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 Fox wants to ensure he has 20 coins left after crossing the bridge four times, and paying a $50$-coin toll each time, determine the number of coins that Fox had at the beginning. | 25 | 0 | 1,799.3125 | -1 | 1,799.3125 | |
Compute
\[\begin{vmatrix} 7 & 3 \\ -1 & 2 \end{vmatrix}.\] | 17 | 0.9375 | 1,844.6875 | 1,421.533333 | 8,192 | |
Let $\#$ be the relation defined by $A \# B = A^2 + B^2$. If $A \# 5 = 169$, what is the positive value of $A$? | 12 | 1 | 1,050.4375 | 1,050.4375 | -1 | |
On each side of a right-angled triangle, a semicircle is drawn with that side as a diameter. The areas of the three semicircles are \( x^{2} \), \( 3x \), and 180, where \( x^{2} \) and \( 3x \) are both less than 180. What is the area of the smallest semicircle? | 144 | 0 | 4,466 | -1 | 4,466 | |
Let $b_n$ be the integer obtained by writing down the integers from 1 to $n$ in reverse, from right to left. Compute the remainder when $b_{39}$ is divided by 125. | 21 | 0 | 7,892.5625 | -1 | 7,892.5625 | |
Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements. | 48 | 0 | 8,192 | -1 | 8,192 | |
If the polynomial $x^3+x^{10}=a_0+a_1(x+1)+\ldots+a_9(x+1)^9+a_{10}(x+1)^{10}$, then $a_2=$ ______. | 42 | 0.8125 | 5,198.4375 | 4,507.615385 | 8,192 | |
Determine the smallest positive integer \(n\) for which there exists positive real numbers \(a\) and \(b\) such that
\[(a + 3bi)^n = (a - 3bi)^n,\]
and compute \(\frac{b}{a}\). | \frac{\sqrt{3}}{3} | 0 | 7,409.375 | -1 | 7,409.375 | |
Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that
(i) $m<2a$;
(ii) $2n|(2am-m^2+n^2)$;
(iii) $n^2-m^2+2mn\leq2a(n-m)$.
For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\]
Determine the maximum and minimum values of $f$. | 2 \text{ and } 3750 |
Let \( a = 2001 \). Consider the set \( A \) of all pairs of integers \((m, n)\) with \( n \neq 0 \) such that:
1. \( m < 2a \),
2. \( 2n \mid (2am - m^2 + n^2) \),
3. \( n^2 - m^2 + 2mn \leq 2a(n - m) \).
For \((m, n) \in A\), let
\[ f(m, n) = \frac{2am - m^2 - mn}{n}. \]
We need to determine the maximum and minimum... | 0 | 8,192 | -1 | 8,192 |
How many distinct three-digit positive integers have only odd digits? | 125 | 0.9375 | 1,492.4375 | 1,478 | 1,709 | |
The blue parabola shown is the graph of the equation \( x = ay^2 + by + c \). The vertex of the parabola is at \( (5, 3) \), and it passes through the point \( (3, 5) \). Find \( c \). | \frac{1}{2} | 1 | 2,533.875 | 2,533.875 | -1 | |
Given a line $y = \frac{\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\sqrt{3}$.
(1) Given a point $P(-1, \sqrt{7})$, and $Q$ is any point on circle $C$, find the maximum value of $|PQ|$.
(2) If a ray is drawn from t... | \frac{1}{3} | 1 | 4,974.25 | 4,974.25 | -1 | |
Two more than three times $B$ is equal to 20. What is the value of $B$? | 6 | 1 | 1,221.5 | 1,221.5 | -1 | |
What is the 43rd digit after the decimal point in the decimal representation of $\frac{1}{13}$? | 0 | 1 | 2,027.1875 | 2,027.1875 | -1 | |
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers. | 200 | 0.4375 | 7,140.9375 | 5,789.571429 | 8,192 | |
Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It is known that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal ... | 60^\circ |
Given a triangle \(\triangle ABC\) with the following properties:
- \( BL \) is the angle bisector of \(\angle ABC\), with \( L \) on \( AC \).
- \( AH \) is the altitude from \( A \) to \( BC \), with \( H \) on \( BC \).
- \( M \) is the midpoint of \( AB \).
Furthermore, we are informed that the midpoints of seg... | 0.6875 | 6,158.5625 | 5,307.545455 | 8,030.8 |
An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of flavors? | 92 | 0.8125 | 4,034.1875 | 3,442.384615 | 6,598.666667 | |
Two sectors of a circle of radius $12$ are placed side by side, as shown. Determine the $\textit{area}$ of figure $ABCD.$ [asy]
draw((0,0)--(12,0)..(10.3923,6)..(6,10.3923)--(-6,10.3923)..(-4.3923,4.3923)..(0,0),black+linewidth(1));
draw((0,0)--(6,10.3923),black+linewidth(1)+dashed);
label("$A$",(-6,10.3923),NW);
label... | 48\pi | 0.125 | 7,861.75 | 6,893 | 8,000.142857 | |
Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$? | 18 | 1. **Identify the Geometry of the Problem**: Given that $A$, $B$, and $C$ are on the graph $y = x^2$, and $AB$ is parallel to the $x$-axis, we know that $A$ and $B$ have the same $y$-coordinate. Since $\triangle ABC$ is a right triangle with area $2008$, we need to determine the position of $C$.
2. **Determine the Rig... | 0.9375 | 3,739.375 | 3,442.533333 | 8,192 |
How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have? | 18 |
To find the number of pairs of parallel edges in a cube, we can consider the cube's structure and symmetry. A cube has 12 edges, and each edge has exactly one parallel counterpart in each of the three dimensions (length, width, height).
1. **Identify Parallel Edges in One Dimension:**
- Consider the front face of ... | 0.8125 | 5,230.5625 | 4,547.153846 | 8,192 |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | 1 | 0.5625 | 6,636.75 | 5,427.111111 | 8,192 | |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that 2(tanA + tanB) = $\frac{\text{tanA}}{\text{cosB}} + \frac{\text{tanB}}{\text{cosA}}$.
(1) Find the value of $\frac{a+b}{c}$;
(2) If c = 2 and C = $\frac{\pi}{3}$, find the area of triangle ABC. | \sqrt{3} | 1 | 3,832 | 3,832 | -1 | |
Let \\(\triangle ABC\\) have internal angles \\(A\\), \\(B\\), and \\(C\\) opposite to sides of lengths \\(a\\), \\(b\\), and \\(c\\) respectively, and it satisfies \\(a^{2}+c^{2}-b^{2}= \sqrt {3}ac\\).
\\((1)\\) Find the size of angle \\(B\\);
\\((2)\\) If \\(2b\cos A= \sqrt {3}(c\cos A+a\cos C)\\), and the median... | \sqrt {3} | 0 | 7,005.8125 | -1 | 7,005.8125 | |
Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{5}{13}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[|a_0 a_1 \dot... | 108 | 0.375 | 7,296.3125 | 5,803.5 | 8,192 | |
Given a sequence $\{a\_n\}$, for any $k \in \mathbb{N}^*$, when $n = 3k$, $a\_n = a\_{\frac{n}{3}}$; when $n \neq 3k$, $a\_n = n$. The 10th occurrence of 2 in this sequence is the \_\_\_\_\_\_th term. | 2 \cdot 3^{9} | 0 | 6,038.5625 | -1 | 6,038.5625 | |
In a regular decagon $ABCDEFGHIJ$, points $K$, $L$, $M$, $N$, $O$, $P$, $Q$, $R$, and $S$ are selected on the sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, $\overline{EF}$, $\overline{FG}$, $\overline{GH}$, $\overline{HI}$, and $\overline{IJ}$ respectively. Each of these points divides their... | \frac{3\sqrt{3}}{40} | 0 | 8,192 | -1 | 8,192 | |
$r(x)$ has domain $\{-1,0,1,2\}$ and range $\{0,2,4,6\}$. $s(x)$ has domain $\{1,2,3,4\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$? | 8 | 0.9375 | 2,867.875 | 2,763.066667 | 4,440 | |
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions such that $a_1 = 50, b_1 = 100$, and $a_{50} + b_{50} = 850$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$ | 25000 | 0.875 | 3,451.8125 | 3,112.642857 | 5,826 | |
Bill draws two circles which intersect at $X,Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3,4$ and t... | 4807 | 0.5 | 6,845.0625 | 5,835.625 | 7,854.5 | |
If the line $y=kx+t$ is a tangent line to the curve $y=e^x+2$ and also a tangent line to the curve $y=e^{x+1}$, find the value of $t$. | 4-2\ln 2 | 0.8125 | 4,069.125 | 3,268.615385 | 7,538 | |
Let $ABC$ be a triangle with $AB=5, BC=4$ and $AC=3$. Let $\mathcal{P}$ and $\mathcal{Q}$ be squares inside $ABC$ with disjoint interiors such that they both have one side lying on $AB$. Also, the two squares each have an edge lying on a common line perpendicular to $AB$, and $\mathcal{P}$ has one vertex on $AC$ and $\... | \frac{144}{49} | Let the side lengths of $\mathcal{P}$ and $\mathcal{Q}$ be $a$ and $b$, respectively. Label two of the vertices of $\mathcal{P}$ as $D$ and $E$ so that $D$ lies on $AB$ and $E$ lies on $AC$, and so that $DE$ is perpendicular to $AB$. The triangle $ADE$ is similar to $ACB$. So $AD=\frac{3}{4}a$. Using similar arguments,... | 0 | 7,979.375 | -1 | 7,979.375 |
Write a twelve-digit number that is not a perfect cube. | 100000000000 | 0.1875 | 7,141.9375 | 6,380.333333 | 7,317.692308 | |
Given $tan({θ+\frac{π}{4}})=2tanθ-7$, determine the value of $\sin 2\theta$. | \frac{4}{5} | 1 | 2,181.3125 | 2,181.3125 | -1 | |
Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\frac{x}{\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$. | 4\sqrt{e}-2 | 0.4375 | 6,823.8125 | 5,666.857143 | 7,723.666667 | |
In a two-day problem-solving competition, Gamma and Delta participated and attempted questions worth a total of 500 points. On the first day, Gamma scored 180 points out of 280 points attempted, and on the second day, he scored 120 points out of 220 points attempted. Delta, who also divided his attempts across the two ... | \frac{409}{500} | 0 | 8,054.375 | -1 | 8,054.375 | |
Suppose the function $f(x) = ax + \frac{x}{x-1}$ where $x > 1$.
(1) If $a > 0$, find the minimum value of the function $f(x)$.
(2) If $a$ is chosen from the set \{1, 2, 3\} and $b$ is chosen from the set \{2, 3, 4, 5\}, find the probability that $f(x) > b$ always holds true. | \frac{5}{6} | 0.9375 | 4,667 | 4,700.933333 | 4,158 | |
How many ways can you arrange 15 dominoes (after removing all dominoes with five or six pips) in a single line according to the usual rules of the game, considering arrangements from left to right and right to left as different? As always, the dominoes must be placed such that matching pips (e.g., 1 to 1, 6 to 6, etc.... | 126760 | 0 | 8,192 | -1 | 8,192 | |
In triangle $XYZ$, $\angle X = 90^\circ$ and $\sin Y = \frac{3}{5}$. Find $\cos Z$. | \frac{3}{5} | 0.9375 | 3,016.125 | 2,671.066667 | 8,192 | |
The perimeter of a semicircle with an area of ______ square meters is 15.42 meters. | 14.13 | 0.1875 | 3,980.4375 | 5,359 | 3,662.307692 | |
Rectangle \( EFGH \) is 10 cm by 6 cm. \( P \) is the midpoint of \( \overline{EF} \), and \( Q \) is the midpoint of \( \overline{GH} \). Calculate the area of region \( EPGQ \).
** | 30 | 0.9375 | 4,758.125 | 4,529.2 | 8,192 | |
Compute $\tan \left (\operatorname{arccot} \frac{4}{7} \right).$ | \frac{7}{4} | 1 | 1,813.4375 | 1,813.4375 | -1 | |
A certain company, in response to the national call for garbage classification, has launched a project with the support of the research department to innovate technologies. The project processes kitchen waste into reusable chemical products. It is known that the daily processing capacity $x$ (unit: tons) of the company... | 1800 | 0 | 6,463.875 | -1 | 6,463.875 | |
$2018$ people (call them $A, B, C, \ldots$ ) stand in a line with each permutation equally likely. Given that $A$ stands before $B$ , what is the probability that $C$ stands after $B$ ? | 1/3 | 0.0625 | 7,310.0625 | 5,043 | 7,461.2 | |
Let \( p, q, r, s \) be distinct real numbers such that the roots of \( x^2 - 12px - 13q = 0 \) are \( r \) and \( s \), and the roots of \( x^2 - 12rx - 13s = 0 \) are \( p \) and \( q \). Find the value of \( p + q + r + s \). | 2028 | 0.125 | 7,799.25 | 5,050 | 8,192 | |
What is the diameter in centimeters of a circle whose area is $100\pi \text{cm}^2$? | 20 | 1 | 996.4375 | 996.4375 | -1 | |
Three real numbers $x, y, z$ are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of $x-y$ and $x-z$ is greater than $-\frac{1}{2}$ and less than $\frac{1}{2}$? | \frac{7}{12} | Consider a $1 \times 1 \times 1$ cube. We associate a triple $(x, y, z)$ of real numbers with $0 \leq x \leq 1$ and $0 \leq y \leq 1$ and $0 \leq z \leq 1$ with a point inside this cube by letting $x$ be the perpendicular distance of a point from the left face, $y$ the perpendicular distance of a point from the front f... | 0.5625 | 7,043 | 6,423.777778 | 7,839.142857 |
Let $\mathcal{P}$ be the parabola given by the equation \( y = x^2 \). Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are \((-4,16)\), \((1,1)\), and \((6,36)\), find the sum of the distances from the directrix of the parabola to all four intersection points. | 63 | 0.9375 | 4,752.4375 | 4,523.133333 | 8,192 | |
Real numbers \(a, b, c\) and positive number \(\lambda\) make the function \(f(x) = x^3 + ax^2 + bx + c\) have three real roots \(x_1, x_2, x_3\), such that (1) \(x_2 - x_1 = \lambda\); (2) \(x_3 > \frac{1}{2}(x_1 + x_2)\). Find the maximum value of \(\frac{2a^3 + 27c + 9ab}{\lambda^3}\). | \frac{3\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
Beatriz loves odd numbers. How many numbers between 0 and 1000 can she write using only odd digits? | 155 | 0.5 | 5,960.625 | 4,914.625 | 7,006.625 | |
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expec... | 16/3 | 0.0625 | 7,616.25 | 6,530 | 7,688.666667 | |
The price of an article was increased $p\%$. Later the new price was decreased $p\%$. If the last price was one dollar, the original price was: | \frac{10000}{10000-p^2} | 1. **Identify the variables and setup the equation:**
Let $x$ be the original price of the article. The price is first increased by $p\%$, and then the new price is decreased by $p\%$. We need to find the value of $x$ such that after these changes, the final price is one dollar.
2. **Calculate the price after the i... | 1 | 3,449.9375 | 3,449.9375 | -1 |
At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probabil... | 39 | 0.6875 | 6,523.3125 | 5,997.363636 | 7,680.4 | |
Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. | -10 | 1 | 4,528.625 | 4,528.625 | -1 | |
Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ Find $\lfloor P\rfloor.$ | 45 | If we don't see the fourth power, we can always factor the LHS to try to create a quadratic substitution. Checking, we find that $x=0$ and $x=2$ are both roots. Synthetic division gives $(x^2-2x)(x^2-2x+2)=2005$. We now have our quadratic substitution of $y=x^2-2x+1=(x-1)^2$, giving us $(y-1)(y+1)=2005$. From here we p... | 0.5625 | 7,228.875 | 6,479.777778 | 8,192 |
Let $\mathbf{D}$ be the $2 \times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7. Find $\det \mathbf{D}.$ | 49 | 1 | 1,221.9375 | 1,221.9375 | -1 | |
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his money is gone after cros... | 35 | Let's denote the amount of money Fox has at the beginning as $x$ coins. We will analyze the changes in the amount of money Fox has after each crossing and payment.
1. **First Crossing:**
- Before crossing: Fox has $x$ coins.
- After doubling: $2x$ coins.
- After paying toll: $2x - 40$ coins.
2. **Second Cros... | 1 | 2,400.25 | 2,400.25 | -1 |
In the Cartesian coordinate plane, a polar coordinate system is established with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that point A has polar coordinates $$( \sqrt{2}, \frac{\pi}{4})$$, and the parametric equation of line $l$ is:
$$\begin{cases} x= \frac{3}{2} - \... | \frac{4\sqrt{2}}{5} | 0 | 6,133.125 | -1 | 6,133.125 | |
Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \mathrm{~km} / \mathrm{h}$.
What is the average speed at which Vendelín ... | 15 | 0.0625 | 7,896.9375 | 5,972 | 8,025.266667 | |
A hollow silver sphere with an outer diameter of $2 R = 1 \mathrm{dm}$ is exactly half-submerged in water. What is the thickness of the sphere's wall if the specific gravity of silver is $s = 10.5$? | 0.008 | 0 | 7,989.5 | -1 | 7,989.5 | |
We color certain squares of an $8 \times 8$ chessboard red. How many squares can we color at most if we want no red trimino? How many squares can we color at least if we want every trimino to have at least one red square? | 32 | 0 | 8,192 | -1 | 8,192 | |
To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed? | 1092 | Note that 2 and 9 are equivalent $\bmod 7$. So we will replace the 9 with a 2 for now. Since 7 is a divisor of 21, a four digit multiple of 7 consisting of $2,0,1$, and 2 cannot have a 2 followed by a 1 (otherwise we could subtract a multiple of 21 to obtain a number of the form $2 \cdot 10^{k}$). Thus our number eithe... | 0.0625 | 8,127.5 | 7,160 | 8,192 |
Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? | 112 | 0.5 | 6,648 | 5,104 | 8,192 | |
Given that $a_{1}, a_{2}, \cdots, a_{10}$ are ten different positive integers satisfying the equation $\left|a_{i+1}-a_{i}\right|=2 \text { or } 3$, where $i=1,2, \cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximum number among $a_{1}, a_{2}, \cdots, a_{10}$ a... | 14 | 0 | 8,192 | -1 | 8,192 | |
A factory produces two types of products, A and B, with profits P and Q (in ten thousand yuan), respectively. The relationship between the profits and the invested capital m (in ten thousand yuan) follows the empirical formulas P = (1/3)m + 65 and Q = 76 + 4√m. Now, 150 ten thousand yuan of capital will be invested in ... | 203 | 0.8125 | 3,934 | 3,463.076923 | 5,974.666667 | |
Let $S$ be the set of all 3-digit numbers with all digits in the set $\{1,2,3,4,5,6,7\}$ (so in particular, all three digits are nonzero). For how many elements $\overline{a b c}$ of $S$ is it true that at least one of the (not necessarily distinct) 'digit cycles' $\overline{a b c}, \overline{b c a}, \overline{c a b}$ ... | 127 | Since the value of each digit is restricted to $\{1,2, \ldots, 7\}$, there is exactly one digit representative of each residue class modulo 7. Note that $7 \mid \overline{a b c}$ if and only if $100 a+10 b+c \equiv 0(\bmod 7)$ or equivalently $2 a+3 b+c \equiv 0$. So we want the number of triples of residues $(a, b, c)... | 0.25 | 7,623.25 | 6,768 | 7,908.333333 |
Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2) = ( \ )$. | -9 | 1 | 2,076.1875 | 2,076.1875 | -1 | |
Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$ ? | 37 | 0.4375 | 6,620.4375 | 6,327.714286 | 6,848.111111 | |
Let $a>0$ and $b>0,$ and define two operations:
$$a \nabla b = \dfrac{a + b}{1 + ab}$$
$$a \Delta b = \dfrac{a - b}{1 - ab}$$
Calculate $3 \nabla 4$ and $3 \Delta 4$. | \frac{1}{11} | 1 | 1,996.4375 | 1,996.4375 | -1 | |
Given that the magnitude of vector $\overrightarrow {a}$ is 1, the magnitude of vector $\overrightarrow {b}$ is 2, and the magnitude of $\overrightarrow {a}+ \overrightarrow {b}$ is $\sqrt {7}$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac {\pi}{3} | 0.0625 | 3,949.875 | 2,649 | 4,036.6 | |
By definition, a polygon is regular if all its angles and sides are equal. Points \( A, B, C, D \) are consecutive vertices of a regular polygon (in that order). It is known that the angle \( ABD = 135^\circ \). How many vertices does this polygon have? | 12 | 0.125 | 7,448.5 | 5,473.5 | 7,730.642857 | |
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play? | 14 | 0 | 8,192 | -1 | 8,192 | |
Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB=11$, $BC=5$, $CD=19$, and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$? | $30\sqrt{3}$ | 1. **Identify the properties of points $P$ and $Q$:**
- $P$ is the intersection of the angle bisectors of $\angle A$ and $\angle D$. By the Angle Bisector Theorem, $P$ is equidistant from the sides $\overline{AB}$, $\overline{AD}$, and $\overline{CD}$.
- Similarly, $Q$ is the intersection of the angle bisectors o... | 0 | 8,130.4375 | -1 | 8,130.4375 |
Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once. | \frac{149}{12} | Suppose Mark has already rolled $n$ unique numbers, where $1 \leq n \leq 5$. On the next roll, there are 5 possible numbers he could get, with $6-n$ of them being new. Therefore, the probability of getting another unique number is $\frac{6-n}{5}$, so the expected number of rolls before getting another unique number is ... | 0 | 7,921.3125 | -1 | 7,921.3125 |
Compute \( 105 \times 95 \). | 9975 | 1 | 691.125 | 691.125 | -1 | |
Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \). | 1 - \ln 2 | 0.125 | 8,049.5625 | 7,052.5 | 8,192 | |
Given the function $f(x)=2\ln x - ax^2 + 3$,
(1) Discuss the monotonicity of the function $y=f(x)$;
(2) If there exist real numbers $m, n \in [1, 5]$ such that $f(m)=f(n)$ holds when $n-m \geq 2$, find the maximum value of the real number $a$. | \frac{\ln 3}{4} | 0.125 | 8,172.25 | 8,034 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$, and $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}| (k > 0)$. Find the maximum value of the angle between vectors $\overrightarrow{a}$ and $\overr... | \frac{\pi}{3} | 0.6875 | 5,730.8125 | 4,736.727273 | 7,917.8 | |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 0.125 | 7,841.5 | 5,800 | 8,133.142857 | |
The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$? | 28 | 1 | 3,036.4375 | 3,036.4375 | -1 | |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, ... | 20\sqrt{5} | 0 | 8,192 | -1 | 8,192 | |
We plotted the graph of the function \( f(x) = \frac{1}{x} \) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \( g(x) = \frac{2}{x} \)? | \frac{\sqrt{2}}{2} | 0 | 7,038.75 | -1 | 7,038.75 | |
How many numbers are in the list $250, 243, 236, \ldots, 29, 22?$ | 34 | 0 | 8,138.0625 | -1 | 8,138.0625 | |
In a $10 \times 5$ grid, an ant starts from point $A$ and can only move right or up along the grid lines but is not allowed to pass through point $C$. How many different paths are there from point $A$ to point $B$? | 1827 | 0.75 | 5,817.125 | 5,516.416667 | 6,719.25 | |
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 929 | We consider how many ways we can have 2*2 grid
$(1)$: All the girds are red--$1$ case
$(2)$: One unit square is blue--The blue lies on the center of the bigger square, makes no 2*2 grid $9-1=8$ cases
$(3)$: Two unit squares are blue--one of the squares lies in the center of the bigger square, makes no 2*2 grid, $8$ cas... | 0.0625 | 7,150 | 5,663 | 7,249.133333 |
What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome? | 13 | 1 | 4,653.125 | 4,653.125 | -1 | |
On a blackboard, the number 123456789 is written. Select two adjacent digits from this number, and if neither of them is 0, subtract 1 from each and swap their positions. For example: \( 123456789 \rightarrow 123436789 \rightarrow \cdots \). After performing this operation several times, what is the smallest possible n... | 101010101 | 0 | 8,192 | -1 | 8,192 | |
Calculate: $(128)^{\frac{7}{3}}$ | 65536 \cdot \sqrt[3]{2} | 0 | 7,190.4375 | -1 | 7,190.4375 | |
In how many ways can the set of ordered pairs of integers be colored red and blue such that for all $a$ and $b$, the points $(a, b),(-1-b, a+1)$, and $(1-b, a-1)$ are all the same color? | 16 | Let $\varphi_{1}$ and $\varphi_{2}$ be $90^{\circ}$ counterclockwise rotations about $(-1,0)$ and $(1,0)$, respectively. Then $\varphi_{1}(a, b)=(-1-b, a+1)$, and $\varphi_{2}(a, b)=(1-b, a-1)$. Therefore, the possible colorings are precisely those preserved under these rotations. Since $\varphi_{1}(1,0)=(-1,2)$, the c... | 0 | 8,192 | -1 | 8,192 |
The equations
\[60x^4 + ax^3 + bx^2 + cx + 20 = 0\]and
\[20x^5 + dx^4 + ex^3 + fx^2 + gx + 60 = 0\]have a common rational root $r$ which is not an integer, and which is positive. What is $r?$ | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120... | 672 | Suppose we label the points as shown here. By radical axis, the tangents to $\omega$ at $D$ and $E$ intersect on $AB$. Thus $PDQE$ is harmonic, so the tangents to $\omega$ at $P$ and $Q$ intersect at $X \in DE$. Moreover, $OX \parallel O_1O_2$ because both $OX$ and $O_1O_2$ are perpendicular to $AB$, and $OX = 2OP$ bec... | 0 | 8,192 | -1 | 8,192 |
Peter has three times as many sisters as brothers. His sister Louise has twice as many sisters as brothers. How many children are there in the family? | 13 | 0.1875 | 5,769.5625 | 2,969.333333 | 6,415.769231 | |
Subtract $123.45$ from $567.89.$ Express the result as a decimal to the nearest hundredth. | 444.44 | 1 | 2,483.9375 | 2,483.9375 | -1 | |
If $x \geq 0$, then $\sqrt{x\sqrt{x\sqrt{x}}} =$ | $\sqrt[8]{x^7}$ | 1. **Understanding the expression**: We start with the expression $\sqrt{x\sqrt{x\sqrt{x}}}$. This involves nested square roots, which can be simplified using the property that $\sqrt{y} = y^{\frac{1}{2}}$ for all $y \geq 0$.
2. **Simplifying the innermost square root**: Begin by simplifying the innermost square root:... | 0 | 4,925.9375 | -1 | 4,925.9375 |
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 | To solve this problem, we need to find the common sum of the numbers in each row, column, and diagonal of a $5 \times 5$ square matrix using the integers from $-10$ to $14$ inclusive.
1. **Calculate the total sum of all integers from $-10$ to $14$:**
The sum of an arithmetic series can be calculated using the formu... | 1 | 2,910.1875 | 2,910.1875 | -1 |
The spinner shown is divided into 6 sections of equal size. Determine the probability of landing on a section that contains the letter Q using this spinner. | \frac{2}{6} | 0 | 301.625 | -1 | 301.625 | |
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$ . Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle a... | 30 | 0 | 8,143.25 | -1 | 8,143.25 | |
How many distinct solutions are there to the equation $|x-7| = |x+1|$? | 1 | 1 | 1,761.8125 | 1,761.8125 | -1 | |
The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group will discuss their choices with the other ... | \frac{13}{15} | 0.625 | 4,945.3125 | 3,836.9 | 6,792.666667 | |
Given the right focus $F$ and the right directrix $l$ of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with eccentricity $e = \frac{\sqrt{5}}{5}$. Draw $AM \perp l$ through the vertex $A(0, b)$, with $M$ as the foot of the perpendicular. Find the slope of the line $FM$. | \frac{1}{2} | 1 | 3,189.9375 | 3,189.9375 | -1 | |
A cauldron has the shape of a paraboloid of revolution. The radius of its base is \( R = 3 \) meters, and the depth is \( H = 5 \) meters. The cauldron is filled with a liquid, the specific weight of which is \( 0.8 \Gamma / \text{cm}^3 \). Calculate the work required to pump the liquid out of the cauldron. | 294300\pi | 0 | 7,862.9375 | -1 | 7,862.9375 |
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