problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Given that $\cos ( \frac {π}{6}+α) \cdot \cos ( \frac {π}{3}-α)=- \frac {1}{4}$, where $α \in ( \frac {π}{3}, \frac {π}{2})$, find the value of $\sin 2α$ and the value of $\tan α - \frac {1}{\tan α}$. | \frac{2\sqrt{3}}{3} | 0 | 3,712.125 | -1 | 3,712.125 | |
A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people. | 12 | 0.25 | 572.8125 | 620.5 | 556.916667 | |
Find the value of $x$ between 0 and 180 such that
\[\tan (120^\circ - x^\circ) = \frac{\sin 120^\circ - \sin x^\circ}{\cos 120^\circ - \cos x^\circ}.\] | 100 | 1 | 4,117.375 | 4,117.375 | -1 | |
Given a sequence $\{a_n\}$ satisfying $a_1=81$ and $a_n= \begin{cases} -1+\log_{3}a_{n-1}, & n=2k \\ 3^{a_{n-1}}, & n=2k+1 \end{cases}$ (where $k\in\mathbb{N}^*$), find the maximum value of the sum of the first $n$ terms of the sequence, $S_n$. | 127 | 0.1875 | 8,103.25 | 7,718.666667 | 8,192 | |
A sequence \(a_1\), \(a_2\), \(\ldots\) of non-negative integers is defined by the rule \(a_{n+2}=|a_{n+1}-a_n|\) for \(n\geq1\). If \(a_1=1010\), \(a_2<1010\), and \(a_{2023}=0\), how many different values of \(a_2\) are possible? | 399 | 0 | 8,192 | -1 | 8,192 | |
In a bag containing 12 green marbles and 8 purple marbles, Phil draws a marble at random, records its color, replaces it, and repeats this process until he has drawn 10 marbles. What is the probability that exactly five of the marbles he draws are green? Express your answer as a decimal rounded to the nearest thousandt... | 0.201 | 0.75 | 6,589.125 | 6,165.916667 | 7,858.75 | |
On each side of a square, a point is taken. It turns out that these points are the vertices of a rectangle whose sides are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6. | 12 | 0.3125 | 7,443.0625 | 5,795.4 | 8,192 | |
It is known that ship $A$ is located at $80^{\circ}$ north by east from lighthouse $C$, and the distance from $A$ to $C$ is $2km$. Ship $B$ is located at $40^{\circ}$ north by west from lighthouse $C$, and the distance between ships $A$ and $B$ is $3km$. Find the distance from $B$ to $C$ in $km$. | \sqrt {6}-1 | 0 | 6,693.625 | -1 | 6,693.625 | |
Phoenix hiked the Rocky Path Trail last week. It took four days to complete the trip. The first two days she hiked a total of 22 miles. The second and third days she averaged 13 miles per day. The last two days she hiked a total of 30 miles. The total hike for the first and third days was 26 miles. How many miles long ... | 52 | 0.9375 | 2,387.1875 | 2,000.2 | 8,192 | |
Find the number of ordered triples of divisors $(d_{1}, d_{2}, d_{3})$ of 360 such that $d_{1} d_{2} d_{3}$ is also a divisor of 360. | 800 | Since $360=2^{3} \cdot 3^{2} \cdot 5$, the only possible prime divisors of $d_{i}$ are 2,3 , and 5 , so we can write $d_{i}=2^{a_{i}} \cdot 3^{b_{i}} \cdot 5^{c_{i}}$, for nonnegative integers $a_{i}, b_{i}$, and $c_{i}$. Then, $d_{1} d_{2} d_{3} \mid 360$ if and only if the following three inequalities hold. $$\begin{... | 0.3125 | 7,475.5625 | 5,899.4 | 8,192 |
Find the set of all attainable values of $\frac{ab+b^{2}}{a^{2}+b^{2}}$ for positive real $a, b$. | \left(0, \frac{1+\sqrt{2}}{2}\right] | Suppose that $k=\frac{ab+b^{2}}{a^{2}+b^{2}}$ for some positive real $a, b$. We claim that $k$ lies in $\left(0, \frac{1+\sqrt{2}}{2}\right]$. Let $x=\frac{a}{b}$. We have that $\frac{ab+b^{2}}{a^{2}+b^{2}}=\frac{\frac{a}{b}+1}{\left(\frac{a}{b}\right)^{2}+1}=\frac{x+1}{x^{2}+1}$. Thus, $x+1=k\left(x^{2}+1\right)$, so ... | 0 | 7,481.375 | -1 | 7,481.375 |
Two numbers in the $4 \times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).
What is the sum of these two numbers?
A 12
B 15
C 22
D 26
E 28
\begin{tabular}{|c|c|c|c|}
\hline
9 & 6 & 3 & 16 \\
\hline
4 & 13 & 10 & 5 \\
\hline
14 & 1 & 8 ... | 28 | 0 | 7,926.75 | -1 | 7,926.75 | |
Calculate the circulation of the vector field:
a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;
b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$. | -3 | 0.375 | 7,619.375 | 7,135.5 | 7,909.7 | |
Find the total number of sets of positive integers \((x, y, z)\), where \(x, y\) and \(z\) are positive integers, with \(x < y < z\) such that
$$
x + y + z = 203.
$$ | 3333 | 0.0625 | 8,132.375 | 7,888 | 8,148.666667 | |
For which values of the parameter \(a\) does the equation \(x^{3} + 16x^{2} + ax + 64 = 0\) have three distinct real roots that form a geometric progression? | 64 | 0.75 | 6,033.4375 | 5,313.916667 | 8,192 | |
Compute the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{4^{5 x}-9^{-2 x}}{\sin x-\operatorname{tg}(x^{3})}
$$ | \ln (1024 \cdot 81) | 0 | 7,393.375 | -1 | 7,393.375 | |
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$ .
*2017 CCA Math Bonanza Lightning Round #2.4* | 13 | 0.25 | 7,750.1875 | 6,424.75 | 8,192 | |
Let $P_0(x) = x^3 + 313x^2 - 77x - 8\,$. For integers $n \ge 1\,$, define $P_n(x) = P_{n - 1}(x - n)\,$. What is the coefficient of $x\,$ in $P_{20}(x)\,$? | 763 | Notice the transformation of $P_{n-1}(x)\to P_n(x)$ adds $n$ to the roots. Thus, all these transformations will take the roots and add $1+2+\cdots+20=210$ to them. (Indeed, this is very easy to check in general.)
Let the roots be $r_1,r_2,r_3.$ Then $P_{20}(x)=(x-r_1-210)(x-r_2-210)(x-r_3-210).$ By Vieta's/expanding/c... | 0.375 | 7,525.0625 | 6,413.5 | 8,192 |
A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is | 72 | 1. **Calculate the original surface area of the cube**:
The cube has 6 faces, and each face is a square with side length 3 meters. The area of one face is therefore $3^2 = 9$ square meters. Thus, the total surface area of the cube before any modifications is:
\[
6 \times 9 = 54 \text{ square meters}
\]
2.... | 0 | 5,990.875 | -1 | 5,990.875 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $2\sin 2A+\sin (A-B)=\sin C$, and $A\neq \frac{\pi}{2}$.
- (I) Find the value of $\frac{a}{b}$;
- (II) If $c=2$ and $C= \frac{\pi}{3}$, find the area of $\triangle ABC$. | \frac{2 \sqrt {3}}{3} | 0 | 5,662.5625 | -1 | 5,662.5625 | |
Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | \frac{\sqrt{3}}{4} | 0 | 8,192 | -1 | 8,192 | |
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$? | 248 | The sum is equivalent to $\sum_{i | 10^6}^{} \frac{i}{1000}$ Therefore, it's the sum of the factors of $10^6$ divided by $1000$. The sum is $\frac{127 \times 19531}{1000}$ by the sum of factors formula. The answer is therefore $\boxed{248}$ after some computation. - whatRthose | 0 | 8,147 | -1 | 8,147 |
Elective 4-5: Selected Topics on Inequalities. Given the function $f(x) = |2x-1| + |2x+3|$.
$(1)$ Solve the inequality $f(x) \geqslant 6$;
$(2)$ Let the minimum value of $f(x)$ be $m$, and let the positive real numbers $a, b$ satisfy $2ab + a + 2b = m$. Find the minimum value of $a + 2b$. | 2\sqrt{5} - 2 | 1 | 5,287.875 | 5,287.875 | -1 | |
Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Then $b$ is: | \frac{1}{2} | 0.1875 | 7,756.8125 | 6,989 | 7,934 | |
Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions? | 15 | 0.1875 | 7,843.125 | 6,331.333333 | 8,192 | |
The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. | \sqrt{3} | 0.6875 | 6,428.125 | 5,626.363636 | 8,192 | |
On graph paper, a stepwise right triangle was drawn with legs equal to 6 cells each. Then, all grid lines inside the triangle were outlined. What is the maximum number of rectangles that can be found in this drawing? | 126 | 0 | 8,024.4375 | -1 | 8,024.4375 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c \sin A = \sqrt{3}a \cos C$ and $(a-c)(a+c)=b(b-c)$, find the period and the monotonically increasing interval of the function $f(x) = 2 \sin x \cos (\frac{\pi}{2} - x) - \sqrt{3} \sin (\pi + x) \cos x + ... | \frac{5}{2} | 1 | 5,311.75 | 5,311.75 | -1 | |
Given 6 teachers who will be allocated to two classes, where the maximum number of teachers in each class is 4, determine the number of different arrangements. | 50 | 0.1875 | 7,822.9375 | 6,223.666667 | 8,192 | |
In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 points. For Topic B, answering correctly y... | 36 | 0.0625 | 8,032.5625 | 7,789 | 8,048.8 | |
First, factorize 42 and 30 into prime factors, then answer the following questions:
(1) 42= , 30= .
(2) The common prime factors of 42 and 30 are .
(3) The unique prime factors of 42 and 30 are .
(4) The greatest common divisor (GCD) of 42 and 30 is .
(5) The least common multiple (LCM) of 4... | 210 | 0.375 | 1,523.0625 | 1,520.5 | 1,524.6 | |
Person A and person B each have a certain number of books. If person A gives 10 books to person B, then the total number of books between the two of them will be equal. If person B gives 10 books to person A, then the number of books person A has will be twice the number of books person B has left. Find out how many bo... | 50 | 0.9375 | 2,788.0625 | 2,431.266667 | 8,140 | |
Given that the solution set for the inequality $ax^2+ax+2>0$ is $\mathbb{R}$ (the set of all real numbers), let the set of all numerical values of the real number $a$ be denoted as $M$.
(1) Find the set $M$.
(2) If $t>0$, for all $a \in M$, it holds that $(a^2-2a)t \leq t^2 + 3t - 46$. Find the minimum value of $t$. | 46 | 0.6875 | 6,248.5625 | 5,365.181818 | 8,192 | |
The midpoints of the sides of a triangle with area $T$ are joined to form a triangle with area $M$. What is the ratio of $M$ to $T$? Express your answer as a common fraction. | \frac{1}{4} | 1 | 3,966.4375 | 3,966.4375 | -1 | |
For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$. | 736 | Note: Archimedes15 Solution which I added an answer here are two cases. Either the $4$ and $10$ are around an obtuse angle or the $4$ and $10$ are around an acute triangle. If they are around the obtuse angle, the area of that triangle is $<20$ as we have $\frac{1}{2} \cdot 40 \cdot \sin{\alpha}$ and $\sin$ is at most ... | 0 | 8,154.5 | -1 | 8,154.5 |
For natural numbers $m$ greater than or equal to 2, the decomposition of their cube powers can be represented as follows: $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$, $4^3 = 13 + 15 + 17 + 19$. Then,
(1) The smallest number in the decomposition of $8^3$ is;
(2) Following the above pattern, the $n$-th equation can be represen... | 57 | 0.5625 | 6,481.3125 | 6,266.222222 | 6,757.857143 | |
A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo needs to complete to earn 30 study points. | 90 | 0 | 1,249.125 | -1 | 1,249.125 | |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$, respectively. If $\overline{AD}$ and $\overline{BE}$ intersect at $T$ so that $AT/DT=3$ and $BT/ET=4$, what is $CD/BD$?
[asy]
pair A,B,C,D,I,T;
A=(0,0);
B=(6,8);
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D=(9.33,2.66);
I=(7.5,0);
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label("$T$",T,NW);
label("$... | \frac{4}{11} | 0.375 | 6,904 | 4,868 | 8,125.6 | |
A triangle $H$ is inscribed in a regular hexagon $S$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$? | 3/8 | 0 | 8,192 | -1 | 8,192 | |
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa'... | 85 | 0.0625 | 8,027.8125 | 7,248 | 8,079.8 | |
$\frac{\text{华杯赛}}{\text{少} \times \text{俊} + \text{金坛} + \text{论} \times \text{数}} = 15$
In the above equation, different Chinese characters represent different digits between $1$ and $9$. When the three-digit number "华杯赛" reaches its maximum value, please write a solution where the equation holds. | 975 | 0 | 8,192 | -1 | 8,192 | |
Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and Buddy wins otherwise. If the probability that Ainsley wins is $\frac{a}{b}$ for relatively prime positive integers $a... | 109 | We let $X$ be the event of a multiple of 3 being rolled and $Y$ be the event of a nonmultiple of 3 being rolled. In order for Ainsley to win, she needs event $X$ to happen consecutively; meanwhile, Buddy just needs $Y$ then $X$ to occur. Thus, if $Y$ occurs in the first two rolls, Buddy will be guaranteed to win, since... | 0 | 7,707.9375 | -1 | 7,707.9375 |
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence? | 236 | 0.25 | 6,345.25 | 4,651.75 | 6,909.75 | |
In the geometric sequence with a first term of $6$ and a second term of $-6$, what is the $205^{th}$ term? | 6 | 1 | 1,303.75 | 1,303.75 | -1 | |
Given a circle $O: x^2 + y^2 = 6$, and $P$ is a moving point on circle $O$. A perpendicular line $PM$ is drawn from $P$ to the x-axis at $M$, and $N$ is a point on $PM$ such that $\overrightarrow{PM} = \sqrt{2} \overrightarrow{NM}$.
(Ⅰ) Find the equation of the trajectory $C$ of point $N$;
(Ⅱ) If $A(2,1)$ and $B(3,0... | -2 | 0.5625 | 6,308.6875 | 4,853.444444 | 8,179.714286 | |
A frog starts climbing from the bottom of a 12-meter deep well at 8:00 AM. For every 3 meters it climbs up, it slides down 1 meter due to the slippery walls. The time to slide down 1 meter is one-third the time taken to climb up 3 meters. At 8:17 AM, the frog reaches 3 meters from the well's top for the second time. De... | 22 | 0 | 8,014.6875 | -1 | 8,014.6875 | |
The reciprocal of $\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______. | 2.5 | 0.5 | 368.625 | 422.625 | 314.625 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.4375 | 1,836.5625 | 1,885.571429 | 1,798.444444 | |
Given that the function $f(x)=\sin (ωx+φ)(ω > 0,0 < φ < π)$ has a distance of $\frac {π}{2}$ between adjacent symmetry axes, and the function $y=f(x+ \frac {π}{2})$ is an even function.
1. Find the analytical expression of $f(x)$.
2. If $α$ is an acute angle, and $f(\frac {α}{2}+ \frac {π}{12})= \frac {3}{5}$, find the... | \frac {24+7 \sqrt {3}}{50} | 0 | 5,927.25 | -1 | 5,927.25 | |
How many even integers between 3000 and 6000 have four different digits? | 784 | 0.375 | 7,384.0625 | 6,152.166667 | 8,123.2 | |
Given a positive number \(r\) such that the set \(T=\left\{(x, y) \mid x, y \in \mathbf{R}\right.\) and \(\left.x^{2}+(y-7)^{2} \leqslant r^{2}\right\}\) is a subset of the set \(S=\{(x, y) \mid x, y \in \mathbf{R}\right.\) and for any \(\theta \in \mathbf{R}\), \(\cos 2\theta + x \cos \theta + y \geqslant 0\},\) deter... | 4 \sqrt{2} | 0.1875 | 7,995.8125 | 7,566 | 8,095 | |
Let \( a \) and \( b \) be positive real numbers. Given that \(\frac{1}{a} + \frac{1}{b} \leq 2\sqrt{2}\) and \((a - b)^2 = 4(ab)^3\), find \(\log_a b\). | -1 | 0.4375 | 7,626.875 | 6,900.285714 | 8,192 | |
Find the greatest common divisor of 75 and 360. | 15 | 0.9375 | 1,231 | 1,275.666667 | 561 | |
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the s... | 225 | 0 | 8,192 | -1 | 8,192 | |
Side $\overline{A B}$ of $\triangle A B C$ is the diameter of a semicircle, as shown below. If $A B=3+\sqrt{3}, B C=3 \sqrt{2}$, and $A C=2 \sqrt{3}$, then the area of the shaded region can be written as $\frac{a+(b+c \sqrt{d}) \pi}{e}$, where $a, b, c, d, e$ are integers, $e$ is positive, $d$ is square-free, and $\ope... | 147938 | Drop an altitude to point $D$ on $\overline{A B}$ from $C$ and let $x=A D$. Solving for $x$, we find $$\begin{aligned} 12-x^{2}=18-(3+\sqrt{3}-x)^{2} & \Rightarrow 12=18-9-6 \sqrt{3}-3+2(3+\sqrt{3}) x-x^{2} \\ & \Rightarrow 6+6 \sqrt{3}=(6+2 \sqrt{3}) x \\ & \Rightarrow x=\sqrt{3} \end{aligned}$$ So $A C=2 A D$, from w... | 0 | 7,966.125 | -1 | 7,966.125 |
In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$. | 20 | We find that, as before, $AE = \sqrt{7}$, and also the area of $\Delta DAE$ is 1/3 the area of $\Delta ABC$. Thus, using the area formula, $1/2 \cdot 7 \cdot \sin(\angle EAD) = 3\sqrt{3}/4$, and $\sin(\angle EAD) = \dfrac{3\sqrt{3}}{14}$. Therefore, $a + b + c = \boxed{020}.$ | 1 | 3,303.625 | 3,303.625 | -1 |
For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals: | \frac{2}{7} | To solve for the sum $S$ of the series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$, we first observe the pattern in the series. The series can be grouped into terms of three as follows:
\[
S = \left(1 - \frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{8} - \frac{1}{16} - \fr... | 0.4375 | 7,051.25 | 5,584.571429 | 8,192 |
In a bookshelf, there are four volumes of Astrid Lindgren's collected works in order, each containing 200 pages. A little worm living in these volumes burrowed a path from the first page of the first volume to the last page of the fourth volume. How many pages did the worm burrow through? | 400 | 0 | 554.1875 | -1 | 554.1875 | |
What is the least value of $y$ such that $3y^2 + 5y + 2 = 4$? | -2 | 1 | 2,039 | 2,039 | -1 | |
Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
[asy]
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draw(unitcircle,dg);
for(int i = 0; i < 12; ++i) {
draw(dir(30*i... | 300 | 0.75 | 4,517 | 3,292 | 8,192 | |
For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$? | 500 | 0.6875 | 6,191.8125 | 5,282.636364 | 8,192 | |
Evaluate $\log_5625$. | 4 | 1 | 1,672.5625 | 1,672.5625 | -1 | |
There is a settlement $C$ (a point) located at the intersection of roads $A$ and $B$ (straight lines). Sasha walks along road $A$ towards $C$, taking 45 steps per minute with a step length of 60 cm. At the start, Sasha is 290 m away from $C$. Dania walks along road $B$ towards $C$ at a rate of 55 steps per minute with ... | 57 | 0 | 7,520.1875 | -1 | 7,520.1875 | |
Let $p_{i}$ be the $i$th prime. Let $$f(x)=\sum_{i=1}^{50} p_{i} x^{i-1}=2+3x+\cdots+229x^{49}$$ If $a$ is the unique positive real number with $f(a)=100$, estimate $A=\lfloor 100000a\rfloor$. An estimate of $E$ will earn $\max (0,\lfloor 20-|A-E| / 250\rfloor)$ points. | 83601 | Note $f(x)$ is increasing. Since $f(0)=2$ and $f(1) \approx 50000$, we have $0<a<1$. Since we know that $p_{50}=229$, we can crudely bound $$f(x) \lesssim \sum_{i=1}^{\infty} 5i x^{i-1}=\frac{5}{(1-x)^{2}}$$ Setting this equal to 100 yields $x=1-20^{-1 / 2} \approx 0.78$, so this is a good lower bound for $a$, though j... | 0 | 8,192 | -1 | 8,192 |
A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digit of $M$, then we call this number $M$ a "... | 5883 | 0.25 | 7,881.4375 | 6,949.75 | 8,192 | |
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$ | m = 9,n = 6 |
To solve for the values of \( m \) and \( n \), we have the given conditions:
1. \( a^2 + b^2 + c^2 + d^2 = 1989 \)
2. \( a + b + c + d = m^2 \)
3. The largest of \( a, b, c, d \) is \( n^2 \)
We need to find positive integers \( m \) and \( n \) that satisfy these equations.
### Step 1: Analyze the range for \( m ... | 0 | 8,192 | -1 | 8,192 |
Consider the graph of \( y = g(x) \), with \( 1 \) unit between grid lines, where \( g(x) = \frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \), defined only on the shown domain.
Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.
[asy]
size(150);
real f(real ... | -5 | 0 | 8,192 | -1 | 8,192 | |
How many 4-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, and $E$? (Note that $B$, $C$, and $D$ are consonants, any word is valid, not just English language words, and letters may be used more than once.) | 609 | 0.875 | 2,984.6875 | 2,430.714286 | 6,862.5 | |
A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle.
Find the area of the quadrilateral \(ABCD\). | 60 | 0 | 8,192 | -1 | 8,192 | |
Seven identical bowling balls weigh the same as three identical canoes. If one of the canoes weighs a total of 28 pounds, how many pounds does one of the bowling balls weigh? | 12 | 1 | 1,148.1875 | 1,148.1875 | -1 | |
Arrange the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) in some order to form a nine-digit number \(\overline{\text{abcdefghi}}\). If \(A = \overline{\text{abc}} + \overline{\text{bcd}} + \overline{\text{cde}} + \overline{\text{def}} + \overline{\text{efg}} + \overline{\text{fgh}} + \overline{\text{ghi}}\), find the maximum p... | 4648 | 0 | 8,192 | -1 | 8,192 | |
There are positive integers $x$ and $y$ that satisfy the system of equations \begin{align*} \log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) &= 60\\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*} Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n... | 880 | Let $x=10^a$ and $y=10^b$ and $a<b$. Then the given equations become $3a=60$ and $3b=570$. Therefore, $x=10^{20}=2^{20}\cdot5^{20}$ and $y=10^{190}=2^{190}\cdot5^{190}$. Our answer is $3(20+20)+2(190+190)=\boxed{880}$. | 0.3125 | 6,952.375 | 4,767.8 | 7,945.363636 |
Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at... | 270 | $AB^2 = 4 AM^2 =2x(2x+ 2 XY) =(XP - XY) (XP + XY) = XP^2 - XY^2 = XC \cdot XD - XY^2 = 67 \cdot 37 - 47^2 = \boxed{270}.$
vladimir.shelomovskii@gmail.com, vvsss
~MathProblemSolvingSkills.com | 0 | 8,192 | -1 | 8,192 |
At a community gathering there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{3}{7}$. Calculate the fraction of the people in the gathering who are married men. | \frac{4}{11} | 0.875 | 2,052.3125 | 1,888.857143 | 3,196.5 | |
Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\frac{1}{4}$ of the candies. Then, he shared $\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike had. Later, Mike’s sist... | 64 | 0 | 6,864.0625 | -1 | 6,864.0625 | |
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.
For all positive integers $n$ and $t$, determi... | \lceil \frac{n}{t} \rceil |
In a sports league, each team uses a set of at most \( t \) signature colors. A set \( S \) of teams is color-identifiable if one can assign each team in \( S \) one of their signature colors, such that no team in \( S \) is assigned any signature color of a different team in \( S \).
For all positive integers \( n \... | 0 | 8,192 | -1 | 8,192 |
The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ and circle B has radius $42,$ find the ... | 147 | 0.375 | 7,383.9375 | 6,037.166667 | 8,192 | |
At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar.
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?... | 9 |
To solve this problem, we need to maximize the number of people at a round table who can truthfully say: "Both of my neighbors are knights." Considering the rules:
- Knights always tell the truth.
- Liars always lie.
- At least one knight and one liar are present.
Let's analyze the configuration of people around the ... | 0 | 8,190.3125 | -1 | 8,190.3125 |
Express $249_{11}+3AB_{12}$ as a base 10 integer. Here $A$ and $B$ denote the digits whose values are 10 and 11, respectively, in base 12. | 858 | 1 | 2,379.3125 | 2,379.3125 | -1 | |
Given that the function $f(x)$ satisfies $f(x+y)=f(x)+f(y)$ for all real numbers $x, y \in \mathbb{R}$, and $f(x) < 0$ when $x > 0$, and $f(3)=-2$.
1. Determine the parity (odd or even) of the function.
2. Determine the monotonicity of the function on $\mathbb{R}$.
3. Find the maximum and minimum values of $f(x)$ on $[... | -8 | 1 | 2,505.3125 | 2,505.3125 | -1 | |
One of the roots of
\[ax^3 + 3x^2 + bx - 65 = 0,\]is $-2 - 3i,$ where $a$ and $b$ are real numbers. Find the real root of this cubic polynomial. | \frac{5}{2} | 0.875 | 4,935.5625 | 4,470.357143 | 8,192 | |
Find the sum of the digits of the number \( A \), if \( A=2^{63} \cdot 4^{25} \cdot 5^{106}-2^{22} \cdot 4^{44} \cdot 5^{105}-1 \). | 959 | 0.375 | 7,216.6875 | 5,724.833333 | 8,111.8 | |
The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form one large equilateral triangular corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answer as a common fraction.
[asy]
draw((0,0)... | \frac{1}{4} | 0.5625 | 5,714.875 | 4,666.333333 | 7,063 | |
Riley has 64 cubes with dimensions \(1 \times 1 \times 1\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \(4 \times 4 \times 4\) cube. Riley determines the total of the numbers on the outside of the \(4 \times 4 \times 4... | 49 | 0 | 8,177.8125 | -1 | 8,177.8125 | |
What is the value of $1234 + 2341 + 3412 + 4123$ | 11110 | To solve the problem, we need to add the numbers $1234$, $2341$, $3412$, and $4123$. We can do this by aligning the numbers vertically and adding them column by column:
\[
\begin{array}{c}
\phantom{+}1234 \\
+2341 \\
+3412 \\
+4123 \\
\hline
\end{array}
\]
We start from the units digit and move to the left:
1. **Uni... | 0.875 | 3,658.25 | 3,010.571429 | 8,192 |
Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but now the price of wheat has risen to 1.6 yu... | 20 | 0.1875 | 6,291.0625 | 4,397.333333 | 6,728.076923 | |
Given that the central angle of a sector is $\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \_\_\_\_\_\_ cm, and the area of the sector is \_\_\_\_\_\_ cm<sup>2</sup>. | 27 | 0.875 | 1,356.4375 | 1,468.285714 | 573.5 | |
John wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property? | 456976 | 0.875 | 3,085.25 | 2,709.357143 | 5,716.5 | |
An integer has exactly 4 prime factors, and the sum of the squares of these factors is 476. Find this integer. | 1989 | 0 | 8,192 | -1 | 8,192 | |
A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ | 77 | 1. **Understanding the problem**: Each bench section can hold either $7$ adults or $11$ children. When $N$ bench sections are connected, the same number of adults and children must fill all the seats exactly.
2. **Setting up the equation**: Let $x$ be the number of adults and $y$ be the number of children. Since the n... | 0 | 3,157.6875 | -1 | 3,157.6875 |
Find all prime numbers $p,q,r$ , such that $\frac{p}{q}-\frac{4}{r+1}=1$ | \[
(7, 3, 2), (3, 2, 7), (5, 3, 5)
\] | The given equation can be rearranged into the below form:
$4q = (p-q)(r+1)$
$Case 1: 4|(p-q)$
then we have
$q = ((p-q)/4)(r+1)$ $=> (p-q)/4 = 1$ and $q = r + 1$ $=> r = 2, q = 3$ and $p = 7$
$Case 2: 4|(r+1)$
then we have
$q = (p-q)((r+1)/4)$ $=> (p-q) = 1$ and $q = (r + 1)/4$ $=> p = q + 1 => q = 2, p = 3$... | 0 | 7,120.625 | -1 | 7,120.625 |
Given a random variable $X \sim B(10, 0.6)$, calculate the values of $E(X)$ and $D(X)$. | 2.4 | 0.9375 | 2,501.5 | 2,122.133333 | 8,192 | |
The average of 12, 21 and $x$ is 18. What is the value of $x$? | 21 | 1 | 1,351.4375 | 1,351.4375 | -1 | |
Let the sequence \(b_1, b_2, b_3, \dots\) be defined such that \(b_1 = 24\), \(b_{12} = 150\), and for all \(n \geq 3\), \(b_n\) is the arithmetic mean of the first \(n - 1\) terms. Find \(b_2\). | 276 | 0.4375 | 7,022.25 | 5,718.428571 | 8,036.333333 | |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) grid such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? | 25 | 0 | 7,652.3125 | -1 | 7,652.3125 | |
Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity? | 225 | 0.375 | 6,519.875 | 6,586.333333 | 6,480 | |
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: | \frac{40}{3} | 1. **Identify Given Information and Draw Auxiliary Lines**:
- Triangle $ABC$ has side $AB = 8$ inches.
- Line $DEF$ is parallel to $AB$, with $D$ on $AC$, $E$ on $BC$, and $DE = 5$ inches.
- Line $AE$ extended bisects angle $FEC$.
2. **Use of Parallel Lines and Angle Properties**:
- Since $DEF$ is paralle... | 0.125 | 8,098.6875 | 7,815 | 8,139.214286 |
In the equilateral triangle \(ABC\), point \(T\) is its centroid, point \(R\) is the reflection of \(T\) across the line \(AB\), and point \(N\) is the reflection of \(T\) across the line \(BC\).
Determine the ratio of the areas of triangles \(ABC\) and \(TRN\). | 3:1 | 0.0625 | 6,539.0625 | 7,952 | 6,444.866667 | |
If 6 students want to sign up for 4 clubs, where students A and B do not join the same club, and every club must have at least one member with each student only joining one club, calculate the total number of different registration schemes. | 1320 | 0.125 | 7,885.3125 | 7,100.5 | 7,997.428571 | |
What is the sum of the squares of the lengths of the $\textbf{medians}$ of a triangle whose side lengths are $10,$ $10,$ and $12$? | 258 | 0.9375 | 3,875.0625 | 3,587.266667 | 8,192 |
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