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 |
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In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.2 times the probability of earning a B. Assuming that all grades are A, B, or C, how many B's will there be in a mathematics class of 40 students? | 14 | 0.25 | 5,552.625 | 679.25 | 7,177.083333 | |
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible. | 95 |
To solve this problem, we need to find the smallest positive integer \( n \) such that all 73 fractions of the form \(\frac{k}{n+k+20}\) for \( k = 19, 20, \ldots, 91 \) are irreducible. A fraction \(\frac{a}{b}\) is irreducible if and only if \(\gcd(a, b) = 1\).
For the fractions \(\frac{k}{n+k+20}\) to be irreducib... | 0.125 | 7,419.25 | 5,905 | 7,635.571429 |
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults? | 28 | 1 | 2,053.375 | 2,053.375 | -1 | |
What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4? | 10 | 1 | 3,066.6875 | 3,066.6875 | -1 | |
Find all integers \( z \) for which exactly two of the following five statements are true, and three are false:
1) \( 2z > 130 \)
2) \( z < 200 \)
3) \( 3z > 50 \)
4) \( z > 205 \)
5) \( z > 15 \) | 16 | 0.1875 | 7,884.3125 | 6,551 | 8,192 | |
Given a sequence \(\{a_n\}\) such that: \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{(n+1)(a_n + 1)}\) for \(n \in \mathbb{Z^+}\), find the value of \(\lim_{n \rightarrow +\infty} n! \cdot a_n\). | \frac{1}{e} | 0.6875 | 6,668 | 5,975.272727 | 8,192 | |
What is the probability that exactly one person gets their hat back when 6 people randomly pick hats? | \frac{11}{30} | There are 6 people that could get their hat back, so we must multiply 6 by the number of ways that the other 5 people can arrange their hats such that no one gets his/her hat back. So, the number of ways this will happen is ( $6 \cdot$ derangement of 5 ), or $6 * 44=264$. Since there are $6!=720$ possible arrangements ... | 0.6875 | 4,972.1875 | 3,508.636364 | 8,192 |
A square sheet of paper has area $6 \text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your answer in simplest radic... | 2\sqrt{2} | 0 | 7,973.5 | -1 | 7,973.5 | |
In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections. | 222 | 0.125 | 7,783.8125 | 6,710 | 7,937.214286 | |
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs? | 60 | 1. **Identify the constraints for the professors' seating**: Professors Alpha, Beta, and Gamma must each be seated between two students. This means they cannot occupy the first or last chair in the row, as these positions do not allow a student to be seated on both sides.
2. **Determine the possible seats for the prof... | 0.5 | 6,083 | 5,311.875 | 6,854.125 |
If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$? | 703 | 0.1875 | 7,062.9375 | 5,316.333333 | 7,466 | |
The function $f$ satisfies
\[
f(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3
\]
for all real numbers $x, y$. Determine the value of $f(10)$. | -37 | 0.875 | 4,152 | 3,913.785714 | 5,819.5 | |
Find the ordered pair $(a,b)$ of integers such that
\[\sqrt{9 - 8 \sin 50^\circ} = a + b \csc 50^\circ.\] | (3,-1) | 0.4375 | 7,029.8125 | 5,535.571429 | 8,192 | |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$
$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.
$(2)$ Find the sum of the first $20$ te... | 300 | 0.6875 | 5,437.1875 | 4,729.727273 | 6,993.6 | |
Given a point $P^{}_{}$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P_{}^{}$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P^{}_{}$ is one of the vertices, do not intersect. Suppos... | 597 | Let $O_{AB}$ be the intersection of the perpendicular bisectors (in other words, the intersections of the creases) of $\overline{PA}$ and $\overline{PB}$, and so forth. Then $O_{AB}, O_{BC}, O_{CA}$ are, respectively, the circumcenters of $\triangle PAB, PBC, PCA$. According to the problem statement, the circumcenters ... | 0 | 8,192 | -1 | 8,192 |
When two standard dice and one 8-sided die (with faces showing numbers from 1 to 8) are tossed, the numbers \(a, b, c\) are obtained respectively where \(a\) and \(b\) are from the standard dice and \(c\) is from the 8-sided die. Find the probability that \((a-1)(b-1)(c-1) \neq 0\). | \frac{175}{288} | 0.875 | 3,978 | 3,700.714286 | 5,919 | |
In some cells of a \(10 \times 10\) board, there are fleas. Every minute, the fleas jump simultaneously to an adjacent cell (along the sides). Each flea jumps strictly in one of the four directions parallel to the sides of the board, maintaining its direction as long as possible; otherwise, it changes to the opposite d... | 40 | 0 | 7,978.5 | -1 | 7,978.5 | |
Let \( w \) be a complex number such that \( |w - 3 + 2i| = 4 \). Find the minimum value of
\[
|w + 1 + 2i|^2 + |w - 7 - 2i|^2.
\] | 48 | 0.375 | 7,369.6875 | 6,030.166667 | 8,173.4 | |
Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that
- each member in $S$ is a positive integer not exceeding $2002$ ,
- if $a,b\in S$ (not necessarily different), then $ab\not\in S$ .
| 1958 | 0.0625 | 8,030.75 | 5,612 | 8,192 | |
Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that $$ \begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\ & g(h(f(x)))=f(h(g(x)))=h(x), \text { and } \\ & h(f(g(x)))=g(f(h(x)))=f(x) \end{aligned} $$ for all $x \in\{1,2,3,4,5\}$. | 146 | Let $f g$ represent the composition of permutations $f$ and $g$, where $(f g)(x)=f(g(x))$ for all $x \in\{1,2,3,4,5\}$. Evaluating fghfh in two ways, we get $$ f=g f h=(f g h) f h=f g h f h=f(g h f) h=f h h, $$ so $h h=1$. Similarly, we get $f, g$, and $h$ are all involutions. Then $$ f g h=g \Longrightarrow f g=g h $$... | 0 | 7,781.25 | -1 | 7,781.25 |
In the ellipse $\dfrac {x^{2}}{36}+ \dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\overrightarrow{KM} \cdot \overrightarrow{KN} = 0$, find the minimum value of $\overrightarrow{KM} \cdot \overrightarrow{NM}$. | \dfrac{23}{3} | 0.0625 | 8,082.375 | 6,669 | 8,176.6 | |
Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $d$. | \frac{5 + \sqrt{105}}{2} | 0 | 6,486.0625 | -1 | 6,486.0625 | |
Suppose $x$ is a random real number between $1$ and $4$ , and $y$ is a random real number between $1$ and $9$ . If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, ... | 1112 | 0.75 | 4,625.375 | 3,817.583333 | 7,048.75 | |
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \] | n+1 |
Let \( S \) be a finite set with \( |S| = n \). We are asked to find the maximum size of the image of a function \( f \) in the set \(\mathcal{F}\), where \(\mathcal{F}\) is the set of all functions \( f : \mathcal{P}(S) \to \mathbb{R} \) satisfying the condition that for all subsets \( X, Y \subseteq S \), we have:
\... | 0 | 8,192 | -1 | 8,192 |
Given a triangle \(A B C\) with \(A B = A C\) and \(\angle A = 110^{\circ}\). Inside the triangle, a point \(M\) is chosen such that \(\angle M B C = 30^{\circ}\) and \(\angle M C B = 25^{\circ}\). Find \(\angle A M C\). | 85 | 0.3125 | 7,538.5625 | 6,101 | 8,192 | |
Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces
$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$ | 4\pi | 0.5625 | 5,349.8125 | 4,385.777778 | 6,589.285714 | |
Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side lengths. The shortest possible length of the longest of the eight sticks is: | 21 | 0.25 | 7,248.625 | 5,063.5 | 7,977 | |
In an acute-angled triangle $ABC$ , the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $ . Find the value of the smallest angle of triangle $ ABC $ .
(Black Maxim) | 30 | 0 | 7,625.0625 | -1 | 7,625.0625 | |
Given $α \in \left( \frac{π}{2}, π \right)$, and $\sin α = \frac{1}{3}$.
$(1)$ Find the value of $\sin 2α$;
$(2)$ If $\sin (α+β) = -\frac{3}{5}$, and $β \in (0, \frac{π}{2})$, find the value of $\sin β$. | \frac{6\sqrt{2}+4}{15} | 0 | 7,194.4375 | -1 | 7,194.4375 | |
On a Cartesian plane, consider the points \(A(-8, 2), B(-4, -2)\) and \(X(1, y)\) and \(Y(10, 3)\). If segment \(AB\) is parallel to segment \(XY\), find the value of \(y\). | 12 | 1 | 1,355.5 | 1,355.5 | -1 | |
Given a quadratic function in terms of \\(x\\), \\(f(x)=ax^{2}-4bx+1\\).
\\((1)\\) Let set \\(P=\\{1,2,3\\}\\) and \\(Q=\\{-1,1,2,3,4\\}\\), randomly pick a number from set \\(P\\) as \\(a\\) and from set \\(Q\\) as \\(b\\), calculate the probability that the function \\(y=f(x)\\) is increasing in the interval \\([1,+∞... | \dfrac{961}{1280} | 0 | 7,961.1875 | -1 | 7,961.1875 | |
For all positive integers $x$, let \[f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases}\] and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$. Let $d(x)$ be the smallest $n$ such that $x_n=1$. (For exam... | 511 | We backcount the number of ways. Namely, we start at $x_{20} = 1$, which can only be reached if $x_{19} = 10$, and then we perform $18$ operations that either consist of $A: (-1)$ or $B: (\times 10)$. We represent these operations in a string format, starting with the operation that sends $f(x_{18}) = x_{19}$ and so fo... | 0 | 8,192 | -1 | 8,192 |
Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM:MC = 1:2 \). The line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1. | \frac{11}{24} | 0.5625 | 7,614.5 | 7,165.333333 | 8,192 | |
If two of the roots of \[2x^3 + 8x^2 - 120x + k = 0\]are equal, find the value of $k,$ given that $k$ is positive. | \tfrac{6400}{27} | 1 | 4,631.875 | 4,631.875 | -1 | |
Consider an arithmetic sequence where the first four terms are $x+2y$, $x-2y$, $2xy$, and $x/y$. Determine the fifth term of the sequence. | -27.7 | 0 | 8,179.6875 | -1 | 8,179.6875 | |
Determine the tens digit of $17^{1993}$. | 3 | 0.75 | 6,389.125 | 5,788.166667 | 8,192 | |
There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.
$(1)$ How many unique three-digit numbers can be formed without repetition?
$(2)$ How many unique four-digit even numbers can be formed without repetition? | 2296 | 0.3125 | 7,293.1875 | 6,118.4 | 7,827.181818 | |
Mateo receives $20 every hour for one week, and Sydney receives $400 every day for one week. Calculate the difference between the total amounts of money that Mateo and Sydney receive over the one week period. | 560 | 0 | 408.3125 | -1 | 408.3125 | |
Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. | 124 | 1 | 1,999.3125 | 1,999.3125 | -1 | |
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=4$ and $\frac{\cos B}{\cos C} = \frac{4}{2a - c}$.
(1) Find the measure of angle $B$;
(2) Find the maximum area of $\triangle ABC$. | 4\sqrt{3} | 0.6875 | 6,012.25 | 5,478.818182 | 7,185.8 | |
Given a sequence $\{a_n\}$ with $a_1=1$ and $a_{n+1}= \frac{2a_n}{a_n+2}$, find the value of $a_{10}$. | \frac{2}{11} | 1 | 3,602.3125 | 3,602.3125 | -1 | |
Determine how many perfect cubes exist between \(3^6 + 1\) and \(3^{12} + 1\), inclusive. | 72 | 0.9375 | 3,541.25 | 3,306 | 7,070 | |
On some cells of a 10x10 board, there is a flea. Every minute, the fleas jump simultaneously, each one to a neighboring cell (adjacent by side). Each flea jumps strictly in one of the four directions parallel to the board's sides and maintains this direction as long as possible; otherwise, it changes to the opposite di... | 40 | 0 | 7,779.8125 | -1 | 7,779.8125 | |
In a bike shed, there are bicycles (two wheels), tricycles, and cars (four wheels). The number of bicycles is four times the number of cars. Several students counted the total number of wheels in the shed, but each of them obtained a different count: $235, 236, 237, 238, 239$. Among these, one count is correct. Smart k... | 19 | 0 | 5,076.8125 | -1 | 5,076.8125 | |
On the side \( BC \) of triangle \( ABC \), point \( A_1 \) is taken such that \( BA_1 : A_1C = 2:1 \). In what ratio does median \( CC_1 \) divide segment \( AA_1 \)? | 3:1 | 0.6875 | 5,566.1875 | 4,993.454545 | 6,826.2 | |
If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals | 1 | 1. Start with the given equation:
\[
1 - \frac{4}{x} + \frac{4}{x^2} = 0
\]
2. Multiply each term by \(x^2\) to clear the denominators:
\[
x^2 \cdot 1 - 4x \cdot x + 4 = x^2 - 4x + 4 = 0
\]
3. Factor the quadratic equation:
\[
x^2 - 4x + 4 = (x - 2)^2 = 0
\]
4. Solve for \(x\) by setting t... | 0.9375 | 2,202.75 | 1,803.466667 | 8,192 |
Compute
\[\frac{2 + 6}{4^{100}} + \frac{2 + 2 \cdot 6}{4^{99}} + \frac{2 + 3 \cdot 6}{4^{98}} + \dots + \frac{2 + 98 \cdot 6}{4^3} + \frac{2 + 99 \cdot 6}{4^2} + \frac{2 + 100 \cdot 6}{4}.\] | 200 | 0.4375 | 7,104.875 | 5,707.142857 | 8,192 | |
Given $0 \le x_0 < 1$, let
\[x_n = \begin{cases} 2x_{n-1} & \text{ if } 2x_{n-1} < 1 \\ 2x_{n-1} - 1 & \text{ if } 2x_{n-1} \ge 1 \end{cases}\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? | 31 | 1. **Understanding the Sequence**: The sequence defined by $x_n$ is a binary sequence where each term is generated by doubling the previous term and subtracting 1 if the result is at least 1. This can be interpreted as a shift and truncate operation in binary representation.
2. **Binary Representation**: Let's represe... | 0.25 | 7,617.8125 | 6,726 | 7,915.083333 |
A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3... | 99 | 0.9375 | 4,268.625 | 4,007.066667 | 8,192 | |
How many digits are there in the number \(N\) if \(N=2^{12} \times 5^{8}\) ?
If \(\left(2^{48}-1\right)\) is divisible by two whole numbers between 60 and 70, find them.
Given \(2^{\frac{1}{2}} \times 9^{\frac{1}{9}}\) and \(3^{\frac{1}{3}} \times 8^{\frac{1}{8}}\), what is the greatest number? | 3^{\frac{1}{3}} \times 8^{\frac{1}{8}} | 0 | 7,865.625 | -1 | 7,865.625 | |
A box of 100 personalized pencils costs $\$30$. How many dollars does it cost to buy 2500 pencils? | \$750 | 1 | 877.1875 | 877.1875 | -1 | |
(1) Calculate: $\left( \frac{1}{8} \right)^{-\frac{1}{3}} - 3\log_{3}^{2}(\log_{3}4) \cdot (\log_{8}27) + 2\log_{\frac{1}{6}} \sqrt{3} - \log_{6}2$
(2) Calculate: $27^{\frac{2}{3}} - 2^{\log_{2}3} \times \log_{2}\frac{1}{8} + 2\lg \left( \sqrt{3+\sqrt{5}} + \sqrt{3-\sqrt{5}} \right)$ | 19 | 0.125 | 7,623.25 | 6,480.5 | 7,786.5 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\sin A = \sqrt{7}a$.
(1) Find the value of $\sin B$;
(2) If $a$, $b$, and $c$ form an arithmetic sequence with a positive common difference, find the value of $\cos A - \cos C$. | \frac{\sqrt{7}}{2} | 0 | 7,706 | -1 | 7,706 | |
The output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \_\_\_\_\_\_\_\_. (Keep one decimal place, ta... | 6.6 | 0.125 | 3,685 | 7,348 | 3,161.714286 | |
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.
[i]Greece[/i] | 1 |
To find the area of the pentagon \(ABCDE\), we will use the given conditions:
1. \(AB = AE = CD = 1\),
2. \(\angle ABC = \angle DEA = 90^\circ\),
3. \(BC + DE = 1\).
We start by placing the pentagon in the coordinate plane to simplify calculations:
- Let \(A\) be at the origin \((0, 0)\).
- Since \(AB = 1\) and \(\... | 0 | 8,192 | -1 | 8,192 |
A bag contains 8 red balls, a number of white balls, and no other balls. If $\frac{5}{6}$ of the balls in the bag are white, then how many white balls are in the bag? | 40 | Since $\frac{5}{6}$ of the balls are white and the remainder of the balls are red, then $\frac{1}{6}$ of the balls are red. Since the 8 red balls represent $\frac{1}{6}$ of the total number of balls and $\frac{5}{6} = 5 \cdot \frac{1}{6}$, then the number of white balls is $5 \cdot 8 = 40$. | 1 | 1,569.25 | 1,569.25 | -1 |
People have long been exploring the numerical solution of high-degree equations. Newton gave a numerical solution method for high-degree algebraic equations in his book "Fluxions." This method for finding the roots of equations has been widely used in the scientific community. For example, to find an approximate soluti... | -\frac{11}{8} | 0 | 4,960.75 | -1 | 4,960.75 | |
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [asy]
/* AMC8 2003 #15 Problem */
draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);
draw((0,1)--(1,1)--(1,0));
draw((4,... | 4 | 0.1875 | 7,722.75 | 6,695 | 7,959.923077 | |
In the sequence $\{a_{n}\}$, $a_{1}=1$, $\sqrt{{a}_{n+1}}-\sqrt{{a}_{n}}=1$ ($n\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\{b_{n}\}$ is $S_{n}=2^{n}-m$. For $n\in N^{*}$, the smallest value of the real number $\lambda$ that satisfies $\lambda b_{n}\geqslant a_{n}$ for all $n$ is ______. | \frac{9}{4} | 1 | 4,139.9375 | 4,139.9375 | -1 | |
Let
\[\mathbf{M} = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}\]be a matrix with complex entries such that $\mathbf{M}^2 = \mathbf{I}.$ If $abc = 1,$ then find the possible values of $a^3 + b^3 + c^3.$ | 2,4 | 0 | 6,745.4375 | -1 | 6,745.4375 | |
Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other.
The man at $R$ walks uniformly at the rate of $4\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant
rate of $3\tfrac{1}{4}$ miles per hour for the first hour, at $3\tfrac{3}{4}$ miles per hour for the se... | 4 | 1. **Calculate the distance each person walks**:
- The man starting from $R$ walks at a rate of $4.5$ miles per hour. Therefore, in $h$ hours, he walks a distance of $4.5h$ miles.
- The man starting from $S$ walks at a rate that increases by $0.5$ miles per hour every hour, starting at $3.25$ miles per hour. The... | 1 | 3,537 | 3,537 | -1 |
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries,... | 20 | 0 | 8,192 | -1 | 8,192 | |
Let $\overrightarrow{m} = (\sin(x - \frac{\pi}{3}), 1)$ and $\overrightarrow{n} = (\cos x, 1)$.
(1) If $\overrightarrow{m} \parallel \overrightarrow{n}$, find the value of $\tan x$.
(2) If $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$, where $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of $f... | 1 - \frac{\sqrt{3}}{2} | 0 | 6,690.4375 | -1 | 6,690.4375 | |
Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green... | \frac{5}{16} | Clearly Reimu will always color a side red and Sanae will always color a side green, because their situation is never worse off when a side of a coin changes to their own color. Since the number of red-only coins is always equal to the number of green-only coins, no matter how Reimu and Sanae color the coins, they will... | 0 | 8,149.875 | -1 | 8,149.875 |
A circle with center $A$ and radius three inches is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),6),gray,linewidth(2));
filldraw(circle(3dir(-30),3),white,linewidth(2)... | 27\pi | 0.9375 | 2,670.6875 | 2,488.733333 | 5,400 | |
Find the value of the expression \(\sum_{i=0}^{1009}(2 k+1)-\sum_{i=1}^{1009} 2 k\). | 1010 | 0.875 | 4,127.125 | 3,910.571429 | 5,643 | |
Find the slope angle of the tangent line to the curve $f(x)=\frac{1}{3}{x}^{3}-{x}^{2}+5$ at $x=1$. | \frac{3\pi}{4} | 0.75 | 2,826.375 | 3,026.416667 | 2,226.25 | |
Given $f(x)= \begin{cases} \sin \frac{\pi}{3}x, & x\leqslant 2011, \\ f(x-4), & x > 2011, \end{cases}$ find $f(2012)$. | -\frac{\sqrt{3}}{2} | 0 | 7,181.4375 | -1 | 7,181.4375 | |
The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ | 9 | 1. **Check divisibility by 5**: The number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ ends in digit $A$. If $A = 5$, the number ends in $5$ and is divisible by $5$. Therefore, option $\textbf{(C)}\ 5$ can be eliminated.
2. **Check divisibility by 3**: The sum of the digi... | 0.1875 | 8,029.8125 | 7,327 | 8,192 |
A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \frac{\text{km}}{\text{hour}}$ on a horizontal track with a friction coefficient of $\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $e=0.030$?
(Note: In this problem, $\rho$ is ... | 10.3 | 0.125 | 7,681.4375 | 7,362.5 | 7,727 | |
In $\triangle ABC$, $\angle A=55^\circ$, $\angle C=75^\circ$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle{BED} =$ | 65^\circ | 1. **Calculate $\angle B$ in $\triangle ABC$**:
Given $\angle A = 55^\circ$ and $\angle C = 75^\circ$, we use the fact that the sum of angles in a triangle is $180^\circ$. Therefore,
\[
\angle B = 180^\circ - \angle A - \angle C = 180^\circ - 55^\circ - 75^\circ = 50^\circ.
\]
2. **Analyze $\triangle BED$*... | 0.6875 | 6,737.1875 | 6,075.909091 | 8,192 |
The quadratic $8x^2+12x-14$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms. | \frac{23}{4} | 1 | 2,279.125 | 2,279.125 | -1 | |
The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent? | 7 | 1 | 1,728.625 | 1,728.625 | -1 | |
Calculate the product $\left(\frac{3}{6}\right)\left(\frac{6}{9}\right)\left(\frac{9}{12}\right)\cdots\left(\frac{2001}{2004}\right)$. Express your answer as a common fraction. | \frac{1}{668} | 0.5625 | 4,338.5625 | 2,729.111111 | 6,407.857143 | |
The hypotenuse of an isosceles right triangle is $4\sqrt{2}$ units. How many square units are in the area of the triangle? | 8 | 1 | 1,696.875 | 1,696.875 | -1 | |
Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes. | 12 | 0.0625 | 6,624.3125 | 5,755 | 6,682.266667 | |
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$. | 70 | 1. **Identify the Relationship Between Angles and Sides**:
Given that $\angle APB = \angle CPD$ and $\angle APD = \angle BPC$, and using the property that $\sin(\theta) = \sin(180^\circ - \theta)$, we can apply the sine area formula for triangles. This leads to the equation:
\[
BP \cdot AP + CP \cdot DP = BP \... | 0 | 8,192 | -1 | 8,192 |
In triangle $\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\overrightarrow{AM} = x\overrightarrow{AB}$ and $\overrightarrow{AN} = y\overrightarrow{AC}$ ($x, y \neq 0$), then the minimum value of $4x+y$ is \_\_\_\_\_\_. | \frac{9}{4} | 0.8125 | 5,630.9375 | 5,039.923077 | 8,192 | |
Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. | \frac{1}{2} | Writing $a=x+2 y^{2}$, the given quickly becomes $4 y^{2} a+2 x^{2} a+a+x=x^{2}+1$. We can rewrite $4 y^{2} a$ for further reduction to $a(2 a-2 x)+2 x^{2} a+a+x=x^{2}+1$, or $$\begin{equation*} 2 a^{2}+\left(2 x^{2}-2 x+1\right) a+\left(-x^{2}+x-1\right)=0 \tag{*} \end{equation*}$$ The quadratic formula produces the d... | 0.25 | 7,613.375 | 5,877.5 | 8,192 |
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | 1728 | 0 | 7,889.6875 | -1 | 7,889.6875 | |
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whe... | 2021 |
Peter has 2022 pieces of magnetic railroad cars, which are of two types:
- Type 1: The front with north polarity and the rear with south polarity.
- Type 2: The rear with north polarity and the front with south polarity.
To determine whether there is the same number of both types of cars, Peter can try to fit two ca... | 0 | 8,120.875 | -1 | 8,120.875 |
It is known that
$$
\sqrt{9-8 \sin 50^{\circ}}=a+b \sin c^{\circ}
$$
for exactly one set of positive integers \((a, b, c)\), where \(0 < c < 90\). Find the value of \(\frac{b+c}{a}\). | 14 | 0.125 | 6,995.6875 | 5,497.5 | 7,209.714286 | |
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what is $m+n$ ?
*2016 CCA Math Bonanza Lightning... | 1002 | 0.375 | 6,385.5625 | 5,142.166667 | 7,131.6 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.3125 | 1,738.125 | 1,465.8 | 1,861.909091 | |
There are 20 chairs arranged in a circle. There are \(n\) people sitting in \(n\) different chairs. These \(n\) people stand, move \(k\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \((n, k)\) with \(1 \leq n \leq 20\) and \(1 \leq k \leq 20\) ... | 72 | 0.0625 | 7,926.0625 | 8,121 | 7,913.066667 | |
Find the smallest positive solution to
\[\tan 2x + \tan 3x = \sec 3x\]in radians. | \frac{\pi}{14} | 0.1875 | 7,482.375 | 4,407.333333 | 8,192 | |
The perimeter of a rectangle is 48. What is the largest possible area of the rectangle? | 144 | 1 | 1,766.4375 | 1,766.4375 | -1 | |
Let S$_{n}$ denote the sum of the first $n$ terms of the arithmetic sequence {a$_{n}$} with a common difference d=2. The terms a$_{1}$, a$_{3}$, and a$_{4}$ form a geometric sequence. Find the value of S$_{8}$. | -8 | 1 | 2,408.8125 | 2,408.8125 | -1 | |
Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the maximum distance is 1... | 24 | 0 | 7,735.625 | -1 | 7,735.625 | |
The smallest sum one could get by adding three different numbers from the set $\{ 7,25,-1,12,-3 \}$ is | 3 | To find the smallest sum possible by adding three different numbers from the set $\{7, 25, -1, 12, -3\}$, we need to consider the smallest numbers in the set, as adding smaller numbers will result in a smaller sum.
1. **Identify the smallest numbers in the set**: The three smallest numbers in the set are $-3$, $-1$, a... | 1 | 3,061.375 | 3,061.375 | -1 |
Let $P$ be a point inside triangle $ABC$ such that
\[\overrightarrow{PA} + 2 \overrightarrow{PB} + 3 \overrightarrow{PC} = \mathbf{0}.\]Find the ratio of the area of triangle $ABC$ to the area of triangle $APC.$ | 3 | 0.9375 | 6,181.0625 | 6,047 | 8,192 | |
A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops b... | 240 | 0 | 8,158.5 | -1 | 8,158.5 | |
A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
*Ray Li.* | 441 | 0 | 8,125.25 | -1 | 8,125.25 | |
On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. | 21 | 0 | 7,476.6875 | -1 | 7,476.6875 | |
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not neces... | 2 \sum_{i=0}^{8} \binom{n}{i} - 1 |
To determine the maximum number of moves that Sir Alex could have made in this game, we need to analyze the operations and how each affects the game state.
Sir Alex has 9 cells initially empty. The objective is to have one cell contain the number \(2^n\) at the end, while all others are empty. During the game, Sir Al... | 0 | 8,192 | -1 | 8,192 |
Calculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(-2, 0, -4)$
$A_{2}(-1, 7, 1)$
$A_{3}(4, -8, -4)$
$A_{4}(1, -4, 6)$ | 5\sqrt{2} | 0.5 | 5,310.1875 | 3,822 | 6,798.375 | |
Find the distance between the points (0,4) and (3,0). | 5 | 1 | 1,584.0625 | 1,584.0625 | -1 | |
Consider a $10 \times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10 \%$ of the burrito's original size and accidentally throw it into a rando... | 71.8 | Label the squares using coordinates, letting the top left corner be $(0,0)$. The burrito will end up in 10 (not necessarily different) squares. Call them $p_{1}=\left(x_{1}, y_{1}\right)=(0,0), p_{2}=\left(x_{2}, y_{2}\right), \ldots, p_{10}=\left(x_{10}, y_{10}\right)$. $p_{2}$ through $p_{10}$ are uniformly distribut... | 0 | 8,156.0625 | -1 | 8,156.0625 |
Calculate $\sin 9^\circ \sin 45^\circ \sin 69^\circ \sin 81^\circ.$ | \frac{0.6293 \sqrt{2}}{4} | 0 | 8,192 | -1 | 8,192 | |
How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?
[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15... | 20 | 1 | 3,253.5 | 3,253.5 | -1 |
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