked one quarter of the way along each side of a triangle. What fraction of the area of the triangle is shaded? | $\frac{5}{8}$ | 0 | 7,880.25 | -1 | 7,880.25 | |
A set of $36$ square blocks is arranged into a $6 \times 6$ square. How many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column? | 5400 | 0.8125 | 5,665.5 | 5,082.461538 | 8,192 | |
What is the largest integer that must divide the product of any $5$ consecutive integers? | 240 | 0 | 6,639.8125 | -1 | 6,639.8125 | |
Let $m$ be the smallest positive three-digit integer congruent to 7 (mod 13). Let $n$ be the smallest positive four-digit integer congruent to 7 (mod 13). What is the value of $n - m$? | 897 | 0.9375 | 3,024.875 | 2,940.866667 | 4,285 | |
A particle is located on the coordinate plane at $(5,0)$. Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest integer le... | 19 | Let $T:\begin{pmatrix}x\\y\end{pmatrix}\rightarrow R(\frac{\pi}{4})\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}10\\0\end{pmatrix}$. We assume that the rotation matrix $R(\frac{\pi}{4}) = R$ here. Then we have
$T^{150}\begin{pmatrix}5\\0\end{pmatrix}=R(R(...R(R\begin{pmatrix}5\\0\end{pmatrix}+\begin{pmatrix}10\\0\en... | 0.125 | 7,905.125 | 7,740 | 7,928.714286 |
In the $xy$ -coordinate plane, the $x$ -axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$ -axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$ . They are $(126, 0... | 111 | 0.125 | 8,120.3125 | 7,618.5 | 8,192 | |
We know the following to be true:
$\bullet$ 1. $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$
$\bullet$ 2. $Z$ = $K \times K^2.$
What is the value of $K$ for which $Z$ is a perfect square? | 9 | 0.875 | 1,170.125 | 1,213.285714 | 868 | |
In the plane Cartesian coordinate system \( xOy \), the circle \( \Omega \) intersects the parabola \( \Gamma: y^{2} = 4x \) at exactly one point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \). | \frac{4 \sqrt{3}}{9} | 0 | 7,197.375 | -1 | 7,197.375 | |
Given that the side lengths of a convex quadrilateral are $a=4, b=5, c=6, d=7$, find the radius $R$ of the circumscribed circle around this quadrilateral. Provide the integer part of $R^{2}$ as the answer. | 15 | 0.5 | 7,543.1875 | 6,894.375 | 8,192 | |
Given a quadratic equation \( x^{2} + bx + c = 0 \) with roots 98 and 99, within the quadratic function \( y = x^{2} + bx + c \), if \( x \) takes on values 0, 1, 2, 3, ..., 100, how many of the values of \( y \) are divisible by 6? | 67 | 0.5625 | 6,497.6875 | 5,463.444444 | 7,827.428571 | |
Let the three-digit number \( n = abc \). If the digits \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle, how many such three-digit numbers exist? | 165 | 0 | 8,192 | -1 | 8,192 | |
In the diagram, square $PQRS$ has side length 2. Points $M$ and $N$ are the midpoints of $SR$ and $RQ$, respectively. The value of $\cos (\angle MPN)$ is | $\frac{4}{5}$ | 0 | 2,427.5625 | -1 | 2,427.5625 | |
Class 2 of the second grade has 42 students, including $n$ male students. They are numbered from 1 to $n$. During the winter vacation, student number 1 called 3 students, student number 2 called 4 students, student number 3 called 5 students, ..., and student number $n$ called half of the students. Determine the number... | 23 | 0.125 | 7,619.625 | 4,002 | 8,136.428571 | |
Find the domain of the function $f(x) = \tan(\arccos(x^2)).$ | [-1,0) \cup (0,1] | 1 | 3,283.375 | 3,283.375 | -1 | |
Suppose that $f(x)$ is a function such that
\[f(xy) + x = xf(y) + f(x)\]for all real numbers $x$ and $y.$ If $f(-1) = 5$ then compute $f(-1001).$ | 2005 | 1 | 2,050.75 | 2,050.75 | -1 | |
Jack and Jill run 10 km. They start at the same point, run 5 km up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when th... | \frac{35}{27} | 1. **Set up the equations for Jack and Jill's movements:**
- Jack starts 10 minutes (or $\frac{1}{6}$ hours) before Jill.
- Jack's speed uphill is 15 km/hr and downhill is 20 km/hr.
- Jill's speed uphill is 16 km/hr and downhill is 22 km/hr.
2. **Calculate the time Jack and Jill take to reach the top of the ... | 0.0625 | 7,599.125 | 7,571 | 7,601 |
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R} $$ For $f \in \mathcal{F},$ let $$ I(f)=\int_0^ef(x) dx $$ Determine $\min_{f \in \mathcal{F}}I(f).$ *Liviu Vlaicu* | \frac{3}{2} | 0.25 | 6,974.5625 | 3,884 | 8,004.75 | |
Among the 100 natural numbers from 1 to 100, how many numbers can be represented as \( m \cdot n + m + n \) where \( m \) and \( n \) are natural numbers? | 74 | 0.125 | 7,648.8125 | 6,589.5 | 7,800.142857 | |
If \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) are integers for which \(729x^3+64 = (ax^2 + bx + c)(dx^2 + ex + f)\) for all \(x\), then what is \(a^2+b^2+c^2+d^2+e^2+f^2\)? | 8210 | 0.125 | 7,187.875 | 2,742 | 7,823 | |
The circumference of a circle $A$ is 60 feet. How many feet long is $\widehat{BC}$? [asy]
import markers;
import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.9));
draw(Circle(origin,1));
draw(dir(90)--origin--dir(30));
label("$B$",dir(90),N);
label("$A$",origin,S);
label("$C$",dir(30),E);... | 10 | 0.9375 | 1,447.0625 | 1,268.933333 | 4,119 | |
Given that $\cos(\alpha - \beta) = \frac{3}{5}$, $\sin(\beta) = -\frac{5}{13}$, where $\alpha \in \left(0, \frac{\pi}{2} \right)$, $\beta \in \left(-\frac{\pi}{2}, 0 \right)$, find the value of $\sin(\alpha)$. | \frac{33}{65} | 0.625 | 6,873.625 | 6,131.6 | 8,110.333333 | |
Given a sector with a radius of 16, and the arc length of the sector is $16\pi$, calculate the central angle and the area of the sector. | 128\pi | 1 | 1,290.125 | 1,290.125 | -1 | |
We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either
red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.
A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour.
Suppose that there are $... | 859 |
Given a regular polygon \( P \) with 43 vertices, each segment (sides and diagonals) of this polygon is colored either red or blue. We know the following conditions:
- Every vertex is an endpoint of 20 red segments.
- Every vertex is an endpoint of 22 blue segments.
Since every vertex is connected to every other vert... | 0.0625 | 8,159.125 | 8,025 | 8,168.066667 |
Express the quotient $1121_5 \div 12_5$ in base $5$. | 43_5. | 0 | 4,492.9375 | -1 | 4,492.9375 | |
Each of the eight vertices of a rectangular prism is truncated, and each face of the prism is divided by a diagonal cut into two triangles. Calculate the total number of edges of the altered shape. | 42 | 0.0625 | 7,735 | 6,322 | 7,829.2 | |
Given the function $f(x)=\sqrt{3}\sin x\cos x-{\cos }^2x$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ If $f(x)=-1$, find the value of $\cos \left(\dfrac{2\pi }{3}-2x\right)$. | -\dfrac{1}{2} | 1 | 5,319.5 | 5,319.5 | -1 | |
A round cake is cut into \( n \) pieces with 3 cuts. Find the product of all possible values of \( n \). | 840 | 0.25 | 6,184.5 | 5,140.25 | 6,532.583333 | |
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next $n$ shots are bullseyes she will be guaranteed victory. What is the minimu... | 42 | 1. **Identify the current situation**: Chelsea is leading by 50 points after 50 shots. Let $k$ be the number of points Chelsea has scored so far.
2. **Determine the maximum possible score for the opponent**: Since the opponent is 50 points behind Chelsea, if the opponent scores bullseyes (10 points each) for the remai... | 0.375 | 5,143.5625 | 3,854 | 5,917.3 |
Given the function $f(x)=kx^{2}+2kx+1$ defined on the interval $[-3,2]$, the maximum value of the function is $4$. Determine the value of the real number $k$. | -3 | 0.5 | 7,485.25 | 6,906.125 | 8,064.375 | |
Calculate $[x]$, where $x = -3.7 + 1.5$. | -3 | 0.9375 | 1,563.375 | 1,602.733333 | 973 | |
Can we find \( N \) such that all \( m \times n \) rectangles with \( m, n > N \) can be tiled with \( 4 \times 6 \) and \( 5 \times 7 \) rectangles? | 840 | 0 | 8,192 | -1 | 8,192 | |
A garden can be watered by any of three sprinklers X, Y, or Z. Sprinklers X and Y together take 5 hours to water the garden. Sprinklers X and Z together take 6 hours to water the garden. Sprinklers Y and Z together take 7 hours to water the garden. How many hours does it take sprinklers X, Y, and Z working together to ... | 3.93 | 0.875 | 5,382.8125 | 5,180.285714 | 6,800.5 | |
The lengths of the three sides of a triangle are \( 10 \), \( y+5 \), and \( 3y-2 \). The perimeter of the triangle is \( 50 \). What is the length of the longest side of the triangle? | 25.75 | 0.0625 | 7,876.8125 | 6,262 | 7,984.466667 | |
There is a committee composed of 10 members who meet around a table: 7 women, 2 men, and 1 child. The women sit in indistinguishable rocking chairs, the men on indistinguishable stools, and the child on a bench, also indistinguishable from any other benches. How many distinct ways are there to arrange the seven rocking... | 360 | 0.0625 | 4,038.875 | 1,700 | 4,194.8 | |
How many numbers, divisible by 4 and less than 1000, do not contain any of the digits 6, 7, 8, 9, or 0? | 31 | 0.25 | 6,839.25 | 6,162.25 | 7,064.916667 | |
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: | 5 | 1. **Identify the nature of the sequence**: Given the recurrence relation $u_{n+1} - u_n = 3 + 4(n-1)$, we can simplify this to $u_{n+1} - u_n = 4n - 1$. This indicates that the sequence $\{u_n\}$ is defined by a quadratic polynomial because the difference between consecutive terms is a linear function.
2. **Determine... | 1 | 2,945.125 | 2,945.125 | -1 |
At a school cafeteria, Jenny wants to buy a meal consisting of one main dish, one drink, one dessert, and one side dish. The list below contains Jenny's preferred choices available:
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Main Dishes} & \textbf{Drinks} & \textbf{Desserts} & \textbf{Side Dishes} \\
\hline
Spaghetti... | 48 | 0.0625 | 1,371.1875 | 430 | 1,433.933333 | |
\(f(x)\) is a linear function, and the equation \(f(f(x)) = x + 1\) has no solutions. Find all possible values of \(f(f(f(f(f(2022)))))-f(f(f(2022)))-f(f(2022))\). | -2022 | 0.5625 | 5,638.6875 | 3,652.777778 | 8,192 | |
What integer $n$ satisfies $0\le n<9$ and $$-1111\equiv n\pmod 9~?$$ | 5 | 1 | 3,191.375 | 3,191.375 | -1 | |
In the polar coordinate system, the curve $C_1$: $\rho=2\cos\theta$, and the curve $$C_{2}：\rho\sin^{2}\theta=4\cos\theta$$．Establish a Cartesian coordinate system xOy with the pole as the origin and the polar axis as the positive half-axis of x, the parametric equation of curve C is $$\begin{cases} x=2+ \frac {1}{2}t ... | \frac {11}{3} | 0.1875 | 7,734.375 | 7,049 | 7,892.538462 | |
Calculate: $(243)^{\frac35}$ | 27 | 0.9375 | 3,354.875 | 3,032.4 | 8,192 | |
What is the sum of the even, positive integers less than 62? | 930 | 1 | 2,346.8125 | 2,346.8125 | -1 | |
In $\triangle ABC$, $\angle C= \frac{\pi}{2}$, $\angle B= \frac{\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\sqrt{2}$. The surface area of the circumscribed sphere of the tetrahedron $M-ABC$ is \_\_\_\_\_\_. | 16\pi | 0.0625 | 8,170.875 | 7,854 | 8,192 | |
1. Solve the trigonometric inequality: $\cos x \geq \frac{1}{2}$
2. In $\triangle ABC$, if $\sin A + \cos A = \frac{\sqrt{2}}{2}$, find the value of $\tan A$. | -2 - \sqrt{3} | 0.625 | 6,230.1875 | 5,544.4 | 7,373.166667 |
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