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 |
|---|---|---|---|---|---|---|
If $F(a, b, c, d) = a^b + c \times d$, what is the value of $x$ such that $F(2, x, 4, 11) = 300$? | 8 | 1 | 1,009.5625 | 1,009.5625 | -1 | |
Convert the point $(6,2 \sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 4 \sqrt{3}, \frac{\pi}{6} \right) | 1 | 1,782.8125 | 1,782.8125 | -1 | |
It is currently 3:00:00 PM, as shown on a 12-hour digital clock. In 300 hours, 55 minutes, and 30 seconds, what will the time be and what is the sum of the hours, minutes, and seconds? | 88 | 0.3125 | 7,256.125 | 6,651.8 | 7,530.818182 | |
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probabili... | 125 | Let $A$ be Azar, $C$ be Carl, $J$ be Jon, and $S$ be Sergey. The $4$ circles represent the $4$ players, and the arrow is from the winner to the loser with the winning probability as the label.
This problem can be solved by using $2$ cases.
$\textbf{Case 1:}$ $C$'s opponent for the semifinal is $A$
The probability $C$... | 0 | 6,096.9375 | -1 | 6,096.9375 |
What is the sum of the six positive integer factors of 30? | 72 | 0.5 | 5,986.3125 | 5,715.75 | 6,256.875 | |
In triangle $\triangle ABC$, it is known that the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $a=\sqrt{6}$, $\sin ^{2}B+\sin ^{2}C=\sin ^{2}A+\frac{2\sqrt{3}}{3}\sin A\sin B\sin C$. Choose one of the following conditions to determine whether triangle $\triangle ABC$ exists. If it exists, find the... | \frac{\sqrt{3}}{2} | 0 | 7,961.1875 | -1 | 7,961.1875 | |
Given an arithmetic sequence $\{a_{n}\}$ and $\{b_{n}\}$ with the sums of the first $n$ terms being $S_{n}$ and $T_{n}$, respectively, if $\frac{S_n}{T_n}=\frac{3n+4}{n+2}$, find $\frac{a_3+a_7+a_8}{b_2+b_{10}}$. | \frac{111}{26} | 0.8125 | 5,526.4375 | 5,502.076923 | 5,632 | |
Given that $E, U, L, S, R,$ and $T$ represent the digits $1, 2, 3, 4, 5, 6$ (each letter represents a unique digit), and the following conditions are satisfied:
1. $E + U + L = 6$
2. $S + R + U + T = 18$
3. $U \times T = 15$
4. $S \times L = 8$
Determine the six-digit number $\overline{EULSRT}$. | 132465 | 0.75 | 3,299.0625 | 3,268.666667 | 3,390.25 | |
Given vectors $\overrightarrow {m}=(a,-1)$, $\overrightarrow {n}=(2b-1,3)$ where $a > 0$ and $b > 0$. If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, determine the value of $\dfrac{2}{a}+\dfrac{1}{b}$. | 8+4\sqrt {3} | 0 | 7,979.1875 | -1 | 7,979.1875 | |
$\triangle PQR$ is inscribed inside $\triangle XYZ$ such that $P, Q, R$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle PYZ, \triangle QXR, \triangle RQP$ have centers $O_4, O_5, O_6$, respectively. Also, $XY = 29, YZ = 35, XZ=28$, and $\stackrel{\frown}{YR} = \stackrel{\frown}{QZ},\ \stackrel{\frown... | 31 | 0.0625 | 7,979.625 | 4,794 | 8,192 | |
Given that triangle XYZ is a right triangle with two altitudes of lengths 6 and 18, determine the largest possible integer length for the third altitude. | 12 | 0 | 8,019.375 | -1 | 8,019.375 | |
Let the function $f(x)=\ln x-\frac{1}{2} ax^{2}-bx$.
$(1)$ When $a=b=\frac{1}{2}$, find the maximum value of the function $f(x)$;
$(2)$ Let $F(x)=f(x)+\frac{1}{2} x^{2}+bx+\frac{a}{x} (0 < x\leqslant 3)$. If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph is always less than or equal to $... | \frac{1}{2} | 0 | 8,158.0625 | -1 | 8,158.0625 | |
Let $A=3^{7}+\binom{7}{2}3^{5}+\binom{7}{4}3^{3}+\binom{7}{6}3$, $B=\binom{7}{1}3^{6}+\binom{7}{3}3^{4}+\binom{7}{5}3^{2}+1$. Find $A-B$. | 128 | 0.3125 | 7,421.3125 | 5,725.8 | 8,192 | |
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once? | 1/4 | 0.4375 | 6,503.5625 | 6,588.857143 | 6,437.222222 | |
For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$, then what is the minimum possible value of $|A\cap B\cap C|$? | 97 | 1. **Given Information and Basic Calculations**:
- $|A| = |B| = 100$, so $n(A) = n(B) = 2^{100}$.
- The formula for the number of subsets of a set $S$ is $n(S) = 2^{|S|}$.
- Given $n(A) + n(B) + n(C) = n(A \cup B \cup C)$, substituting the values, we get:
\[
2^{100} + 2^{100} + 2^{|C|} = 2^{|A \cup B... | 0.125 | 7,852.9375 | 6,659.5 | 8,023.428571 |
50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. How many students posted exactly one photo? | 46 | 0.0625 | 7,776.0625 | 3,558 | 8,057.266667 | |
What is the area, in square units, of a triangle with vertices at $(0,0)$, $(0,5)$, and $(7,12)$? Express your answer as a decimal to the nearest tenth. | 17.5 | 1 | 3,690.375 | 3,690.375 | -1 | |
What is the greatest possible three-digit number that is divisible by 4 and divisible by 5? | 980 | 0.9375 | 1,122.3125 | 651 | 8,192 | |
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\]What is $f(1)$? | -7007 | 0.875 | 4,228.6875 | 4,015.428571 | 5,721.5 | |
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren? | 146 | 1. **Identify the problem and apply the Principle of Inclusion-Exclusion (PIE):**
We need to find the number of days Mrs. Sanders does not receive a call from any of her grandchildren. Each grandchild calls her every 3, 4, and 5 days respectively. We use PIE to count the days she receives at least one call and subtr... | 0.5625 | 6,571.1875 | 5,310.555556 | 8,192 |
Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number. | 1152 | 0.6875 | 4,453.125 | 3,637.272727 | 6,248 | |
Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$. | 574 | This is mostly a clarification to Solution 1, but let's take the diagram for the origin to $(7,3)$. We have the origin circle and square intersected, then two squares, then the circle and square at $(7,3)$. If we take the circle and square at the origin out of the diagram, we will be able to repeat the resulting segmen... | 0 | 7,052.5625 | -1 | 7,052.5625 |
A particle is placed on the parabola $y = x^2- x -6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $x$-coordinates of... | 4 | To solve this problem, we need to find the $x$-coordinates of points $P$ and $Q$ on the parabola $y = x^2 - x - 6$ where $y = 6$ and $y = -6$, respectively, and then calculate the horizontal distance between these points.
1. **Finding $x$-coordinates for $P$ where $y = 6$:**
\[
x^2 - x - 6 = 6
\]
Simplifyi... | 0 | 5,308.75 | -1 | 5,308.75 |
A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo.
A psychologist noticed the following:
1.... | 7/8 | 0.125 | 6,342.3125 | 5,424 | 6,473.5 | |
Suppose $P(x)$ is a polynomial with real coefficients such that $P(t)=P(1) t^{2}+P(P(1)) t+P(P(P(1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$. | \frac{1}{9} | Let $(a, b, c):=(P(1), P(P(1)), P(P(P(1))))$, so $P(t)=a t^{2}+b t+c$ and we wish to maximize $P(c)$. Then we have that $$\begin{aligned} a & =P(1)=a+b+c \\ b & =P(a)=a^{3}+a b+c \\ c & =P(b)=a b^{2}+b^{2}+c \end{aligned}$$ The first equation implies $c=-b$. The third equation implies $b^{2}(a+1)=0$, so $a=-1$ or $b=0$... | 0.3125 | 7,559.8125 | 6,169 | 8,192 |
Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$.
If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals: | 4a | 1. **Identify the expression for $r$:**
Given that $r$ is the result of doubling both the base and the exponent of $a^b$, we have:
\[ r = (2a)^{2b} \]
2. **Set up the equation using the given condition:**
According to the problem, $r$ equals the product of $a^b$ by $x^b$. Therefore, we can write:
\[ (2... | 0.9375 | 1,008.4375 | 1,039.8 | 538 |
In $\triangle ABC$, it is known that $\cos A= \frac {3}{5},\cos B= \frac {5}{13}$, and $AC=3$. Find the length of $AB$. | \frac {14}{5} | 0.8125 | 5,260.1875 | 4,933.538462 | 6,675.666667 | |
Find the smallest value of $n$ for which the series \[1\cdot 3^1 + 2\cdot 3^2 + 3\cdot 3^3 + \cdots + n\cdot 3^n\] exceeds $3^{2007}$ . | 2000 | 0.0625 | 8,069.375 | 6,230 | 8,192 | |
Let $\mathbb{N}_{>1}$ denote the set of positive integers greater than 1. Let $f: \mathbb{N}_{>1} \rightarrow \mathbb{N}_{>1}$ be a function such that $f(mn)=f(m)f(n)$ for all $m, n \in \mathbb{N}_{>1}$. If $f(101!)=101$!, compute the number of possible values of $f(2020 \cdot 2021)$. | 66 | For a prime $p$ and positive integer $n$, we let $v_{p}(n)$ denote the largest nonnegative integer $k$ such that $p^{k} \mid n$. Note that $f$ is determined by its action on primes. Since $f(101!)=101$!, by counting prime factors, $f$ must permute the set of prime factors of 101!; moreover, if $p$ and $q$ are prime fac... | 0 | 7,895.8125 | -1 | 7,895.8125 |
Calculate the sum:
\[\sum_{N = 1}^{2048} \lfloor \log_3 N \rfloor.\] | 12049 | 0 | 6,733.625 | -1 | 6,733.625 | |
Vasya wrote down 11 natural numbers in one notebook. Petya wrote down the greatest common divisors of each pair of numbers recorded in Vasya's notebook in another notebook. It turned out that every number written in one of the two notebooks is also in the other notebook. What is the maximum number of distinct numbers t... | 10 | 0 | 7,926.625 | -1 | 7,926.625 | |
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$? | 14 | 1. **Identify the form of the terms**: We are interested in terms that include all four variables $a, b, c, d$ each raised to some positive power in the expansion of $(a+b+c+d+1)^N$. This means we are looking for terms of the form $a^x b^y c^z d^w 1^t$ where $x, y, z, w > 0$ and $x + y + z + w + t = N$.
2. **Adjust th... | 0.875 | 3,684.5625 | 3,040.642857 | 8,192 |
The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y+3)^2 = 1$ are tangent. Compute $m$. | \frac{8}{9} | 0 | 5,857.25 | -1 | 5,857.25 | |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x=2+t\cos \alpha \\
y=1+t\sin \alpha
\end{cases}
(t \text{ is the parameter}),
$$
In the polar coordinate system (which uses the same unit length as the Cartesian coordinate system $xOy$, with the origin as the pole and th... | 2\sqrt{7} | 0.875 | 5,772.25 | 5,426.571429 | 8,192 | |
Given $x=1-2a$, $y=3a-4$.
$(1)$ Given that the arithmetic square root of $x$ is $3$, find the value of $a$;
$(2)$ If $x$ and $y$ are both square roots of the same number, find this number. | 25 | 0.6875 | 6,701.6875 | 6,219.363636 | 7,762.8 | |
What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)? | 64 | 0 | 8,192 | -1 | 8,192 | |
To be able to walk to the center $C$ of a circular fountain, a repair crew places a 16-foot plank from $A$ to $B$ and then a 10-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$ . What is the area of the circular base of the fountain? Express your answer in terms of $\pi$. [asy]
size(250); impor... | 164 \pi \mbox{ square feet} | 0 | 1,910.375 | -1 | 1,910.375 | |
Starting with 1, alternately add 4 and 3 to obtain the sequence $1, 5, 8, 12, 15, 19, 22, \ldots \ldots$. In this sequence, the number closest to 2013 is $\qquad$ . | 2014 | 0.25 | 8,119.0625 | 7,900.25 | 8,192 | |
The congruent sides of an isosceles triangle are each 8 cm long, and the perimeter is 26 cm. In centimeters, what is the length of the base? Also, find the area of the triangle. | 5\sqrt{39} | 1 | 2,668.875 | 2,668.875 | -1 | |
Let positive integers \( a \) and \( b \) be such that \( 15a + 16b \) and \( 16a - 15b \) are both perfect squares. Find the smallest possible value of the smaller of these two squares.
| 481^2 | 0 | 8,192 | -1 | 8,192 | |
What is the largest prime factor of the sum of $1579$ and $5464$? | 7043 | 0.3125 | 7,280.4375 | 5,768.6 | 7,967.636364 | |
Assuming that $x$ is a multiple of $18711$, determine the greatest common divisor of $g(x) = (4x+5)(5x+3)(6x+7)(3x+11)$ and $x$. | 1155 | 0 | 5,651.125 | -1 | 5,651.125 | |
You are trying to cross a 6 foot wide river. You can jump at most 4 feet, but you have one stone you can throw into the river; after it is placed, you may jump to that stone and, if possible, from there to the other side of the river. However, you are not very accurate and the stone ends up landing uniformly at random ... | \frac{1}{3} | To be able to cross, the stone must land between 2 and 4 feet from the river bank you are standing on. Therefore the probability is $\frac{2}{6}=\frac{1}{3}$. | 0.5625 | 5,355.8125 | 4,658.666667 | 6,252.142857 |
Mr. and Mrs. Smith have three children. They own a family van with a driver's seat, a passenger seat in the front, and two seats in the back. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible if one of the children insists on sitting in the front passenger seat? | 12 | 0.5625 | 4,094.9375 | 3,664.777778 | 4,648 | |
Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 25$, is $f(n)$ prime? | 5 | 1 | 6,612.25 | 6,612.25 | -1 | |
Luke wants to fence a rectangular piece of land with an area of at least 450 square feet. The length of the land is 1.5 times the width. What should the width of the rectangle be if he wants to use the least amount of fencing? | 10\sqrt{3} | 0.9375 | 2,867.5 | 2,744.333333 | 4,715 | |
In a convex 1950-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides. What is the greatest number of sides it can have? | 1949 | 0.3125 | 7,802.5625 | 6,945.8 | 8,192 | |
Thirty teams play in a league where each team plays every other team exactly once, and every game results in a win or loss with no ties. Each game is independent with a $50\%$ chance of either team winning. Determine the probability that no two teams end up with the same number of total victories, expressed as $\frac{p... | 409 | 0.1875 | 6,590.9375 | 4,515 | 7,070 | |
Given the function $f(x)=e^{-x}+ \frac {nx}{mx+n}$.
$(1)$ If $m=0$, $n=1$, find the minimum value of the function $f(x)$.
$(2)$ If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac {m}{n}$. | \frac {1}{2} | 0.1875 | 7,884.25 | 6,593.333333 | 8,182.153846 | |
In $\triangle ABC, AB = 360, BC = 507,$ and $CA = 780.$ Let $M$ be the midpoint of $\overline{CA},$ and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC.$ Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}.$ Suppose that $\overline{DF}$ meets $\overl... | 289 | Firstly, angle bisector theorem yields $\frac{CD}{AD} = \frac{507}{360} = \frac{169}{120}$. We're given that $AM=MC$. Therefore, the cross ratio
\[(A,C;M,D) = \frac{AM(CD)}{AD(MC)} = \frac{169}{120}\]
We need a fourth point for this cross ratio to be useful, so reflect point $F$ over angle bisector $BD$ to a point $F'... | 0 | 8,192 | -1 | 8,192 |
Let \( a < b < c < d < e \) be real numbers. We calculate all the possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of \( e \). | \frac{55}{2} | 0.1875 | 7,893.3125 | 6,599 | 8,192 | |
Suppose the function \( y= \left| \log_{2} \frac{x}{2} \right| \) has a domain of \([m, n]\) and a range of \([0,2]\). What is the minimum length of the interval \([m, n]\)? | 3/2 | 0.0625 | 7,730.3125 | 8,192 | 7,699.533333 | |
In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.
| 90 | 0.0625 | 8,068.3125 | 6,213 | 8,192 | |
Mr. Wong has 10 grandchildren. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability that Mr. Wong has more grandsons than granddaughters or more granddaughters than grandsons? | \frac{193}{256} | 1 | 3,363.875 | 3,363.875 | -1 | |
$ A$ and $ B$ play the following game with a polynomial of degree at least 4:
\[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0
\]
$ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the res... | B |
In this game, Player \( A \) and Player \( B \) take turns filling in the coefficients of the polynomial
\[ P(x) = x^{2n} + a_{2n-1} x^{2n-1} + a_{2n-2} x^{2n-2} + \ldots + a_1 x + 1. \]
Player \( A \) wins if the resulting polynomial has no real roots, and Player \( B \) wins if it has at least one real root. We ne... | 0 | 8,082.5625 | -1 | 8,082.5625 |
Given the scores of the other five students are $83$, $86$, $88$, $91$, $93$, and Xiaoming's score is both the mode and the median among these six scores, find Xiaoming's score. | 88 | 0.5 | 1,608.1875 | 2,025 | 1,191.375 | |
Given the function $f(x)=x+\sqrt{1-x}$, determine the minimum value of $f(x)$. | \frac{5}{4} | 0.0625 | 7,740.25 | 2,047 | 8,119.8 | |
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left... | f(n) = n |
Given the problem, we seek all functions \( f: \mathbb{N}_{\geq 1} \to \mathbb{N}_{\geq 1} \) such that for all positive integers \( m \) and \( n \), the following holds:
\[
\mathrm{GCD}(f(m), n) + \mathrm{LCM}(m, f(n)) = \mathrm{GCD}(m, f(n)) + \mathrm{LCM}(f(m), n).
\]
To solve this, let's explore the properties ... | 0.0625 | 8,101.25 | 6,740 | 8,192 |
The sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$) is arranged sequentially in brackets such that the first bracket contains one number, the second bracket contains two numbers, the third bracket contains three numbers, the fourth bracket contains four numbers, the fifth bracket contains one number, the sixth bracket contain... | 2072 | 0 | 7,507.6875 | -1 | 7,507.6875 | |
The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\frac{{b}^{2}}{4}+\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$. | \sqrt{14} | 0.875 | 5,477.0625 | 5,089.214286 | 8,192 | |
There are 5 blue chips, 4 red chips and 3 yellow chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction. | \frac{47}{72} | 1 | 3,025.625 | 3,025.625 | -1 | |
If the 3-digit decimal number \( n = \overline{abc} \) satisfies that \( a \), \( b \), and \( c \) form an arithmetic sequence, then what is the maximum possible value of a prime factor of \( n \)? | 317 | 0.125 | 8,154.4375 | 7,891.5 | 8,192 | |
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real n... | 80 | 1. For $n$ to be uniquely determined, $n$ AND $n + 1$ both need to be a multiple of $4, 5,$ or $6.$ Since either $n$ or $n + 1$ is odd, we know that either $n$ or $n + 1$ has to be a multiple of $5.$ We can state the following cases:
1. $n$ is a multiple of $4$ and $n+1$ is a multiple of $5$
2. $n$ is a multiple of $... | 0 | 8,192 | -1 | 8,192 |
In triangle $ABC,$ $\angle B = 45^\circ,$ $AB = 100,$ and $AC = 100$. Find the sum of all possible values of $BC$. | 100 \sqrt{2} | 0.75 | 6,420.625 | 5,830.166667 | 8,192 | |
While moving down an escalator, Walker counted 50 steps. Trotman, who moved three times faster than Walker, counted 75 steps. If the escalator were to stop, how many steps could be counted on its visible part? It is assumed that both individuals moved at a constant speed and that the escalator's speed was constant. | 100 | 0.8125 | 5,176.875 | 4,481.076923 | 8,192 | |
For the positive integer $n$, let $\langle n\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\langle 4\rangle=1+2=3$ and $\langle 12 \rangle =1+2+3+4+6=16$. What is $\langle\langle\langle 6\rangle\rangle\rangle$? | 6 | To solve for $\langle\langle\langle 6\rangle\rangle\rangle$, we need to first determine $\langle 6\rangle$, then $\langle \langle 6 \rangle \rangle$, and finally $\langle \langle \langle 6 \rangle \rangle \rangle$.
1. **Calculate $\langle 6 \rangle$:**
The divisors of $6$ are $1, 2, 3, 6$. The sum of all divisors e... | 1 | 2,021 | 2,021 | -1 |
Compute $a^2 + b^2 + c^2,$ given that $a,$ $b,$ and $c$ are the roots of \[2x^3 - x^2 + 4x + 10 = 0.\] | -\frac{15}4 | 0 | 4,298.875 | -1 | 4,298.875 | |
Given the circle $C: x^2 + y^2 - (6 - 2m)x - 4my + 5m^2 - 6m = 0$, and a fixed line $l$ passing through the point $A(1, 0)$, for any real number $m$, the chord intercepted by circle $C$ on line $l$ always has a constant length $A$. Find the constant value of $A$. | \frac{2\sqrt{145}}{5} | 0 | 7,592.875 | -1 | 7,592.875 | |
A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats? | 840 | 1 | 1,649.8125 | 1,649.8125 | -1 | |
In the figure, triangles $ABC$ and $BCD$ are equilateral triangles. What is the value of $AD \div BC$ when expressed in simplest radical form?
[asy]
draw((0,0)--(5,8.7)--(10,0)--cycle);
draw((10,0)--(15,8.7)--(5,8.7));
label("$A$",(0,0),SW);
label("$B$",(5,8.7),N);
label("$C$",(10,0),SE);
label("$D$",(15,8.7),NE);
[/a... | \sqrt{3} | 0.5625 | 6,527.5 | 5,232.888889 | 8,192 | |
Given that \\(\alpha\\) is an angle in the second quadrant, and \\(\sin (π+α)=- \frac {3}{5}\\), find the value of \\(\tan 2α\\). | - \frac {24}{7} | 0.875 | 3,859.4375 | 3,240.5 | 8,192 | |
Let $[x]$ represent the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies $\left[\frac{1}{15}\right] + \left[\frac{2}{15}\right] + \left[\frac{3}{15}\right] + \cdots + \left[\frac{n-1}{15}\right] + \left[\frac{n}{15}\right] > 2011$, what is the smallest value of $n$? | 253 | 0 | 8,087.5625 | -1 | 8,087.5625 | |
How many such pairs of numbers \((n, k)\) are there, for which \(n > k\) and the difference between the internal angles of regular polygons with \(n\) and \(k\) sides is \(1^{\circ}\)? | 52 | 0.3125 | 7,741.1875 | 6,749.4 | 8,192 | |
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, ot... | 13 | 0.4375 | 6,526.625 | 5,055.714286 | 7,670.666667 | |
A regular $17$ -gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$ . A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ such that the chord now stretches from $V... | 4\pi | 0.375 | 6,572.125 | 4,977 | 7,529.2 | |
A group of friends are sharing a bag of candy. On the first day, they eat $ rac{1}{2}$ of the candies in the bag. On the second day, they eat $ rac{2}{3}$ of the remaining candies. On the third day, they eat $ rac{3}{4}$ of the remaining candies. On the fourth day, they eat $ rac{4}{5}$ of the remaining candies. On the... | 720 | We work backwards through the given information. At the end, there is 1 candy remaining. Since $ rac{5}{6}$ of the candies are removed on the fifth day, this 1 candy represents $ rac{1}{6}$ of the candies left at the end of the fourth day. Thus, there were $6 imes 1=6$ candies left at the end of the fourth day. Since ... | 0.9375 | 2,757.375 | 2,395.066667 | 8,192 |
The difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the same digits, increased by 4/3. If you subtr... | 421 | 0 | 8,192 | -1 | 8,192 | |
A student types the following pattern on a computer (where '〇' represents an empty circle and '●' represents a solid circle): 〇●〇〇●〇〇〇●〇〇〇〇●... If this pattern of circles continues, what is the number of solid circles among the first 2019 circles? | 62 | 0.1875 | 7,184.25 | 5,747.333333 | 7,515.846154 | |
Given the function $f(x) = ax^3 + (a-1)x^2 + 27(a-2)x + b$, its graph is symmetric about the origin. Determine the monotonicity of $f(x)$ on the interval $[-4, 5]$ and find the maximum and minimum values of $f(x)$ on this interval. | -54 | 1 | 2,824.25 | 2,824.25 | -1 | |
How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? | 12 | 1. **Symmetry and Reduction of Cases**:
Without loss of generality (WLOG), assume $a \geq 0$ and $a \geq b$. This assumption is valid because if $(a, b, c)$ is a solution, then $(-a, -b, c)$, $(b, a, c)$, and $(-b, -a, c)$ are also solutions due to the symmetry in the equations. If $a = b$, then $|a+b| = |2a| = 2a$,... | 0 | 8,192 | -1 | 8,192 |
If four consecutive natural numbers are all composite numbers, find the smallest sum of these four numbers. | 102 | 0.875 | 5,160.4375 | 4,727.357143 | 8,192 | |
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?
(The top row of Pascal's Triangle has only a single $1$ and is the $0$th row.) | 62 | 0.6875 | 5,855.0625 | 4,792.818182 | 8,192 | |
In the polar coordinate system, the distance from the center of the circle $\rho=4\cos\theta$ ($\rho\in\mathbb{R}$) to the line $\theta= \frac {\pi}{3}$ can be found using the formula for the distance between a point and a line in polar coordinates. | \sqrt {3} | 0 | 3,182.75 | -1 | 3,182.75 | |
A weightless pulley has a rope with masses of 3 kg and 6 kg. Neglecting friction, find the force exerted by the pulley on its axis. Consider the acceleration due to gravity to be $10 \, \mathrm{m/s}^2$. Give the answer in newtons, rounding it to the nearest whole number if necessary. | 80 | 0 | 5,789.4375 | -1 | 5,789.4375 | |
A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$. | \frac{27}{32} | 1. **Assume the common perimeter**: Let the common perimeter of the square and the equilateral triangle be $P$.
2. **Calculate side lengths**:
- For the square, each side length is $\frac{P}{4}$.
- For the equilateral triangle, each side length is $\frac{P}{3}$.
3. **Determine the diameter of the circle circums... | 1 | 2,707.4375 | 2,707.4375 | -1 |
Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7\}$. What is the probability that the positive difference between the two numbers is $2$ or greater? Express your answer as a common fraction. | \frac{5}{7} | 1 | 3,402.0625 | 3,402.0625 | -1 | |
The roots of the equation $x^2+kx+5 = 0$ differ by $\sqrt{61}$. Find the greatest possible value of $k$. | 9 | 0.9375 | 2,757.4375 | 2,648.2 | 4,396 | |
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? | 0 | 1. **Identify the fixed elements and the problem requirements:**
- Points $A$ and $B$ are fixed in the plane and are $10$ units apart.
- We need to find point $C$ such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units.
2. **Set up a coordinate system:**
... | 0.6875 | 7,519.4375 | 7,213.727273 | 8,192 |
What is the remainder when the sum of the first five primes is divided by the sixth prime? | 2 | 1 | 1,394.75 | 1,394.75 | -1 | |
You and five friends need to raise $1500$ dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? | 250 | 1. **Identify the total amount to be raised and the number of people involved:**
- Total amount to be raised: $1500$ dollars.
- Number of people involved: $6$ (you and five friends).
2. **Calculate the amount each person needs to raise:**
- To find out how much each person needs to raise, divide the total amo... | 1 | 1,816.5625 | 1,816.5625 | -1 |
For a positive integer \( n \), if there exist positive integers \( a \) and \( b \) such that \( n = a + b + a \times b \), then \( n \) is called a "good number". For example, \( 3 = 1 + 1 + 1 \times 1 \), so 3 is a "good number". Among the 100 positive integers from 1 to 100, there are \(\qquad\) "good numbers". | 74 | 0.4375 | 6,481.3125 | 5,698.142857 | 7,090.444444 | |
In the production of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x + y + z = 8 \\
xy + yz + xz = -18
\end{array}\right.
$$
Find the length of the cable. | 4\pi \sqrt{\frac{59}{3}} | 0 | 7,683.5625 | -1 | 7,683.5625 | |
The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_1 = 1\), and for each \(n \in \mathbf{N}^{*}\), \(a_n\) and \(a_{n+1}\) are the roots of the equation \(x^2 + 3nx + b_n = 0\). Find \(\sum_{k=1}^{20} b_k\). | 6385 | 0.125 | 8,044.375 | 7,011 | 8,192 | |
All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have? | 4 | Let the total number of marbles Marcy has be denoted by $n$. According to the problem, the distribution of marbles is as follows:
- Blue marbles: $\frac{1}{3}n$
- Red marbles: $\frac{1}{4}n$
- Green marbles: $6$
We need to find the number of yellow marbles, which can be expressed as:
\[ \text{Yellow marbles} = n - \le... | 1 | 3,276 | 3,276 | -1 |
What is the slope of a line perpendicular to the line containing the points $(4,-7)$ and $(-5,-1)$? Express your answer as a common fraction. | \frac{3}{2} | 1 | 1,583.25 | 1,583.25 | -1 | |
A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{21} | 1 | 3,001.875 | 3,001.875 | -1 | |
Define the function $g$ on the set of integers such that \[g(n)= \begin{cases} n-4 & \mbox{if } n \geq 2000 \\ g(g(n+6)) & \mbox{if } n < 2000. \end{cases}\] Determine $g(172)$. | 2000 | 0 | 7,905.875 | -1 | 7,905.875 | |
A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their respective destinations, immediately return... | 14 | 0.0625 | 8,058.75 | 6,060 | 8,192 | |
Consider the region \(B\) in the complex plane consisting of all points \(z\) such that both \(\frac{z}{50}\) and \(\frac{50}{\overline{z}}\) have real and imaginary parts between 0 and 1, inclusive. Find the area of \(B\). | 2500 - 312.5 \pi | 0 | 7,561.5 | -1 | 7,561.5 | |
Find all 6-digit multiples of 22 of the form $5d5,\!22e$ where $d$ and $e$ are digits. What is the maximum value of $d$? | 8 | 0.9375 | 3,763.8125 | 3,468.6 | 8,192 |
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