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 |
|---|---|---|---|---|---|---|
Yesterday, Sasha cooked soup and added too little salt, so he had to add more salt later. Today he doubled the amount of salt, but still had to add more salt, though only half as much as yesterday. By what factor does Sasha need to increase today's portion of salt so that tomorrow he does not have to add any more salt?... | 1.5 | 0 | 6,834.4375 | -1 | 6,834.4375 | |
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is | 16 | To solve this problem, we need to count all the triangles formed by the lines in the given figure. The figure consists of a square, its diagonals, and segments joining the midpoints of opposite sides. Let's analyze the figure and count the triangles of different sizes.
1. **Identify the vertices and lines:**
- The ... | 0.25 | 8,082.125 | 8,108 | 8,073.5 |
Find the largest real number $k$ , such that for any positive real numbers $a,b$ , $$ (a+b)(ab+1)(b+1)\geq kab^2 $$ | 27/4 | 0.0625 | 8,060.1875 | 6,083 | 8,192 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.4375 | 6,716.25 | 5,785.428571 | 7,440.222222 | |
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$ | $(7, 1), (1, 7), (22,22)$ | To find all ordered pairs of positive integers \((x, y)\) such that \(x^3 + y^3 = x^2 + 42xy + y^2\), we start by rewriting the given equation as follows:
\[
x^3 + y^3 - x^2 - y^2 - 42xy = 0
\]
We rearrange and factor the left-hand side:
\[
(x^3 - x^2) + (y^3 - y^2) = 42xy
\]
This equation can be simplified by fact... | 0 | 8,192 | -1 | 8,192 |
If $(x + y)^2 = 45$ and $xy = 10$, what is $(x - y)^2$? | 5 | 1 | 1,724.875 | 1,724.875 | -1 | |
Given $\{a_{n}\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\_\_\_\_\_\_$. | -2 | 1 | 3,508.625 | 3,508.625 | -1 | |
Consider the line $18x + 9y = 162$ forming a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.
A) 21.21
B) 42.43
C) 63.64
D) 84.85
E) 105.06 | 42.43 | 0 | 8,192 | -1 | 8,192 | |
The legs of a right triangle have lengths $\log_4 27$ and $\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$ | 243 | 1 | 3,041.0625 | 3,041.0625 | -1 | |
Petya was running down an escalator, counting the steps. Exactly halfway down, he tripped and tumbled the rest of the way (he tumbles 3 times faster than he runs). How many steps are on the escalator if Petya counted 20 steps with his feet (before falling) and 30 steps with his sides (after falling)? | 80 | 0.1875 | 6,157.0625 | 2,978 | 6,890.692308 | |
A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$ . For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $... | 15 | 0.0625 | 7,137.5 | 6,947 | 7,150.2 | |
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is | 13 | 1. **Define the problem setup**: Two boys start at the same point A on a circular track and move in opposite directions with speeds of 5 ft/s and 9 ft/s respectively. We need to find the number of times they meet excluding the start and finish.
2. **Calculate relative speed**: Since they are moving in opposite directi... | 0.375 | 6,669.375 | 5,208.333333 | 7,546 |
Given two lines $ l_1: x + my + 6 = 0 $ and $ l_2: (m-2)x + 3y + 2m = 0 $, if $ l_1 \parallel l_2 $, then the distance between $ l_1 $ and $ l_2 $ is __________. | \frac{8\sqrt{2}}{3} | 0 | 5,856.4375 | -1 | 5,856.4375 | |
Mr. Taf takes his 12 students on a road trip. Since it takes two hours to walk from the school to the destination, he plans to use his car to expedite the journey. His car can take at most 4 students at a time, and travels 15 times as fast as traveling on foot. If they plan their trip optimally, what is the shortest am... | 30.4 \text{ or } \frac{152}{5} | A way to plan the trip is to have Mr. Taf drive 4 students to the $80 \%$ mark, then drive back to the $10 \%$ mark to pick up another 4 students to the $90 \%$ mark, and finally drive back to the $20 \%$ mark to pick up the last 4 students to the destination. All students will reach the destination at the same time, a... | 0 | 7,742.125 | -1 | 7,742.125 |
For a positive integer $n$ , let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$ . | 18 | 0.5625 | 6,552.875 | 5,686.777778 | 7,666.428571 | |
The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$? | 78 | 1. **Calculate the area of both pizzas**:
- The area of a circle is given by the formula $A = \pi r^2$.
- For the pizza with radius $4$ inches, the area is:
\[
A_1 = \pi \times 4^2 = 16\pi \text{ square inches}
\]
- For the pizza with radius $3$ inches, the area is:
\[
A_2 = \pi \times... | 1 | 1,462.25 | 1,462.25 | -1 |
Define $\#N$ by the formula $\#N = .5(N) + 1$. Calculate $\#(\#(\#58))$. | 9 | 1 | 3,251.75 | 3,251.75 | -1 | |
In $\triangle ABC$, $\cos A = \frac{5}{13}$, find $\sin 2A = \_\_\_\_\_\_\_\_, \cos 2A = \_\_\_\_\_\_\_\_, \tan 2A = \_\_\_\_\_\_\_\_.$ | -\frac{120}{119} | 1 | 2,471.125 | 2,471.125 | -1 | |
Each term of a geometric sequence $\left\{a_{n}\right\}$ is a real number, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then find $S_{4n}$. | 150 | 0.75 | 6,486.5 | 5,918 | 8,192 | |
What is the sum of the integers that are both greater than 3 and less than 12? | 60 | 0.6875 | 1,118.1875 | 1,473.181818 | 337.2 | |
The military kitchen needs 1000 jin of rice and 200 jin of millet for dinner. Upon arriving at the rice store, the quartermaster finds a promotion: "Rice is 1 yuan per jin, with 1 jin of millet given for every 10 jin purchased (fractions of 10 jins do not count); Millet is 2 yuan per jin, with 2 jins of rice given for ... | 1200 | 0 | 7,736.5 | -1 | 7,736.5 | |
The sum of an infinite geometric series is \( 16 \) times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio? | -\frac{1}{4} | 0 | 5,053.5625 | -1 | 5,053.5625 | |
In a right triangle, side AB is 8 units, and side BC is 15 units. Calculate $\cos C$ and $\sin C$. | \frac{8}{15} | 0.0625 | 4,902.625 | 6,391 | 4,803.4 | |
Given that $|\overrightarrow{a}|=5$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a}\cdot\overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$. | -4 | 0.8125 | 4,191 | 3,754 | 6,084.666667 | |
The matrix
\[\begin{pmatrix} \frac{4}{29} & -\frac{10}{29} \\ -\frac{10}{29} & \frac{25}{29} \end{pmatrix}\]corresponds to projecting onto a certain vector $\begin{pmatrix} x \\ y \end{pmatrix}.$ Find $\frac{y}{x}.$ | -\frac{5}{2} | 0.8125 | 4,910.4375 | 4,153.153846 | 8,192 | |
Find all pairs of integers $a$ and $b$ for which
\[7a+14b=5a^2+5ab+5b^2\] | (0,0),(1,2),(-1,3) |
We are given the equation:
\[
7a + 14b = 5a^2 + 5ab + 5b^2
\]
and tasked with finding all integer pairs \((a, b)\) satisfying it. Let's begin by simplifying and solving the equation.
### Step 1: Simplify the Equation
First, divide the entire equation by 5 for simplicity. This yields:
\[
\frac{7a}{5} + \frac{14b}{... | 0 | 7,066.8125 | -1 | 7,066.8125 |
What is the largest value of $x$ that satisfies the equation $\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form. | \frac{1}{\sqrt[3]{12}} | 0 | 5,788.1875 | -1 | 5,788.1875 | |
Given the sets of consecutive integers $\{1\}$,$\{2, 3\}$,$\{4,5,6\}$,$\{7,8,9,10\}$,$\; \cdots \;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the nth set. The... | 4641 | To solve for $S_{21}$, the sum of the elements in the 21st set, we first need to determine the first and last elements of this set.
1. **Determine the first element of the 21st set:**
Each set starts where the previous one ended, plus one. The $n$-th set starts with the element that is one more than the sum of the ... | 0.8125 | 3,962 | 3,346.307692 | 6,630 |
In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? | 50 | 0.9375 | 4,711.125 | 4,479.066667 | 8,192 | |
In the Cartesian coordinate system, suppose a curve $C_1$ is given by $x^2+y^2=1$. All points on curve $C_1$ have their $x$ and $y$ coordinates stretched by a factor of $\sqrt{2}$ and $\sqrt{3}$, respectively, resulting in a new curve $C_2$.
$(1)$ Write down the parametric equations for curve $C_2$.
$(2)$ Find the maxi... | \frac{5\sqrt{10}}{2} | 0 | 5,552.125 | -1 | 5,552.125 | |
Six different digits from the set \(\{ 1,2,3,4,5,6,7,8,9\}\) are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is | 29 | 1. **Identify the structure of the problem**: We have a vertical column and a horizontal row intersecting at one square. The vertical column has three squares, and the horizontal row has four squares, with one square shared between them.
2. **Set up the equations based on the given sums**:
- Let the numbers in the ... | 0.1875 | 7,526.125 | 5,969.666667 | 7,885.307692 |
Convert the fraction $\frac{123456789}{2^{26} \times 5^{4}}$ to a decimal. How many decimal places does the resulting number have? | 26 | 0.125 | 7,699.4375 | 4,436.5 | 8,165.571429 | |
Find the largest value of $c$ such that $1$ is in the range of $f(x)=x^2-5x+c$. | \frac{29}{4} | 0.9375 | 2,377.75 | 2,356.866667 | 2,691 | |
Let \(a, b\), and \(c\) be the roots of the cubic polynomial \(3x^3 - 4x^2 + 100x - 3\). Compute \[(a+b+2)^2 + (b+c+2)^2 + (c+a+2)^2.\] | 119.888... | 0 | 8,139.875 | -1 | 8,139.875 | |
The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$ | 16:3 | 1. **Given Ratios**: We are given the following ratios:
- $\frac{w}{x} = \frac{4}{3}$
- $\frac{y}{z} = \frac{3}{2}$
- $\frac{z}{x} = \frac{1}{6}$
2. **Express in terms of variables**: From these ratios, we can express the relationships as:
- $3w = 4x$ (from $\frac{w}{x} = \frac{4}{3}$)
- $2y = 3z$ (from... | 0.4375 | 3,132.0625 | 3,235.714286 | 3,051.444444 |
Compute the smallest base-10 positive integer greater than 7 that is a palindrome when written in both base 3 and 5. | 26 | 0.5625 | 6,704.375 | 5,547.333333 | 8,192 | |
What is the smallest positive integer with exactly 12 positive integer divisors? | 288 | 0 | 4,343.875 | -1 | 4,343.875 | |
When $0.42\overline{153}$ is expressed as a fraction in the form $\frac{x}{99900}$, what is the value of $x$? | 42111 | 0.6875 | 6,741.5625 | 6,082.272727 | 8,192 | |
Evaluate the infinite series $\sum_{k = 1}^\infty \frac{k^2}{2^k}.$ | 6 | 0.625 | 6,057.625 | 4,777 | 8,192 | |
Given that points P and Q are on the curve $f(x) = x^2 - \ln x$ and the line $x-y-2=0$ respectively, find the minimum distance between points P and Q. | \sqrt{2} | 0.6875 | 6,793.75 | 6,785.272727 | 6,812.4 | |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AC}$ and $\overline{BC}$ respectively, where $A(0,8)$, $B(0,0)$, and $C(10,0)$. Find the sum of the slope and $y$-intercept of the line passing through the points $C$ and $D$. | \frac{36}{5} | 1 | 2,183.5625 | 2,183.5625 | -1 | |
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? | 20201 | 1. **Identify the pattern**: We observe that the number of unit squares in figures $0$, $1$, $2$, and $3$ are $1$, $5$, $13$, and $25$ respectively. We need to find a formula that describes this sequence.
2. **Recognize the sequence type**: The sequence seems to be quadratic because the differences between consecutive... | 1 | 2,937.5 | 2,937.5 | -1 |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{2}{3} | 0.375 | 7,315.625 | 5,855 | 8,192 | |
Given plane vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(1,-4)$.
$(1)$ If $4\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $k\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$.
$(2)$ If $\theta$ is the angle between $4\overrightarrow{a}+\overrightarrow{b}$ and ... | -\frac{3}{4} | 0.9375 | 3,931 | 4,056.933333 | 2,042 | |
Solve for $c$: \[\frac{c-23}{2} = \frac{2c +5}{7}.\] | 57 | 1 | 1,360.625 | 1,360.625 | -1 | |
Compute $\sin(-30^\circ)$ and verify by finding $\cos(-30^\circ)$, noticing the relationship, and confirming with the unit circle properties. | \frac{\sqrt{3}}{2} | 0 | 2,015.375 | -1 | 2,015.375 | |
Given that $0 < x < \frac{\pi}{2}$ and $\sin(2x - \frac{\pi}{4}) = -\frac{\sqrt{2}}{10}$, find the value of $\sin x + \cos x$. | \frac{2\sqrt{10}}{5} | 0 | 7,089.9375 | -1 | 7,089.9375 | |
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$ , the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$ . | 9/2 | 0.1875 | 8,192 | 8,192 | 8,192 | |
1. The focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $... | 2\sqrt{2} | 0.125 | 7,832.6875 | 5,317.5 | 8,192 | |
Given that there are 6 teachers with IDs $A$, $B$, $C$, $D$, $E$, $F$ and 4 different schools, with the constraints that each school must have at least 1 teacher and $B$ and $D$ must be arranged in the same school, calculate the total number of different arrangements. | 240 | 0.0625 | 8,082.0625 | 8,192 | 8,074.733333 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2B=A+C$ and $a+\sqrt{2}b=2c$, find the value of $\sin C$. | \frac{\sqrt{6}+\sqrt{2}}{4} | 0 | 4,453.75 | -1 | 4,453.75 | |
A bag contains 3 red balls, 2 black balls, and 1 white ball. All 6 balls are identical in every aspect except for color and are well mixed. Balls are randomly drawn from the bag.
(1) With replacement, find the probability of drawing exactly 1 red ball in 2 consecutive draws;
(2) Without replacement, find the probabilit... | \frac{3}{5} | 0.8125 | 4,445.25 | 3,849.615385 | 7,026.333333 | |
Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements. | 45 | 0.5625 | 7,535.5 | 7,072.444444 | 8,130.857143 | |
Robyn has 4 tasks to do and Sasha has 14 tasks to do. How many of Sasha's tasks should Robyn do in order for them to have the same number of tasks? | 5 | Between them, Robyn and Sasha have \(4 + 14 = 18\) tasks to do. If each does the same number of tasks, each must do \(18 \div 2 = 9\) tasks. This means that Robyn must do \(9 - 4 = 5\) of Sasha's tasks. | 0.8125 | 429.1875 | 416.846154 | 482.666667 |
In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given.
Find the side \(AC\), if it is known that the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\). | \sqrt{13} | 0.125 | 7,770.375 | 6,494 | 7,952.714286 | |
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? | \frac{5}{7} |
To solve this problem, we need to calculate the probability that a triangle formed by choosing three vertices of a regular octagon has at least one side that is also a side of the octagon.
#### Total Number of Triangles:
The total number of ways to choose 3 vertices from 8 vertices of the octagon is given by the comb... | 0.375 | 7,312.625 | 6,222.833333 | 7,966.5 |
In a box, 10 smaller boxes are placed. Some of the boxes are empty, and some contain another 10 smaller boxes each. Out of all the boxes, exactly 6 contain smaller boxes. How many empty boxes are there? | 55 | 0 | 6,199.6875 | -1 | 6,199.6875 | |
Using five nines (9), arithmetic operations, and exponentiation, create the numbers from 1 to 13. | 13 | 0 | 8,192 | -1 | 8,192 | |
Given a line $l$ whose inclination angle $\alpha$ satisfies the condition $\sin \alpha +\cos \alpha = \frac{1}{5}$, determine the slope of $l$. | -\frac{4}{3} | 0.25 | 6,452.4375 | 6,740.5 | 6,356.416667 | |
Given acute angles \\(\alpha\\) and \\(\beta\\) satisfy \\((\tan \alpha-1)(\tan \beta-1)=2\\), then the value of \\(\alpha+\beta\\) is \_\_\_\_\_\_. | \dfrac {3\pi}{4} | 0.5 | 3,763.375 | 4,218.625 | 3,308.125 | |
The function $f$ defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$,$b$,$c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $f$. | 58 | Because there are no other special numbers other than $19$ and $97$, take the average to get $\boxed{58}$. (Note I solved this problem the solution one way but noticed this and this probably generalizes to all $f(x)=x, f(y)=y$ questions like these) | 0.625 | 5,818.875 | 4,570 | 7,900.333333 |
Let \(a\) and \(b\) be two natural numbers. If the remainder of the product \(a \cdot b\) divided by 15 is 1, then \(b\) is called the multiplicative inverse of \(a\) modulo 15. Based on this definition, find the sum of all multiplicative inverses of 7 modulo 15 that lie between 100 and 200. | 1036 | 1 | 3,252.4375 | 3,252.4375 | -1 | |
Find the value of the arithmetic series $1-3+5-7+9-11+\cdots +2021-2023+2025$. | 1013 | 0.625 | 6,180.4375 | 5,239.3 | 7,749 | |
In triangle $XYZ$ where $XY=60$ and $XZ=15$, the area of the triangle is given as $225$. Let $W$ be the midpoint of $\overline{XY}$, and $V$ be the midpoint of $\overline{XZ}$. The angle bisector of $\angle YXZ$ intersects $\overline{WV}$ and $\overline{YZ}$ at points $P$ and $Q$, respectively. Determine the area of qu... | 123.75 | 0 | 8,192 | -1 | 8,192 | |
What is the largest integer less than or equal to \(\sqrt[3]{(2010)^{3}+3 \times(2010)^{2}+4 \times 2010+1}\)? | 2011 | 0.4375 | 7,570.875 | 6,772.285714 | 8,192 | |
The solutions to the equation $x^2 - 3|x| - 2 = 0$ are. | \frac{-3 - \sqrt{17}}{2} | 0 | 4,091.1875 | -1 | 4,091.1875 | |
Let $Q$ be the product of the first $50$ positive even integers. Find the largest integer $l$ such that $Q$ is divisible by $2^l$. | 97 | 0.9375 | 3,685.25 | 3,384.8 | 8,192 | |
Observe the following equations:
1. $\cos 2\alpha = 2\cos^2\alpha - 1$;
2. $\cos 4\alpha = 8\cos^4\alpha - 8\cos^2\alpha + 1$;
3. $\cos 6\alpha = 32\cos^6\alpha - 48\cos^4\alpha + 18\cos^2\alpha - 1$;
4. $\cos 8\alpha = 128\cos^8\alpha - 256\cos^6\alpha + 160\cos^4\alpha - 32\cos^2\alpha + 1$;
5. $\cos 10\alpha = m\co... | 962 | 0.625 | 6,978.625 | 6,250.6 | 8,192 | |
For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^\circ$ and $AB + AD = AC$. It is also given that $\angle ACD = 23^\circ$. What is the measure of angle $ABC$ in degrees? | 83 | 0.3125 | 7,686.875 | 6,575.6 | 8,192 | |
Given $f(x)= \frac{1}{4^{x}+2}$, use the method of deriving the sum formula for an arithmetic sequence to find the value of $f( \frac{1}{10})+f( \frac{2}{10})+…+f( \frac{9}{10})$. | \frac{9}{4} | 0.625 | 6,088.375 | 5,505.5 | 7,059.833333 | |
The chess club has 20 members: 12 boys and 8 girls. A 4-person team is chosen at random. What is the probability that the team has at least 2 boys and at least 1 girl? | \frac{4103}{4845} | 0.0625 | 7,666.75 | 7,961 | 7,647.133333 | |
Maria is 54 inches tall, and Samuel is 72 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is each person in centimeters? Additionally, what is the difference in their heights in centimeters? | 45.72 | 0.5625 | 433.1875 | 439.111111 | 425.571429 | |
How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides? | 144 | 0.75 | 6,717.3125 | 6,225.75 | 8,192 | |
Jenny has 8 stamp books that each contain 42 pages. Each page in her books contains 6 stamps. Jenny decides to reorganize her stamp books such that each page contains 10 stamps. This will give her more space to collect stamps without having to buy new books. Under her new system, Jenny fills up 4 complete books, st... | 6 | 1 | 1,820.3125 | 1,820.3125 | -1 | |
The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$? | 13 + 4\sqrt{2} | To solve this problem, we need to calculate the total length of the line segments that form the acronym "AMC" on the given grid. We will count the number of straight line segments and slanted line segments separately, and then compute their total lengths.
1. **Identify the straight and slanted line segments:**
- **... | 0 | 7,036.0625 | -1 | 7,036.0625 |
Gulliver arrives in the land of the Lilliputians with 7,000,000 rubles. He uses all the money to buy kefir at a price of 7 rubles per bottle (an empty bottle costs 1 ruble at that time). After drinking all the kefir, he returns the bottles and uses the refunded money to buy more kefir. During this process, he notices t... | 1166666 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest positive integer whose cube ends in $888$.
| 192 | 0.4375 | 7,224.5 | 5,980.571429 | 8,192 | |
What is the area, in square units, of a triangle with vertices at $A(1, 1), B(6, 1), C(3, 7)$? | 15 | 1 | 3,224.3125 | 3,224.3125 | -1 | |
What is the largest $n$ for which the numbers $1,2, \ldots, 14$ can be colored in red and blue so that for any number $k=1,2, \ldots, n$, there are a pair of blue numbers and a pair of red numbers, each pair having a difference equal to $k$? | 11 | 0 | 8,029.125 | -1 | 8,029.125 | |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{3} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$ | \frac{2 \sqrt{2}}{3} | 0 | 4,043.5 | -1 | 4,043.5 | |
The Bank of Springfield's Super High Yield savings account compounds annually at a rate of one percent. If Lisa invests 1000 dollars in one of these accounts, then how much interest will she earn after five years? (Give your answer to the nearest dollar.) | 51 | 1 | 3,860.75 | 3,860.75 | -1 | |
Suppose $x$ satisfies $x^{3}+x^{2}+x+1=0$. What are all possible values of $x^{4}+2 x^{3}+2 x^{2}+2 x+1 ?$ | 0 | $x^{4}+2 x^{3}+2 x^{2}+2 x+1=(x+1)\left(x^{3}+x^{2}+x+1\right)=0$ is the only possible solution. | 0.875 | 4,038.5625 | 3,445.214286 | 8,192 |
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enemies and 10 Harvard enemies, compute how ma... | 342 | We count the number of MIT-Harvard friendships. Each of the $m$ MIT students has 8 Harvard friends, for a total of $8 m$ friendships. Each of the $h$ Harvard students has $m-7$ MIT friends, for a total of $h(m-7)$ friendships. So, $8 m=h(m-7) \Longrightarrow m h-8 m-7 h=0 \Longrightarrow(m-7)(h-8)=56$. Each MIT student... | 0 | 7,570.4375 | -1 | 7,570.4375 |
How many four-digit positive integers are multiples of 7? | 1286 | 1 | 3,066.75 | 3,066.75 | -1 | |
Triangle \(ABC\) has \(AB = 10\) and \(BC:AC = 35:36\). What is the largest area that this triangle can have? | 1260 | 0 | 8,192 | -1 | 8,192 | |
Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C... | 13 | 0.5625 | 5,596.5625 | 4,451.222222 | 7,069.142857 | |
A right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy]
import three;
size(2.5inch);
currentproje... | 32 | 0.5625 | 5,349.0625 | 3,137.888889 | 8,192 | |
Two positive integers differ by 8 and their product is 168. What is the larger integer? | 14 | 0 | 8,192 | -1 | 8,192 | |
A class of 10 students took a math test. Each problem was solved by exactly 7 of the students. If the first nine students each solved 4 problems, how many problems did the tenth student solve? | 6 | Suppose the last student solved $n$ problems, and the total number of problems on the test was $p$. Then the total number of correct solutions written was $7 p$ (seven per problem), and also equal to $36+n$ (the sum of the students' scores), so $p=(36+n) / 7$. The smallest $n \geq 0$ for which this is an integer is $n=... | 0.1875 | 7,756 | 5,866.666667 | 8,192 |
At the end of a game, each of the five members of a basketball team shake hands with each of the five members of the other team, and all of the players shake hands with the two referees. How many handshakes occur? | 45 | 1 | 3,692.0625 | 3,692.0625 | -1 | |
Find the number of permutations of \( n \) distinct elements \( a_1, a_2, \cdots, a_n \) (where \( n \geqslant 2 \)) such that \( a_1 \) is not in the first position and \( a_2 \) is not in the second position. | 32,527,596 | 0 | 8,066.125 | -1 | 8,066.125 | |
Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$? | 1512 | 0 | 7,657.375 | -1 | 7,657.375 | |
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible. | \sqrt{2} | 0.1875 | 8,089.875 | 7,647.333333 | 8,192 | |
Given that an odd function \( f(x) \) satisfies the condition \( f(x+3) = f(x) \). When \( x \in [0,1] \), \( f(x) = 3^x - 1 \). Find the value of \( f\left(\log_1 36\right) \). | -1/3 | 0 | 7,012.875 | -1 | 7,012.875 | |
How many whole numbers between $200$ and $500$ contain the digit $3$? | 138 | 0.1875 | 7,690.6875 | 5,518.333333 | 8,192 | |
The lines containing the altitudes of the scalene triangle \( ABC \) intersect at point \( H \). Let \( I \) be the incenter of triangle \( ABC \), and \( O \) be the circumcenter of triangle \( BHC \). It is known that point \( I \) lies on the segment \( OA \). Find the angle \( BAC \). | 60 | 0.125 | 7,704.75 | 4,294 | 8,192 | |
For how many pairs $(m, n)$ with $m$ and $n$ integers satisfying $1 \leq m \leq 100$ and $101 \leq n \leq 205$ is $3^{m}+7^{n}$ divisible by 10? | 2625 | The units digits of powers of 3 cycle $3,9,7,1$ and the units digits of powers of 7 cycle $7,9,3,1$. For $3^{m}+7^{n}$ to be divisible by 10, one of the following must be true: units digit of $3^{m}$ is 3 and $7^{n}$ is 7, or 9 and 1, or 7 and 3, or 1 and 9. The number of possible pairs $(m, n)$ is $27 \times 25+26 \ti... | 0.5625 | 6,618.125 | 5,686.777778 | 7,815.571429 |
The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is: | 0 | 1. **Identify the Expression**: The given expression is $(x^2 - 2xy + y^2)^7$.
2. **Recognize the Binomial Expansion**: Observe that $x^2 - 2xy + y^2$ can be rewritten as $(x-y)^2$. Therefore, the expression becomes $((x-y)^2)^7 = (x-y)^{14}$.
3. **Apply the Binomial Theorem**: The binomial theorem states that $(a+b... | 1 | 2,639.8125 | 2,639.8125 | -1 |
Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.
**A)** $20\sqrt{3} + 24\pi$
**B)** $20\sqrt{3} + \frac{70\pi}{3}$
**C)** $30\sqrt{3} + 22\pi$
**D)** $16\sq... | 20\sqrt{3} + \frac{70\pi}{3} | 0 | 8,192 | -1 | 8,192 | |
The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum. | 22.5 | 0.125 | 6,739.6875 | 7,582 | 6,619.357143 |
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