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 |
|---|---|---|---|---|---|---|
Triangle $ABC$ is isosceles with $AB=AC$ . The bisectors of angles $ABC$ and $ACB$ meet at $I$ . If the measure of angle $CIA$ is $130^\circ$ , compute the measure of angle $CAB$ .
*Proposed by Connor Gordon* | 80 | 0.0625 | 4,473 | 1,458 | 4,674 | |
Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$.
(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$.
(b) Suppose $n$ is an odd prime. F... | (n-1)(n-2)^{n-1} - \frac{2^{n-1} - 1}{n} - 1 |
Let \( n \) be a positive integer. Consider sequences \( a_0, a_1, \ldots, a_n \) for which \( a_i \in \{1, 2, \ldots , n\} \) for all \( i \) and \( a_n = a_0 \).
### Part (a)
Suppose \( n \) is odd. We need to find the number of such sequences if \( a_i - a_{i-1} \not\equiv i \pmod{n} \) for all \( i = 1, 2, \ldots... | 0 | 8,192 | -1 | 8,192 |
Consider a month with 31 days, where the number of the month is a product of two distinct primes (e.g., July, represented as 7). Determine how many days in July are relatively prime to the month number. | 27 | 0.5625 | 5,468.1875 | 3,746.777778 | 7,681.428571 | |
Given the polar equation of a line is $ρ\sin(θ+ \frac{π}{4})= \frac{\sqrt{2}}{2}$, and the parametric equation of the circle $M$ is $\begin{cases} x = 2\cosθ \\ y = -2 + 2\sinθ \end{cases}$, where $θ$ is the parameter.
(I) Convert the line's polar equation into a Cartesian coordinate equation;
(II) Determine the minimu... | \frac{3\sqrt{2}}{2} - 2 | 0 | 5,902.125 | -1 | 5,902.125 | |
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$. | 540^\circ |
Let \(ABCD\) be a convex quadrilateral where \(AB = AD\) and \(CB = CD\). Given that the bisector of \(\angle BDC\) intersects \(BC\) at \(L\), and \(AL\) intersects \(BD\) at \(M\), we are informed that \(BL = BM\). We are to determine the value of \(2\angle BAD + 3\angle BCD\).
First, note the following properties ... | 0 | 8,192 | -1 | 8,192 |
Given the real numbers \( x \) and \( y \) that satisfy
\[ x + y = 3 \]
\[ \frac{1}{x + y^2} + \frac{1}{x^2 + y} = \frac{1}{2} \]
find the value of \( x^5 + y^5 \). | 123 | 0.9375 | 4,917 | 4,698.666667 | 8,192 | |
Find the radius of the circle with equation $x^2 - 6x + y^2 + 2y + 6 = 0$. | 2 | 1 | 2,174.375 | 2,174.375 | -1 | |
Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. | 163 | We use Burnside's Lemma. The set being acted upon is the set of integer triples $(a,r,s)$ such that $ars=2010$. Because $r$ and $s$ are indistinguishable, the permutation group consists of the identity and the permutation that switches $r$ and $s$. In cycle notation, the group consists of $(a)(r)(s)$ and $(a)(r \: s)$.... | 0 | 8,192 | -1 | 8,192 |
The square root of $2x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition? | 3 | 1 | 1,886.25 | 1,886.25 | -1 | |
In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take?
Recall that among 99 numbers, the medi... | 54 | 0.125 | 7,803.5625 | 5,556 | 8,124.642857 | |
Eight distinct integers are picked at random from $\{1,2,3,\ldots,15\}$. What is the probability that, among those selected, the third smallest is $5$? | \frac{4}{21} | 0 | 4,935.125 | -1 | 4,935.125 | |
Given a point on the parabola $y^2=6x$ whose distance to the focus is twice the distance to the y-axis, find the x-coordinate of this point. | \frac{3}{2} | 1 | 4,022.3125 | 4,022.3125 | -1 | |
In the convex quadrilateral \(ABCD\), \(\angle ABC=60^\circ\), \(\angle BAD=\angle BCD=90^\circ\), \(AB=2\), \(CD=1\), and the diagonals \(AC\) and \(BD\) intersect at point \(O\). Find \(\sin \angle AOB\). | \frac{15 + 6\sqrt{3}}{26} | 0 | 8,192 | -1 | 8,192 | |
The symphony orchestra has more than 200 members but fewer than 300 members. When they line up in rows of 6, there are two extra members; when they line up in rows of 8, there are three extra members; and when they line up in rows of 9, there are four extra members. How many members are in the symphony orchestra? | 260 | 0 | 8,192 | -1 | 8,192 | |
The first term of a sequence is $3107$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $614^{\text{th}}$ term of the sequence? | 20 | 0.0625 | 5,006 | 5,792 | 4,953.6 | |
The positive divisors of the integer 630 (including 1 and 630) total how many? | 24 | 0.9375 | 2,940.75 | 2,590.666667 | 8,192 | |
Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. Point $A(4,2\sqrt{2})$ lies on the ellipse, and $AF\_2$ is perpendicular to the $x$-axis.
1. Find the equation of the ellipse.
2. A line passing through point $F\_2$ intersects the ellip... | 8\sqrt{2} | 0.125 | 8,173 | 8,192 | 8,170.285714 | |
Let $z$ be a nonreal complex number such that $|z| = 1$. Find the real part of $\frac{1}{z - i}$. | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Let $G$ be the centroid of quadrilateral $ABCD$. If $GA^2 + GB^2 + GC^2 + GD^2 = 116$, find the sum $AB^2 + AC^2 + AD^2 + BC^2 + BD^2 + CD^2$. | 464 | 0.375 | 6,922.4375 | 5,142 | 7,990.7 | |
Find the point on the line defined by
\[\begin{pmatrix} 4 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} -2 \\ 6 \\ -3 \end{pmatrix}\]that is closest to the point $(2,3,4).$ | \left( \frac{170}{49}, \frac{78}{49}, \frac{10}{49} \right) | 0.875 | 4,937.9375 | 4,473.071429 | 8,192 | |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and every head either always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "On my left is a green dra... | 176 | 0 | 8,192 | -1 | 8,192 | |
During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor who wins the election? | 14 | 0 | 8,192 | -1 | 8,192 | |
Determine the remainder when \(1 + 5 + 5^2 + \cdots + 5^{1002}\) is divided by \(500\). | 31 | 0.4375 | 6,375.5625 | 4,629.714286 | 7,733.444444 | |
The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map? | 30:1 | 0.3125 | 454.875 | 440.4 | 461.454545 | |
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121. | 8100 | 0.0625 | 8,030 | 5,600 | 8,192 | |
The triangle $\triangle ABC$ is an isosceles triangle where $AB = 4\sqrt{2}$ and $\angle B$ is a right angle. If $I$ is the incenter of $\triangle ABC,$ then what is $BI$?
Express your answer in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by any perfect square other than $1... | 8 - 4\sqrt{2} | 1 | 5,996.875 | 5,996.875 | -1 | |
A rectangle has a perimeter of 120 inches and each side has an even integer length. How many non-congruent rectangles meet these criteria? | 15 | 0.875 | 5,248.25 | 4,827.714286 | 8,192 | |
Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden and Daryl have 78 marbles, what is the value of $a?$ | 2 | 1 | 1,026.3125 | 1,026.3125 | -1 | |
Find the $y$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair. | (0,4) | 1 | 1,343.0625 | 1,343.0625 | -1 | |
Let $A$ be the greatest common factor and let $B$ be the least common multiple of 8, 12 and 24. What is the value of $A + B$? | 28 | 0.9375 | 3,272.4375 | 2,944.466667 | 8,192 | |
Calculate the value of $(3,1) \nabla (4,2)$ using the operation ' $\nabla$ ' defined by $(a, b) \nabla (c, d)=ac+bd$. | 14 | From the definition, $(3,1) \nabla (4,2)=(3)(4)+(1)(2)=12+2=14$. | 1 | 646 | 646 | -1 |
Given the function $f(x)=2\ln(3x)+8x$, find the value of $\lim_{\triangle x \to 0}\frac{f(1-2\triangle x)-f(1)}{\triangle x}$. | -20 | 1 | 3,060.8125 | 3,060.8125 | -1 | |
Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends. | 6 |
Given 30 students such that each student has at most 5 friends and for every 5 students there is a pair of students that are not friends, we need to determine the maximum \( k \) such that for all such possible configurations, there exists \( k \) students who are all not friends.
In graph theory terms, we are given ... | 0.125 | 8,151.875 | 7,871 | 8,192 |
Simplify first, then choose a suitable value for $x$ from $2$, $-2$, and $-6$ to substitute and evaluate.<br/>$\frac{{x}^{3}+2x^{2}}{{x}^{2}-4x+4}÷\frac{4x+8}{x-2}-\frac{1}{x-2}$. | -1 | 0.5625 | 3,522.6875 | 3,772.333333 | 3,201.714286 | |
Given two vectors $a$ and $b$ in a plane that are orthogonal to each other, with $|a|=2$ and $|b|=1$. Let $k$ and $t$ be real numbers that are not simultaneously zero.
(1) If $x=a+(t-3)b$ and $y=-ka+tb$ are orthogonal, find the functional relationship $k=f(t)$.
(2) Find the minimum value of the function $k=f(t)$. | -\frac{9}{16} | 1 | 2,669.125 | 2,669.125 | -1 | |
When the fraction $\frac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be | 19 | 1. **Simplify the Fraction:** Start by factoring the numerator and the denominator of the fraction $\dfrac{49}{84}$.
- The numerator $49$ can be factored as $7^2$.
- The denominator $84$ can be factored as $2^2 \cdot 3 \cdot 7$.
2. **Reduce the Fraction:** Cancel the common factors in the numerator and the denom... | 1 | 1,227.125 | 1,227.125 | -1 |
Calculate the line integral
$$
\int_{L} \frac{y}{3} d x - 3 x d y + x d z
$$
along the curve \( L \), which is given parametrically by
$$
\begin{cases}
x = 2 \cos t \\
y = 2 \sin t \\
z = 1 - 2 \cos t - 2 \sin t
\end{cases}
\quad \text{for} \quad 0 \leq t \leq \frac{\pi}{2}
$$ | 2 - \frac{13\pi}{3} | 0.25 | 7,375.3125 | 6,417.25 | 7,694.666667 | |
Given that point P is on the left branch of the hyperbola $x^2-y^2=4$, and $F_1$, $F_2$ are the left and right foci of the hyperbola, respectively, then $|PF_1|-|PF_2|$ equals to __. | -4 | 0.8125 | 4,958.25 | 4,587.923077 | 6,563 | |
Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probability that both groups will stop flipping the... | \frac{1}{702} | 0.0625 | 5,605 | 8,192 | 5,432.533333 | |
Evaluate $81^{-\frac{1}{4}} + 16^{-\frac{3}{4}}$. Express your answer as a common fraction. | \frac{11}{24} | 1 | 1,833.3125 | 1,833.3125 | -1 | |
Given that 8 first-year high school students are divided evenly between two companies, A and B, with the condition that two students with excellent English grades cannot be assigned to the same company and three students with computer skills cannot be assigned to the same company, determine the number of different dist... | 36 | 0 | 7,913.4375 | -1 | 7,913.4375 | |
Find the largest positive number \( c \) such that for every natural number \( n \), the inequality \( \{n \sqrt{2}\} \geqslant \frac{c}{n} \) holds, where \( \{n \sqrt{2}\} = n \sqrt{2} - \lfloor n \sqrt{2} \rfloor \) and \( \lfloor x \rfloor \) denotes the integer part of \( x \). Determine the natural number \( n \)... | \frac{1}{2\sqrt{2}} | 0 | 8,192 | -1 | 8,192 | |
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersectio... | R^2 |
Given a circle \( O \) with radius \( R \), and an inscribed acute scalene triangle \( ABC \) where \( AB \) is the largest side, let \( AH_A, BH_B, CH_C \) be the altitudes from \( A, B, C \) to \( BC, CA, AB \) respectively. Let \( D \) be the symmetric point of \( H_A \) with respect to \( H_BH_C \), and \( E \) be... | 0 | 8,192 | -1 | 8,192 |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimeter of $\triangle APR$. | 50 | 0 | 7,500.6875 | -1 | 7,500.6875 | |
Find \( k \) such that, for all \( n \), the following expression is a perfect square:
$$
4 n^{2} + k n + 9
$$ | 12 | 0.125 | 5,540.5 | 6,700.5 | 5,374.785714 | |
Given a line $l$ intersects the hyperbola $x^2 - \frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$. | 4\sqrt{2} | 0.875 | 4,583.0625 | 4,403.285714 | 5,841.5 | |
If \( f(x) = x^{6} - 2 \sqrt{2006} x^{5} - x^{4} + x^{3} - 2 \sqrt{2007} x^{2} + 2 x - \sqrt{2006} \), then find \( f(\sqrt{2006} + \sqrt{2007}) \). | \sqrt{2007} | 0.125 | 7,997.4375 | 6,635.5 | 8,192 | |
Two squares of a $7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible? | 300 | There are 4 cases: 1. The center square is occupied, in which there are $12$ cases. 2. The center square isn't occupied and the two squares that are opposite to each other with respect to the center square, in which there are $12$ cases. 3. The center square isn't occupied and the two squares can rotate to each other w... | 0.75 | 4,575.625 | 3,899.083333 | 6,605.25 |
If three different lines $x+y=1$, $x-y=1$, and $ax+y=1$ cannot form a triangle, then the value of the real number $a$ is. | -1 | 0.6875 | 5,739.5 | 4,995.090909 | 7,377.2 | |
In $\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos B = \frac{4}{5}$ and $b = 2$.
1. Find the value of $a$ when $A = \frac{\pi}{6}$.
2. Find the value of $a + c$ when the area of $\triangle ABC$ is $3$. | 2\sqrt{10} | 0.5625 | 5,825.4375 | 4,759 | 7,196.571429 | |
A club consists initially of 20 total members, which includes eight leaders. Each year, all the current leaders leave the club, and each remaining member recruits three new members. Afterwards, eight new leaders are elected from outside. How many total members will the club have after 4 years? | 980 | 0.25 | 6,584.25 | 4,904 | 7,144.333333 | |
Given the function $f(x)$ with the domain $[1, +\infty)$, and $f(x) = \begin{cases} 1-|2x-3|, & 1\leq x<2 \\ \frac{1}{2}f\left(\frac{1}{2}x\right), & x\geq 2 \end{cases}$, then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1, 2017)$ is \_\_\_\_\_\_. | 11 | 0 | 8,192 | -1 | 8,192 | |
In a triangle with sides \(AB = 4\), \(BC = 2\), and \(AC = 3\), an incircle is inscribed. Find the area of triangle \(AMN\), where \(M\) and \(N\) are the points of tangency of this incircle with sides \(AB\) and \(AC\) respectively. | \frac{25 \sqrt{15}}{64} | 0 | 6,898.4375 | -1 | 6,898.4375 | |
For each vertex of the triangle \(ABC\), the angle between the altitude and the angle bisector drawn from that vertex was determined. It turned out that these angles at vertices \(A\) and \(B\) are equal to each other and are less than the angle at vertex \(C\). What is the measure of angle \(C\) in the triangle? | 60 | 0.0625 | 8,116.875 | 7,236 | 8,175.6 | |
Let $a \neq b$ be positive real numbers and $m, n$ be positive integers. An $m+n$-gon $P$ has the property that $m$ sides have length $a$ and $n$ sides have length $b$. Further suppose that $P$ can be inscribed in a circle of radius $a+b$. Compute the number of ordered pairs $(m, n)$, with $m, n \leq 100$, for which su... | 940 | Letting $x=\frac{a}{a+b}$, we have to solve $$m \arcsin \frac{x}{2}+n \arcsin \frac{1-x}{2}=\pi$$ This is convex in $x$, so if it is to have a solution, we must find that the LHS exceeds $\pi$ at one of the endpoints. Thus $\max (m, n) \geq 7$. If $\min (m, n) \leq 5$ we can find a solution by by the intermediate value... | 0 | 8,192 | -1 | 8,192 |
A reflection takes $\begin{pmatrix} -1 \\ 7 \end{pmatrix}$ to $\begin{pmatrix} 5 \\ -5 \end{pmatrix}.$ Which vector does the reflection take $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$ to? | \begin{pmatrix} 0 \\ -5 \end{pmatrix} | 0.625 | 5,959.8125 | 4,620.5 | 8,192 | |
How many positive integers \( n \) are there such that \( n \) is a multiple of 4, and the least common multiple of \( 4! \) and \( n \) equals 4 times the greatest common divisor of \( 8! \) and \( n \)? | 12 | 0 | 7,601.25 | -1 | 7,601.25 | |
Let $PQRS$ be a convex quadrilateral, and let $H_P,$ $H_Q,$ $H_R,$ $H_S$ denote the centroids of triangles $QRS,$ $PRS,$ $PQS,$ and $PQR,$ respectively. Calculate $\frac{[H_P H_Q H_R H_S]}{[PQRS]}.$ | \frac{1}{9} | 0.25 | 7,754.625 | 6,442.5 | 8,192 | |
The solutions to the equation \( x^3 - 4 \lfloor x \rfloor = 5 \), where \( x \) is a real number, are denoted by \( x_1, x_2, x_3, \ldots, x_k \) for some positive integer \( k \). Find \( \sum_{i=1}^{k} x_{i}^{3} \). | 10 | 0.625 | 6,490.5 | 5,691.7 | 7,821.833333 | |
Let \( x \) and \( y \) be non-zero real numbers such that
\[ \frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}. \]
(1) Find the value of \(\frac{y}{x}\).
(2) In triangle \( \triangle ABC \), if \( \tan C = \frac{y}{x} \), find the maximum value o... | \frac{3}{2} | 0.375 | 7,704 | 6,890.666667 | 8,192 | |
A stock investment increased by 30% in 2006. Starting at this new value, what percentage decrease is needed in 2007 to return the stock to its original price at the beginning of 2006? | 23.077\% | 0.0625 | 659.6875 | 778 | 651.8 | |
If the function $$f(x)=(2m+3)x^{m^2-3}$$ is a power function, determine the value of $m$. | -1 | 0 | 7,904.5 | -1 | 7,904.5 | |
Given quadrilateral $\Box FRDS$ with $\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\angle RFS = \angle FDR$, find the length of RS. | \sqrt{89} | 0.5625 | 5,758 | 4,249.888889 | 7,697 | |
Among the three-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, there are a total of numbers whose digits sum up to 9 (answer in digits). | 16 | 0.625 | 5,866 | 4,750.3 | 7,725.5 | |
Find the area of the region \(D\) bounded by the curves
\[ x^{2} + y^{2} = 12, \quad x \sqrt{6} = y^{2} \quad (x \geq 0) \] | 3\pi + 2 | 0 | 7,993.875 | -1 | 7,993.875 | |
An $8$-cm-by-$8$-cm square is partitioned as shown. Points $A$ and $B$ are the midpoints of two opposite sides of the square. What is the area of the shaded region?
[asy]
draw((0,0)--(10,0));
draw((10,0)--(10,10));
draw((10,10)--(0,10));
draw((0,0)--(0,10));
draw((0,0)--(5,10));
draw((5,10)--(10,0));
draw((0,10)--(5,0... | 16 | 0.625 | 5,533 | 5,309.6 | 5,905.333333 | |
The graph of the function $y=\sin (2x+\varphi)$ is translated to the left by $\dfrac {\pi}{8}$ units along the $x$-axis and results in a graph of an even function, then determine one possible value of $\varphi$. | \dfrac {\pi}{4} | 0.75 | 5,056.625 | 4,011.5 | 8,192 | |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | 12 | 1 | 2,161.125 | 2,161.125 | -1 | |
Given the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$ with three non-collinear points $A$, $B$, $C$ on it. The midpoints of $AB$, $BC$, $AC$ are $D$, $E$, $F$ respectively. If the sum of the slopes of $OD$, $OE$, $OF$ is $-1$, find the value of $\frac{1}{k_{AB}} + \frac{1}{k_{BC}} + \frac{1}{k_{AC}}$. | -2 | 0 | 8,123.4375 | -1 | 8,123.4375 | |
Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? | 190 | To solve the problem, we need to find the number of ways Alice, Becky, and Chris can each receive at least two apples from a total of 24 apples. Let's denote the number of apples received by Alice, Becky, and Chris as $a$, $b$, and $c$ respectively. The condition given is that each person must receive at least two appl... | 0.9375 | 3,135 | 2,797.866667 | 8,192 |
Jack, Jill, and John play a game in which each randomly picks and then replaces a card from a standard 52 card deck, until a spades card is drawn. What is the probability that Jill draws the spade? (Jack, Jill, and John draw in that order, and the game repeats if no spade is drawn.) | \frac{12}{37} | The desired probability is the relative probability that Jill draws the spade. In the first round, Jack, Jill, and John draw a spade with probability $1 / 4,3 / 4 \cdot 1 / 4$, and $(3 / 4)^{2} \cdot 1 / 4$ respectively. Thus, the probability that Jill draws the spade is $$\frac{3 / 4 \cdot 1 / 4}{1 / 4+3 / 4 \cdot 1 /... | 0.375 | 6,719.875 | 6,139 | 7,068.4 |
A chessboard of size $8 \times 8$ is considered.
How many ways are there to place 6 rooks such that no two rooks are ever on the same row or column? | 564480 | 0.875 | 5,838.0625 | 5,501.785714 | 8,192 | |
Find all real numbers $x$ such that the product $(x + 2i)((x + 1) + 2i)((x + 2) + 2i)((x + 3) + 2i)$ is purely imaginary. | -2 | 0 | 7,525.75 | -1 | 7,525.75 | |
In triangle $ABC$, $\angle C=90^\circ$, $AC=6$ and $BC=8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=4$, then what is the length of $BD$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
draw(origin--(6,0)--(6,8)--cycle);... | \frac{20}{3} | 0.875 | 4,702.5625 | 4,442.142857 | 6,525.5 | |
Given that triangle $PQR$ is a right triangle, each side being the diameter of a semicircle, the area of the semicircle on $\overline{PQ}$ is $18\pi$, and the arc of the semicircle on $\overline{PR}$ has length $10\pi$, calculate the radius of the semicircle on $\overline{QR}$. | \sqrt{136} | 0 | 4,613.125 | -1 | 4,613.125 | |
In triangle $ABC$, if $a=2$, $c=2\sqrt{3}$, and $\angle A=30^\circ$, then the area of $\triangle ABC$ is equal to __________. | \sqrt{3} | 0.125 | 6,903 | 3,752.5 | 7,353.071429 | |
A chest of gold coins is divided among 10 pirates where the kth pirate takes k/10 of the remaining coins. Determine the smallest number of coins initially in the chest such that each pirate gets a positive whole number of coins and find the number of coins the 10th pirate receives. | 362880 | 0 | 8,106.625 | -1 | 8,106.625 | |
Given a segment of length $2$ with endpoints $A$ and $B$ sliding respectively on the $x$-axis and $y$-axis, the midpoint $M$ of segment $AB$ traces curve $C$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Point $P(x,y)$ is a moving point on curve $C$, find the range of values for $3x-4y$;
(Ⅲ) Given a fixed point $Q(0, ... | t=\lambda= \frac {3}{2} | 0 | 5,957.4375 | -1 | 5,957.4375 | |
The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? | 28 | 1 | 1,778 | 1,778 | -1 | |
In a particular country, the state of Sunland issues license plates with a format of one letter, followed by three digits, and then two letters (e.g., A123BC). Another state, Moonland, issues license plates where the format consists of two digits, followed by two letters, and then two more digits (e.g., 12AB34). Assumi... | 6084000 | 1 | 3,373.1875 | 3,373.1875 | -1 | |
Given a shooter who has a probability of $\frac{3}{4}$ of hitting target A with a single shot and a probability of $\frac{2}{3}$ of hitting target B with each of two shots, determine the probability that the shooter hits exactly one of the three shots. | \frac{7}{36} | 0.75 | 5,798.375 | 5,358.666667 | 7,117.5 | |
Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$ , where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers. | 7 + 19 | 0 | 6,959.3125 | -1 | 6,959.3125 | |
Compute the least possible value of $ABCD - AB \times CD$ , where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$ , $B$ , $C$ , and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.) | 109 | 0.0625 | 8,167.875 | 7,806 | 8,192 | |
Find the smallest positive integer $n$ such that $$\underbrace{2^{2^{2^{2}}}}_{n 2^{\prime} s}>\underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text { factorials }}$$ | 104 | Note that $2^{2^{2^{2}}}>100^{2}$. We claim that $a>b^{2} \Longrightarrow 2^{a}>(b!)^{2}$, for $b>2$. This is because $$2^{a}>b^{2 b} \Longleftrightarrow a>2 b \log _{2}(b)$$ and $\log _{2}(b)<b^{2} / 2$ for $b>2$. Then since $b^{b}>b$ ! this bound works. Then $$\underbrace{\left(2^{2^{2 \cdots 2}}\right)}_{m 2^{\prime... | 0 | 8,192 | -1 | 8,192 |
There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive... | 107 | Let $x$ be the probability that it rains on Sunday given that it doesn't rain on Saturday. We then have $\dfrac{3}{5}x+\dfrac{2}{5}2x = \dfrac{3}{10} \implies \dfrac{7}{5}x=\dfrac{3}{10}$ $\implies x=\dfrac{3}{14}$. Therefore, the probability that it doesn't rain on either day is $\left(1-\dfrac{3}{14}\right)\left(\dfr... | 0.75 | 4,273.9375 | 3,345.75 | 7,058.5 |
On the shore of a circular island (viewed from above) are the cities $A$, $B$, $C$, and $D$. The straight asphalt road $AC$ divides the island into two equal halves. The straight asphalt road $BD$ is shorter than the road $AC$ and intersects it. The cyclist's speed on any asphalt road is 15 km/h. The island also has st... | 450 | 0 | 8,192 | -1 | 8,192 | |
Points \( P \) and \( Q \) are located on side \( BC \) of triangle \( ABC \), with \( BP: PQ: QC = 1: 2: 3 \). Point \( R \) divides side \( AC \) of this triangle such that \( AR: RC = 1: 2 \). What is the ratio of the area of quadrilateral \( PQST \) to the area of triangle \( ABC \), if \( S \) and \( T \) are the ... | 5/24 | 0.5 | 6,852 | 5,751.5 | 7,952.5 | |
Find the largest real number $k$ such that \[x_1^2 + x_2^2 + \dots + x_{11}^2 \geq kx_6^2\] whenever $x_1, x_2, \ldots, x_{11}$ are real numbers such that $x_1 + x_2 + \cdots + x_{11} = 0$ and $x_6$ is the median of $x_1, x_2, \ldots, x_{11}$. | \frac{66}{5} | 0 | 8,146.875 | -1 | 8,146.875 | |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 10 and at least one die shows a number greater than 3? | \frac{2}{3} | 0.3125 | 7,703.375 | 6,628.4 | 8,192 | |
Given the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfying:
\[
\begin{array}{l}
\left|z_{1}\right| \leq 1, \left|z_{2}\right| \leq 2, \\
\left|2z_{3} - z_{1} - z_{2}\right| \leq \left|z_{1} - z_{2}\right|.
\end{array}
\]
What is the maximum value of \( \left|z_{3}\right| \)? | \sqrt{5} | 0.0625 | 7,823.5625 | 5,383 | 7,986.266667 | |
A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that its length is 75 meters. | 50 | 0.5 | 5,001.0625 | 4,073.375 | 5,928.75 | |
In right triangle $XYZ$, we have $\angle Y = \angle Z$ and $XY = 8\sqrt{2}$. What is the area of $\triangle XYZ$? | 64 | 0.9375 | 2,616.5 | 2,729.466667 | 922 | |
Simplify $(5^7+3^6)(1^5-(-1)^4)^{10}$. | 0 | 1 | 1,466.1875 | 1,466.1875 | -1 | |
The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region? | 800 | 0.4375 | 7,675.1875 | 7,379.857143 | 7,904.888889 | |
A circle of radius $3$ is cut into six congruent arcs. These arcs are then rearranged symmetrically to form a hexagonal star as illustrated below. Determine the ratio of the area of the hexagonal star to the area of the original circle.
A) $\frac{4.5}{\pi}$
B) $\frac{4.5\sqrt{2}}{\pi}$
C) $\frac{4.5\sqrt{3}}{\pi}$
D) $... | \frac{4.5\sqrt{3}}{\pi} | 0 | 8,192 | -1 | 8,192 | |
The sides of a triangle have lengths $13, 17,$ and $m,$ where $m$ is a positive integer. For how many values of $m$ is the triangle obtuse? | 14 | 0.75 | 6,457.5 | 5,879.333333 | 8,192 | |
Given $0 < \beta < \frac{\pi}{2} < \alpha < \pi$ and $\cos \left(\alpha- \frac{\beta}{2}\right)=- \frac{1}{9}, \sin \left( \frac{\alpha}{2}-\beta\right)= \frac{2}{3}$, calculate the value of $\cos (\alpha+\beta)$. | -\frac{239}{729} | 0.5 | 7,599 | 7,006 | 8,192 | |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle and the radius of the circle? | \frac{10}{\pi} | 0 | 8,192 | -1 | 8,192 | |
Given that $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$ and $0 < \alpha < \pi$, find the value of $\tan(\alpha - \frac{\pi}{4})$. | 2\sqrt{2} | 0.75 | 6,448.125 | 6,339.916667 | 6,772.75 | |
Compute $(\cos 185^\circ + i \sin 185^\circ)^{54}.$ | -i | 0.6875 | 6,109.25 | 5,162.545455 | 8,192 |
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