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 |
|---|---|---|---|---|---|---|
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$? | 0 | 1. **Identify the Geometry of the Pentagon**: Given that $ABCDE$ is an equiangular convex pentagon, each interior angle is $\frac{180^\circ \times (5-2)}{5} = 108^\circ$.
2. **Extend the Sides to Form a Star**: Extending the sides of the pentagon forms a five-pointed star. The vertices of this star are the intersectio... | 0 | 8,192 | -1 | 8,192 |
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the... | 8 | 1. **Calculate the time to reach the shore**:
Steve rows at a speed of 4 miles per hour. The distance from the shore is 1 mile. The time \( t \) required to reach the shore is given by the formula:
\[
t = \frac{\text{distance}}{\text{speed}} = \frac{1 \text{ mile}}{4 \text{ miles/hour}} = 0.25 \text{ hours}
... | 1 | 1,448.4375 | 1,448.4375 | -1 |
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are count... | 112 | To solve for $D(96)$, we need to consider all possible ways to write $96$ as a product of integers greater than $1$, where the order of factors matters. We start by noting the prime factorization of $96$:
\[ 96 = 2^5 \cdot 3 \]
We will consider different cases based on the number of terms in the product.
#### Case 1:... | 0 | 8,190.125 | -1 | 8,190.125 |
Find the distance between the points $(2,1,-4)$ and $(5,8,-3).$ | \sqrt{59} | 1 | 1,435.9375 | 1,435.9375 | -1 | |
Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. Then $r$ is: | 10 | 1. **Convert the circumference to miles**: The circumference of the wheel is given as $11$ feet. Since there are $5280$ feet in a mile, the circumference in miles is:
\[
\frac{11}{5280} \text{ miles}
\]
2. **Relate speed, time, and distance**: The speed $r$ in miles per hour and the time $t$ in hours for one ... | 0.8125 | 4,146.3125 | 3,212.692308 | 8,192 |
In the function machine shown, the input is 10. What is the output?
[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height... | 15 | 0.1875 | 4,695.5625 | 1,999 | 5,317.846154 | |
Given the parabola $y^{2}=4x$ and the line $l: x+2y-2b=0$ intersecting the parabola at points $A$ and $B$.
(Ⅰ) If the circle with diameter $AB$ is tangent to the $x$-axis, find the equation of the circle;
(Ⅱ) If the line $l$ intersects the negative half of the $y$-axis, find the maximum area of $\triangle AOB$ ($O$ i... | \frac{32 \sqrt{3}}{9} | 0 | 6,792.1875 | -1 | 6,792.1875 | |
Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$ . On the side $AB$ construct the rhombus $BAFC$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$ . If the area of $BAFE$ is equal to $65$ , calculate the area of $ABCD$ . | 120 | 0 | 8,192 | -1 | 8,192 | |
An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and... | 59 | 0.3125 | 5,607.625 | 5,199 | 5,793.363636 | |
Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse. | (3, \sqrt{39}) \cup (\sqrt{89}, 13) |
To determine the real values of \( x \) such that the triangle with sides 5, 8, and \( x \) is obtuse, we need to apply the properties of an obtuse triangle. In an obtuse triangle, the square of the longest side is greater than the sum of the squares of the other two sides.
Assuming \( x \) is the longest side, the c... | 0.6875 | 4,348.5 | 4,045 | 5,016.2 |
A 10-cm-by-10-cm square is partitioned such that points $A$ and $B$ are on two opposite sides of the square at one-third and two-thirds the length of the sides, respectively. What is the area of the new shaded region formed by connecting points $A$, $B$, and their reflections across the square's diagonal?
[asy]
draw((... | 50 | 0 | 8,186.375 | -1 | 8,186.375 | |
Four different natural numbers, of which one is 1, have the following properties: the sum of any two of them is a multiple of 2, the sum of any three of them is a multiple of 3, and the sum of all four numbers is a multiple of 4. What is the minimum possible sum of these four numbers? | 40 | 0.1875 | 7,487.0625 | 6,935.666667 | 7,614.307692 | |
In the expansion of $(C_4+C_4^1x+C_4^2x^2+C_4^3x^3)^2$, calculate the sum of the coefficients of all terms. | 225 | 0.4375 | 5,882.875 | 4,404.571429 | 7,032.666667 | |
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$. | 1203 | 0 | 8,192 | -1 | 8,192 | |
Let $x,$ $y,$ $z$ be positive real numbers. Find the set of all possible values of
\[f(x,y,z) = \frac{x}{x + y} + \frac{y}{y + z} + \frac{z}{z + x}.\] | (1,2) | 0.125 | 8,192 | 8,192 | 8,192 | |
In which numeral system is 792 divisible by 297? | 19 | 0.4375 | 6,768.75 | 5,539.714286 | 7,724.666667 | |
Winnie has 17 red balloons, 33 white balloons, 65 green balloons, and 83 chartreuse balloons. With no regard to color, Winnie gives each of her 8 nearest and dearest friends some balloons. Each friend gets the same number of balloons and she hands out as many balloons as possible. How many balloons does Winnie end u... | 6 | 0.9375 | 2,104.25 | 1,698.4 | 8,192 | |
Consider positive integers $a \leq b \leq c \leq d \leq e$. There are $N$ lists $a, b, c, d, e$ with a mean of 2023 and a median of 2023, in which the integer 2023 appears more than once, and in which no other integer appears more than once. What is the sum of the digits of $N$? | 28 | Since the median of the list $a, b, c, d, e$ is 2023 and $a \leq b \leq c \leq d \leq e$, then $c=2023$. Since 2023 appears more than once in the list, then it appears 5,4,3, or 2 times. Case 1: 2023 appears 5 times Here, the list is 2023, 2023, 2023, 2023, 2023. There is 1 such list. Case 2: 2023 appears 4 times Here,... | 0 | 8,192 | -1 | 8,192 |
How many parallelograms with sides 1 and 2, and angles \(60^{\circ}\) and \(120^{\circ}\), can be placed inside a regular hexagon with side length 3? | 12 | 0.125 | 7,270.125 | 7,637.5 | 7,217.642857 | |
Someone collected data relating the average temperature x (℃) during the Spring Festival to the sales y (ten thousand yuan) of a certain heating product. The data pairs (x, y) are as follows: (-2, 20), (-3, 23), (-5, 27), (-6, 30). Based on the data, using linear regression, the linear regression equation between sales... | 34.4 | 0 | 4,575.8125 | -1 | 4,575.8125 | |
In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{2}}{2} | 0 | 7,380.125 | -1 | 7,380.125 | |
The positive number $a$ is chosen such that the terms $20, a, \frac{5}{4}$ are the first, second and third terms, respectively, of a geometric sequence. What is the value of $a$, if $a$ is positive? | 5 | 1 | 1,458.4375 | 1,458.4375 | -1 | |
Determine the number of ways to place 7 identical balls into 4 distinct boxes such that each box contains at least one ball. | 20 | 0.75 | 4,707.3125 | 3,545.75 | 8,192 | |
George is trying to find the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,2)$, and $C$ is at $(5,4)$. He guesses that the point is at $P = (3,1)$. Compute the sum of the distances from $P$ to the vertices of $\triangle ABC$. If he obtains $x + y\sqrt{z}$, where $x$, $y$, and $z$ are i... | 16 | 0 | 7,512.9375 | -1 | 7,512.9375 | |
In a sequence of positive integers each term after the first is $\frac{1}{3}$ of the sum of the term that precedes it and the term that follows it in the sequence. What is the 5th term of this sequence if the 1st term is 2 and the 4th term is 34? | 89 | 0.9375 | 3,468.8125 | 3,153.933333 | 8,192 | |
Find the smallest number in which all digits are different and the sum of all digits equals 32. | 26789 | 0.3125 | 7,496.6875 | 6,979.4 | 7,731.818182 | |
A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$? | 15 | 1 | 1,574.0625 | 1,574.0625 | -1 | |
Let $z$ be a complex number with $|z| = 2.$ Find the maximum value of
\[ |(z-2)^3(z+2)|. \] | 24\sqrt{3} | 0 | 7,766.375 | -1 | 7,766.375 | |
Nikita schematically drew the graph of the quadratic polynomial \( y = ax^{2} + bx + c \). It turned out that \( AB = CD = 1 \). Consider the four numbers \(-a, b, c\), and the discriminant of the polynomial. It is known that three of these numbers are equal to \( \frac{1}{4}, -1, -\frac{3}{2} \) in some order. Find th... | -1/2 | 0 | 8,192 | -1 | 8,192 | |
In the triangle \( \triangle ABC \), \( \angle C = 90^{\circ} \), and \( CB > CA \). Point \( D \) is on \( BC \) such that \( \angle CAD = 2 \angle DAB \). If \( \frac{AC}{AD} = \frac{2}{3} \) and \( \frac{CD}{BD} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, then what is \( m + n \)?
(49t... | 14 | 0.375 | 7,443.6875 | 6,196.5 | 8,192 | |
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down... | 35 | 0.8125 | 4,706 | 4,392.076923 | 6,066.333333 | |
Use the bisection method to find an approximate solution for $f(x)=0$. Given that $f(1)=-2$, $f(3)=0.625$, and $f(2)=-0.984$, calculate the next $f(m)$, where $m =$ __________. | 2.5 | 0.25 | 5,269.9375 | 6,914.5 | 4,721.75 | |
Given that the discrete random variable \\(\xi\\) follows a normal distribution \\(N \sim (2,1)\\), and \\(P(\xi < 3) = 0.968\\), then \\(P(1 < \xi < 3) =\\) \_\_\_\_\_\_. | 0.936 | 0.0625 | 8,094.25 | 7,884 | 8,108.266667 | |
Find the largest prime divisor of 11! + 12! | 13 | 0.9375 | 2,324.6875 | 1,933.533333 | 8,192 | |
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $1... | 25 | 1. **Understanding the Problem**: Bernardo writes the base-5 and base-6 representations of a three-digit number $N$ on a blackboard. LeRoy, treating these numbers as base-10 integers, adds them to get $S$. We need to find how many such $N$ exist such that the two rightmost digits of $S$ are the same as those of $2N$.
... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x)= \sqrt{2}\sin \left( 2x- \frac{\pi}{4} \right)$, where $x\in\mathbb{R}$, if the maximum and minimum values of $f(x)$ in the interval $\left[ \frac{\pi}{8}, \frac{3\pi}{4} \right]$ are $a$ and $b$ respectively, then the value of $a+b$ is ______. | \sqrt{2}-1 | 1 | 4,386 | 4,386 | -1 | |
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$ .
*Proposed ... | 2017 | 0 | 7,772.0625 | -1 | 7,772.0625 | |
A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. How many students did not go skating and did not go skiing? | 6 | Since 20 students went skating and 5 students went both skating and skiing, then \( 20-5=15 \) students went skating only. Since 9 students went skiing and 5 students went both skating and skiing, then \( 9-5=4 \) students went skiing only. The number of students who went skating or skiing or both equals the sum of the... | 1 | 466.625 | 466.625 | -1 |
Find all odd natural numbers greater than 500 but less than 1000, each of which has the sum of the last digits of all its divisors (including 1 and the number itself) equal to 33. | 729 | 0.625 | 6,683.5 | 5,778.4 | 8,192 | |
Let \(\omega\) denote the incircle of triangle \(ABC\). The segments \(BC, CA\), and \(AB\) are tangent to \(\omega\) at \(D, E\), and \(F\), respectively. Point \(P\) lies on \(EF\) such that segment \(PD\) is perpendicular to \(BC\). The line \(AP\) intersects \(BC\) at \(Q\). The circles \(\omega_1\) and \(\omega_2\... | 196 | 0 | 8,192 | -1 | 8,192 | |
In a triangle $XYZ$, $\angle XYZ = \angle YXZ$. If $XZ=8$ and $YZ=11$, what is the perimeter of $\triangle XYZ$? | 30 | 0 | 7,508 | -1 | 7,508 | |
Let \( Q \) be a point chosen uniformly at random in the interior of the unit square with vertices at \( (0,0), (1,0), (1,1) \), and \( (0,1) \). Find the probability that the slope of the line determined by \( Q \) and the point \( \left(\frac{3}{4}, \frac{1}{4} \right) \) is less than or equal to \( -1 \). | \frac{1}{2} | 0.0625 | 6,503.25 | 3,912 | 6,676 | |
The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits | 99972 | 0.0625 | 7,867.75 | 7,666 | 7,881.2 | |
Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\] | \frac{1}{3} | 0 | 8,192 | -1 | 8,192 | |
Find the smallest possible value of $x$ in the simplified form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{7x}{5}-2=\frac{4}{x}$, where $a, b, c,$ and $d$ are integers. What is $\frac{acd}{b}$? | -5775 | 0.9375 | 3,918.5625 | 3,950.533333 | 3,439 | |
Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained. | 330 | When $1$ is subtracted from a binary number, the number of digits will remain constant if and only if the original number ended in $10.$ Therefore, every subtraction involving two numbers from $B$ will necessarily involve exactly one number ending in $10.$ To solve the problem, then, we can simply count the instances o... | 0.125 | 7,773 | 6,062 | 8,017.428571 |
In the Cartesian coordinate system $(xOy)$, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y= \sqrt{3}+t\sin a\end{cases} (t\text{ is a parameter})$, where $0\leqslant α < π$. In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the c... | \frac{6 \sqrt{2}}{5} | 0 | 7,746.9375 | -1 | 7,746.9375 | |
Katya is passing time while her parents are at work. On a piece of paper, she absentmindedly drew Cheburashkas in two rows (at least one Cheburashka was drawn in each row).
Afterwards, she drew a Crocodile Gena between every two adjacent Cheburashkas in both rows. Then she drew an Old Lady Shapoklyak to the left of ea... | 11 | 0.1875 | 6,775.1875 | 4,796.666667 | 7,231.769231 | |
A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the wa... | 52 | Clearly we have people moving at speeds of $6,8$ and $10$ feet/second. Notice that out of the three people, Cy is at the largest disadvantage to begin with and since all speeds are close, it is hardest for him to catch up. Furthermore, Bob is clearly the farthest along. Thus it is reasonable to assume that there is som... | 0.4375 | 6,064.3125 | 4,476.714286 | 7,299.111111 |
Zhenya took a $3 \times 3$ board and placed a column of blue and red cubes on each cell. Then he drew a diagram of the resulting arrangement: he wrote down the number of cubes of both colors in each column (the order of the cubes is unknown).
What is the maximum number of blue cubes Zhenya can see if he looks at the c... | 12 | 0 | 8,190.6875 | -1 | 8,190.6875 | |
The polynomial $P(x) = 2x^3 + ax^2 + bx + c$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = P(x)$ is 8. What is $b$? | -38 | 0.8125 | 2,010.5 | 2,030.615385 | 1,923.333333 | |
In a certain high school physical examination for seniors, the heights (in centimeters) of 12 students are $173$, $174$, $166$, $172$, $170$, $165$, $165$, $168$, $164$, $173$, $175$, $178$. Find the upper quartile of this data set. | 173.5 | 0.5625 | 6,333.1875 | 5,196.111111 | 7,795.142857 | |
The same amount of steel used to create eight solid steel balls, each with a radius of 1 inch, is used to create one larger steel ball. What is the radius of the larger ball?
[asy]
size(150);
filldraw(circle((0,0),1),gray);
filldraw(circle((.9,-.8),1),gray);
filldraw(circle((1.8,.9),1),gray);
filldraw(circle((2,0),1),... | 2 | 1 | 1,334.5 | 1,334.5 | -1 | |
Find the remainder when $123456789012$ is divided by $240$. | 132 | 0.5 | 6,323 | 5,384.375 | 7,261.625 | |
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,$ $19,$ $20,$ $25$ and $31,$ in some order. Find the area of the pentagon. | 745 | 0.5625 | 6,643.75 | 5,561.444444 | 8,035.285714 | |
Find the distance between the planes $x - 3y + 3z = 8$ and $2x - 6y + 6z = 2.$ | \frac{7 \sqrt{19}}{19} | 0 | 4,972 | -1 | 4,972 | |
A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\$100$.
Let $x$ and $y$ be the s... | 50 | Let the listed price be $P$, where $P > 100$. We can express $P$ as $P = 100 + p$ where $p > 0$.
1. **Calculate the savings from each coupon:**
- **Coupon A:** This coupon gives $15\%$ off the listed price. Therefore, the savings from Coupon A are:
\[
0.15P = 0.15(100 + p) = 15 + 0.15p
\]
- **Coup... | 0.875 | 876.25 | 904.071429 | 681.5 |
In triangle $ABC$, we have $AB=1$ and $AC=2$. Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$? Express your answer in simplest radical form. | \sqrt{2} | 0.9375 | 3,627.4375 | 3,323.133333 | 8,192 | |
Four boys, four girls, and a coach are positioned on a circular track. Each girl is diametrically opposite to one of the boys. The length of the track is 50 meters. On the coach's signal, they all run towards the coach by the shortest path along the track. What is the total distance run by all the children together? | 100 | 0.0625 | 7,072.3125 | 6,308 | 7,123.266667 | |
A circle passes through the vertices of a triangle with side-lengths $7\tfrac{1}{2},10,12\tfrac{1}{2}.$ The radius of the circle is: | \frac{25}{4} | 1. **Identify the Type of Triangle**:
Given the side lengths of the triangle are $7\tfrac{1}{2}$, $10$, and $12\tfrac{1}{2}$. We first convert these mixed fractions to improper fractions for easier calculation:
\[
7\tfrac{1}{2} = \frac{15}{2}, \quad 12\tfrac{1}{2} = \frac{25}{2}
\]
We need to check if ... | 0.375 | 2,644 | 3,414.833333 | 2,181.5 |
Attach a single digit to the left and right of the eight-digit number 20222023 so that the resulting 10-digit number is divisible by 72. (Specify all possible solutions.) | 3202220232 | 0.625 | 6,334.3125 | 5,994 | 6,901.5 | |
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? | 8 | 1. **Define the variables**: Let the two whole numbers be $x$ and $y$.
2. **Set up the equations based on the problem statement**:
- The product of the numbers is 24: \( xy = 24 \)
- The sum of the numbers is 11: \( x + y = 11 \)
3. **Express one variable in terms of the other**:
- From the sum equation, sol... | 1 | 1,850.75 | 1,850.75 | -1 |
Compute the sum of all positive integers $n$ for which $9 \sqrt{n}+4 \sqrt{n+2}-3 \sqrt{n+16}$ is an integer. | 18 | For the expression to be an integer at least one of $n$ and $n+2$ must be a perfect square. We also note that at most one of $n$ and $n+2$ can be a square, so exactly one of them is a square. Case 1: $n$ is a perfect square. By our previous observation, it must be that $4 \sqrt{n+2}=3 \sqrt{n+16} \Rightarrow n=16$. Cas... | 0 | 8,192 | -1 | 8,192 |
If two lines $l$ and $m$ have equations $y = -x + 6$, and $y = -4x + 6$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth.
[asy]
import cse5; import olympiad;
size(150);
add(grid(8,8));
draw... | 0.75 | 0.4375 | 7,008.0625 | 5,876.714286 | 7,888 | |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer.
| 600 | 0.6875 | 6,508.8125 | 6,087.636364 | 7,435.4 | |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the equations:
\[
\left\{
\begin{array}{l}
a_{1} b_{1} + a_{2} b_{3} = 1 \\
a_{1} b_{2} + a_{2} b_{4} = 0 \\
a_{3} b_{1} + a_{4} b_{3} = 0 \\
a_{3} b_{2} + a_{4} b_{4} = 1
\end{array}
\right.
\]
Given that \( a_{2} b_{3} = 7 \... | -6 | 0.375 | 6,741.9375 | 4,793.166667 | 7,911.2 | |
What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$? | 16 | 1 | 1,756.4375 | 1,756.4375 | -1 | |
Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 3000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$? | 750 | 0 | 8,192 | -1 | 8,192 | |
Given that $x+y=12$, $xy=9$, and $x < y$, find the value of $\frac {x^{ \frac {1}{2}}-y^{ \frac {1}{2}}}{x^{ \frac {1}{2}}+y^{ \frac {1}{2}}}=$ ___. | - \frac { \sqrt {3}}{3} | 0 | 4,066.6875 | -1 | 4,066.6875 | |
A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$ . We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$ . Determine the grea... | 6050 | 0 | 8,192 | -1 | 8,192 | |
The base $9$ representation of a positive integer is $AB$ and its base $7$ representation is $BA$. What is the integer expressed in base $10$? | 31 | 1 | 2,063.6875 | 2,063.6875 | -1 | |
What is the value of $a$ for which $\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1$? | 24 | 1 | 2,478.5625 | 2,478.5625 | -1 | |
A function, defined on the set of positive integers, is such that \( f(x y) = f(x) + f(y) \) for all \( x \) and \( y \). It is known that \( f(10) = 14 \) and \( f(40) = 20 \).
What is the value of \( f(500) \)? | 39 | 0.9375 | 3,017.4375 | 2,672.466667 | 8,192 | |
If $a @ b = \frac{a \times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$? | \frac{10}{3} | 1. **Define the operation**: Given the operation $a @ b = \frac{a \times b}{a + b}$, where $a$ and $b$ are positive integers.
2. **Substitute the values**: Substitute $a = 5$ and $b = 10$ into the operation:
\[
5 @ 10 = \frac{5 \times 10}{5 + 10}
\]
3. **Perform the calculations**:
- Calculate the numerat... | 1 | 2,374.375 | 2,374.375 | -1 |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 800$ and $AD = 1600$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$. | 400 | 0.875 | 5,006.3125 | 4,551.214286 | 8,192 | |
Calculate the sum of the geometric series $1+\left(\frac{1}{5}\right)+\left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^3 + \dots$. Express your answer as a common fraction. | \frac{5}{4} | 1 | 1,796.1875 | 1,796.1875 | -1 | |
Compute $\arccos (\cos 3).$ All functions are in radians. | 3 - 2\pi | 0 | 7,120.375 | -1 | 7,120.375 | |
Given the sequence $\{a\_n\}$, where $a\_1=1$, $a\_2=3$, and $2nS\_n=(n+1)S\_{n+1}+(n-1)S\_{n-1}$ ($n\geqslant 2,n\in N$), find $S\_{30}$. | \frac{34}{5} | 0.4375 | 7,380.8125 | 6,337.857143 | 8,192 | |
Let the set \( P = \{1, 2, \ldots, 2014\} \) and \( A \subseteq P \). If the difference between any two numbers in the set \( A \) is not a multiple of 99, and the sum of any two numbers in the set \( A \) is also not a multiple of 99, then the set \( A \) can contain at most how many elements? | 50 | 0 | 8,181.0625 | -1 | 8,181.0625 | |
What is the largest prime factor of 999? | 37 | 1 | 1,776.875 | 1,776.875 | -1 | |
The triangle $\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \, \text{cm}$, $PB = 4 \, \text{cm}$, and $PC = 5 \, \text{cm}$. Calculate the length of the sides of the triangle $\triangle ABC$. | \sqrt{25 + 12 \sqrt{3}} | 0 | 7,492.9375 | -1 | 7,492.9375 | |
A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$, respectively. Which of the following describes the graph of $y=(f+g)(x)$?
(A) a parabola tangent to the $x... | \text{(D)} | 0 | 3,272.75 | -1 | 3,272.75 | |
For real numbers $x$, let \[f(x) = \left\{
\begin{array}{cl}
x+2 &\text{ if }x>3, \\
2x+a &\text{ if }x\le 3.
\end{array}
\right.\]What must the value of $a$ be to make the piecewise function continuous (which means that its graph can be drawn without lifting your pencil from the paper)? | -1 | 1 | 1,103.3125 | 1,103.3125 | -1 | |
Find the number of real solutions to
\[(x^{2006} + 1)(x^{2004} + x^{2002} + x^{2000} + \dots + x^2 + 1) = 2006x^{2005}.\] | 1 | 0.0625 | 8,121.75 | 7,068 | 8,192 | |
Tom's favorite number is between $100$ and $150$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Tom's favorite number? | 143 | 0.1875 | 4,526.1875 | 6,179.333333 | 4,144.692308 | |
Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$. | \frac{\sqrt{3}}{4} | 0 | 8,192 | -1 | 8,192 | |
Let $f(x)$ be an odd function on $R$, $f(x+2)=-f(x)$. When $0\leqslant x\leqslant 1$, $f(x)=x$. Find $f(5.5)$. | 0.5 | 0.5 | 6,593.3125 | 5,418 | 7,768.625 | |
Find the square root of $\dfrac{8!}{70}$. (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$.) | 24 | 1 | 1,985.1875 | 1,985.1875 | -1 | |
Compute $\sqrt{2^2\cdot 3^4}$. | 18 | 1 | 1,852.8125 | 1,852.8125 | -1 | |
Isabella writes the expression $\sqrt{d}$ for each positive integer $d$ not exceeding 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Vidur simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime numbe... | 534810086 | Let $\sqrt{n}$ simplifies to $a_{n} \sqrt{b_{n}}$, and replace 8 ! by $x$. First, notice that $\sum_{n \leq x} a_{n}$ is small $\left(O\left(x^{3 / 2}\right)\right.$ in particular) because each term cannot exceed $\sqrt{x}$. On the other hand, $\sum_{n \leq x} b_{n}$ will be large; we have $b_{n}=n$ when $n$ is squaref... | 0 | 7,357.0625 | -1 | 7,357.0625 |
If \( m \) and \( n \) are the roots of the quadratic equation \( x^2 + 1994x + 7 = 0 \), then the value of \((m^2 + 1993m + 6)(n^2 + 1995n + 8)\) is | 1986 | 1 | 2,553 | 2,553 | -1 | |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]
unitsize(2mm); defaultpen(... | \frac{1}{5} | 1. **Identify the dimensions of the triangles:**
The problem states that the yard contains two congruent isosceles right triangles and the parallel sides of the trapezoidal remainder are $15$ meters and $25$ meters in length. The difference in length between the parallel sides of the trapezoid is $25 - 15 = 10$ mete... | 0.9375 | 3,864.3125 | 3,575.8 | 8,192 |
Given that the first four terms of a geometric sequence $\{a\_n\}$ have a sum of $S\_4=5$, and $4a\_1,\;\; \frac {3}{2}a\_2\;,\;a\_2$ form an arithmetic sequence.
(I) Find the general term formula for $\{a\_n\}$;
(II) Let $\{b\_n\}$ be an arithmetic sequence with first term $2$ and common difference $-a\_1$. Its first ... | 13 | 0.6875 | 4,949.25 | 4,742.363636 | 5,404.4 | |
Given that the perimeter of a rectangular paper $ABCD$ is 10, and it is cut twice parallel to its length and width into 9 smaller rectangles of unequal sizes, what is the total perimeter of these 9 rectangles? | 30 | 0.375 | 5,569 | 3,890.166667 | 6,576.3 | |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1$ and $\|\mathbf{c}\| = 2.$ Find the maximum value of
\[\|\mathbf{a} - 2 \mathbf{b}\|^2 + \|\mathbf{b} - 2 \mathbf{c}\|^2 + \|\mathbf{c} - 2 \mathbf{a}\|^2.\] | 42 | 0.0625 | 8,009.9375 | 8,103 | 8,003.733333 | |
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? | \frac{1}{4} | 1. **Define the circles**: Consider a circle with radius $2$ (outer circle) and a concentric circle with radius $1$ (inner circle). The center of both circles is the same point.
2. **Determine the significance of the inner circle**: The inner circle of radius $1$ consists of all points that are exactly $1$ unit away f... | 1 | 3,430.4375 | 3,430.4375 | -1 |
In triangle \(ABC\), the angle bisector \(AD\) divides side \(BC\) in the ratio \(BD : DC = 2 : 1\). In what ratio does the median from vertex \(C\) divide this angle bisector? | 3:1 | 0.5625 | 5,477.25 | 5,764.444444 | 5,108 | |
A high school with 2000 students held a "May Fourth" running and mountain climbing competition in response to the call for "Sunshine Sports". Each student participated in only one of the competitions. The number of students from the first, second, and third grades participating in the running competition were \(a\), \(... | 36 | 0.125 | 4,038.8125 | 4,972.5 | 3,905.428571 | |
Given point $A(0,2)$, and $P$ is any point on the ellipse $\frac{x^2}{4}+y^2=1$, then the maximum value of $|PA|$ is ______. | \frac{2\sqrt{21}}{3} | 0 | 5,827.9375 | -1 | 5,827.9375 | |
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$ . Compute the area of the pentagon. | \[ 1 \] | Solution 1
Let $BC = a, ED = 1 - a$
Let $\angle DAC = X$
Applying cosine rule to $\triangle DAC$ we get:
$\cos X = \frac{AC ^ {2} + AD ^ {2} - DC ^ {2}}{ 2 \cdot AC \cdot AD }$
Substituting $AC^{2} = 1^{2} + a^{2}, AD ^ {2} = 1^{2} + (1-a)^{2}, DC = 1$ we get:
$\cos^{2} X = \frac{(1 - a - a ^ {2}) ^ {2}}{(1 + a^{2})... | 0 | 8,192 | -1 | 8,192 |
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