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 |
|---|---|---|---|---|---|---|
A math competition consists of three problems, each of which receives an integer score from 0 to 7. For any two competitors, it is known that there is at most one problem in which they received the same score. Find the maximum number of competitors in this competition. | 64 | 0 | 8,192 | -1 | 8,192 | |
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? | \frac{281}{13} | 1. **Find the intercepts of the line with the axes:**
- Set $y = 0$ in the equation $12x + 5y = 60$:
\[
12x = 60 \implies x = \frac{60}{12} = 5
\]
- Set $x = 0$ in the equation $12x + 5y = 60$:
\[
5y = 60 \implies y = \frac{60}{5} = 12
\]
The line intersects the x-axis at $(5,0)$ ... | 0.9375 | 4,811.5 | 4,586.133333 | 8,192 |
Compute $95^2$ in your head. | 9025 | 1 | 2,906.3125 | 2,906.3125 | -1 | |
Let \[f(x) =
\begin{cases}
2x^2 - 3&\text{if } x\le 2, \\
ax + 4 &\text{if } x>2.
\end{cases}
\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper). | \frac{1}{2} | 1 | 1,251.6875 | 1,251.6875 | -1 | |
The difference when the product of 4 and a number is subtracted from 16 is greater than 10. How many positive integers satisfy the given conditions? | 1 | 1 | 2,840.375 | 2,840.375 | -1 | |
A flock of geese was flying over several lakes. On each lake, half of the geese and an additional half goose landed, while the rest continued flying. All the geese landed after seven lakes. How many geese were in the flock? | 127 | 0.5625 | 5,714.5 | 4,803.888889 | 6,885.285714 | |
8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up? | 105 | We create the pairs one at a time. The first person has 7 possible partners. Set this pair aside. Of the remaining six people, pick a person. He or she has 5 possible partners. Set this pair aside. Of the remaining four people, pick a person. He or she has 3 possible partners. Set this pair aside. Then the last two mus... | 1 | 2,836.3125 | 2,836.3125 | -1 |
Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue.
Determine the number of colo... | 3^n + 1 |
Let \( S = \{ 1, 2, \ldots, n \} \) with \( n \geq 1 \). We are tasked with finding the number of ways to color each of the \( 2^n \) subsets of \( S \) either red or blue, such that the condition
\[
f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2)
\]
holds for any subsets \( T_1, T_2 \subseteq S \), where \( f(T) \) rep... | 0 | 8,132.1875 | -1 | 8,132.1875 |
In-Young generates a string of $B$ zeroes and ones using the following method:
- First, she flips a fair coin. If it lands heads, her first digit will be a 0, and if it lands tails, her first digit will be a 1.
- For each subsequent bit, she flips an unfair coin, which lands heads with probability $A$. If the coin land... | 2 | Since each digit is dependent on the previous, and the first digit is random, we note that the probability that In Young obtains a particular string is the same probability as that she obtains the inverse string (i.e. that where the positions of the 0 s and 1 s are swapped). Consequently, we would expect that half of h... | 0 | 6,870.5 | -1 | 6,870.5 |
In the acute triangle \(KLM\), \(V\) is the intersection of its heights, and \(X\) is the foot of the height onto side \(KL\). The angle bisector of angle \(XVL\) is parallel to side \(LM\), and the angle \(MKL\) measures \(70^\circ\).
What are the measures of the angles \(KLM\) and \(KML\)? | 55 | 0 | 7,929.8125 | -1 | 7,929.8125 | |
Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$ .
How many solutions (including University Mathematics )are there for the problem?
Any advice would be appreciated. :) | 1/8 | 0.5 | 6,283.1875 | 5,772 | 6,794.375 | |
Find the area of the triangle with vertices $(6,5,3),$ $(3,3,1),$ and $(15,11,9).$ | 0 | 0.6875 | 6,264.5 | 6,227.818182 | 6,345.2 | |
In a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length 1, $E$ is the midpoint of $AB$, and $F$ is the midpoint of $CC_{1}$. Find the distance from $D$ to the plane passing through points $D_{1}$, $E$, and $F$.
| \frac{4 \sqrt{29}}{29} | 0 | 5,914.375 | -1 | 5,914.375 | |
Let \( r(x) \) have a domain of \(\{ -2, -1, 0, 1 \}\) and a range of \(\{ 1, 3, 5, 7 \}\). Let \( s(x) \) have a domain of \(\{ 0, 1, 2, 3, 4, 5 \}\) and be defined by \( s(x) = 2x + 1 \). What is the sum of all possible values of \( s(r(x)) \)? | 21 | 0.8125 | 3,703.25 | 3,365.076923 | 5,168.666667 | |
Given a point P on the hyperbola $C_1: \frac{x^2}{16} - \frac{y^2}{9} = 1$, a point Q on the circle $C_2: (x - 5)^2 + y^2 = 1$, and a point R on the circle $C_3: (x + 5)^2 + y^2 = 1$, find the maximum value of $|PQ| - |PR|$. | 10 | 0.125 | 8,086.125 | 7,345 | 8,192 | |
In the diagram, $\mathrm{ABCD}$ is a right trapezoid with $\angle \mathrm{DAB} = \angle \mathrm{ABC} = 90^\circ$. A rectangle $\mathrm{ADEF}$ is constructed externally along $\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\mathrm{BE}$ intersects $\mathrm{AD}$ at point $\mathrm{P}$, and line $\mathrm{PC}$ ... | 3.18 | 0 | 8,192 | -1 | 8,192 | |
At a conference, there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{3}{7}$. What fraction of the people in the conference are married men? | \frac{4}{11} | 0.9375 | 2,876.875 | 2,522.533333 | 8,192 | |
From 5 students, 4 are to be selected to participate in competitions in four subjects: mathematics, physics, chemistry, and biology, with each competition having only 1 participant. If student A does not participate in the biology competition, then the total number of different selection schemes is \_\_\_\_\_\_. | 96 | 0.5625 | 6,734.625 | 5,814.111111 | 7,918.142857 | |
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? | 91 | 1. **Calculate the cost of one T-shirt**:
Given that each T-shirt costs $5 more than a pair of socks, and each pair of socks costs $4, the cost of one T-shirt is:
\[
4 + 5 = 9 \text{ dollars}
\]
2. **Calculate the total cost for one member**:
Each member requires 2 pairs of socks and 2 T-shirts. The... | 1 | 1,178.8125 | 1,178.8125 | -1 |
Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$. | \frac{1}{4} | 0.125 | 7,940.5 | 7,326 | 8,028.285714 | |
What are the last two digits in the sum of factorials of the first 100 positive integers? | 13 | 0.9375 | 3,683.6875 | 3,383.133333 | 8,192 | |
Let $A=(0,1),$ $B=(2,5),$ $C=(5,2),$ and $D=(7,0).$ A figure is created by connecting $A$ to $B,$ $B$ to $C,$ $C$ to $D,$ and $D$ to $A.$ The perimeter of $ABCD$ can be expressed in the form $a\sqrt2+b\sqrt{5}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? | 12 | 1 | 2,246.4375 | 2,246.4375 | -1 | |
Add $1_3 + 12_3 + 212_3 + 2121_3.$ Express your answer in base 3. | 10200_3 | 0.25 | 7,437.0625 | 5,172.25 | 8,192 | |
The right triangles \(MDC\) and \(ADK\) have a common right angle at \(D\). Point \(K\) lies on \(CD\) and divides it in the ratio \(2:3\) from point \(C\). Point \(M\) is the midpoint of side \(AD\). Find the sum of the degree measures of angles \(AKD\) and \(MCD\), given that \(AD : CD = 2:5\). | 45 | 0.75 | 5,369.3125 | 4,428.416667 | 8,192 | |
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8? | 4 |
#### Step-by-step Analysis:
**Solution 1:**
1. **Observation of Pattern:** Notice that $3^0 + 3^1 + 3^2 + 3^3 = 1 + 3 + 9 + 27 = 40$, which is divisible by 8. This implies that any sum of four consecutive powers of 3 starting from a power of 0 modulo 4 is divisible by 8.
2. **Grouping Terms:** The sum $3^0 + 3^1 + 3^... | 0.875 | 4,652.1875 | 4,146.5 | 8,192 |
Consider a $3 \times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $33$. Determine the value of the center squa... | 15 | 0.9375 | 4,441.1875 | 4,191.133333 | 8,192 | |
Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$. The average value of $S$ (if all possible orders of these 20 people are considered) is closest... | 9 | 0 | 7,732 | -1 | 7,732 | |
How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point. | 15 | 0 | 7,921.875 | -1 | 7,921.875 | |
In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for? | \frac{2101}{625} | The probability that a chosen candidate will receive no votes at all is $\left(\frac{4}{5}\right)^{5}$, or the probability that every person will vote for someone other than that one candidate. Then the probability that a chosen candidate will receive at least one vote is $1-\left(\frac{4}{5}\right)^{5}=\frac{2101}{312... | 0.8125 | 6,025.875 | 5,713.153846 | 7,381 |
As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} ... | 1.2 | 0 | 8,192 | -1 | 8,192 | |
Determine how many different ways there are to assign the elements of the set \( M = \{1, 2, 3, 4, 5\} \) into three ordered sets \( A, B,\) and \( C \) such that the following conditions are satisfied: each element must belong to at least one of the sets, the intersection of all three sets is empty, and the intersecti... | 1230 | 0 | 7,806.8125 | -1 | 7,806.8125 | |
The volume of the box is 360 cubic units where $a, b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$? | 12 | 0.375 | 7,951.8125 | 7,551.5 | 8,192 | |
Given a point P $(x, y)$ on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$. | \sqrt{5} | 0.8125 | 5,574.25 | 4,970.153846 | 8,192 | |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=- \dfrac { \sqrt {3}}{2}t \\ y=1+ \dfrac {1}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate system is established. The eq... | \dfrac {3+ \sqrt {3}}{2} | 0 | 7,898.4375 | -1 | 7,898.4375 | |
Calculate $8 \cdot 9\frac{2}{5}$. | 75\frac{1}{5} | 0.6875 | 1,411.9375 | 1,299.272727 | 1,659.8 | |
If $(x-1)(x+3)(x-4)(x-8)+m$ is a perfect square, find the value of $m$. | 196 | 0.9375 | 4,317.125 | 4,058.8 | 8,192 | |
A fair standard six-sided dice is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one "3" is tossed?
A) $\frac{1}{6}$
B) $\frac{6}{17}$
C) $\frac{9}{17}$
D) $\frac{1}{2}$
E) $\frac{1}{3}$ | \frac{9}{17} | 0 | 8,192 | -1 | 8,192 | |
Simplify $(3-2i)-(5-2i)$. | -2 | 1 | 1,415.0625 | 1,415.0625 | -1 | |
$2.46\times 8.163\times (5.17+4.829)$ is approximately equal to what value? | 200 | 0.375 | 335.3125 | 382 | 307.3 | |
What is the tenth term in the geometric sequence $9,3,1,\frac 13, \ldots$? | \frac{1}{2187} | 1 | 2,151.125 | 2,151.125 | -1 | |
The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out of the tournament. If 3 of th... | $\frac{4}{5}$ | We note that the probability there are not two Spanish teams in the final two is the probability that the 3 of them have already competed against each other in previous rounds. Note that the random pairings in each round is equivalent, by the final round, to dividing the 16 into two groups of 8 and taking a winner from... | 0 | 7,946.5 | -1 | 7,946.5 |
On Monday, Knight Milivoj traveled 25 miles and spent the night in Zubín. The next day, Tuesday, he reached Veselín. On the way back, he traveled 6 miles more on Thursday than on Monday and spent the night in Kostín. On Friday, he traveled the remaining 11 miles to Rytířov. Determine the distance between Zubín and Vese... | 17 | 0 | 5,874.125 | -1 | 5,874.125 | |
Mary has a sequence $m_{2}, m_{3}, m_{4}, \ldots$, such that for each $b \geq 2, m_{b}$ is the least positive integer $m$ for which none of the base-$b$ logarithms $\log _{b}(m), \log _{b}(m+1), \ldots, \log _{b}(m+2017)$ are integers. Find the largest number in her sequence. | 2188 | It is not difficult to see that for all of the logarithms to be non-integers, they must lie strictly between $n$ and $n+1$ for some integer $n$. Therefore, we require $b^{n+1}-b^{n}>2018$, and so $m_{b}=b^{n}+1$ where $n$ is the smallest integer that satisfies the inequality. In particular, this means that $b^{n}-b^{n-... | 0 | 8,192 | -1 | 8,192 |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a} - 2\overrightarrow{b}| \leqslant 2$, find the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$. | -\frac{1}{2} | 0.5625 | 7,380 | 6,748.444444 | 8,192 | |
Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled? | 20 | 1. **Calculate the number of potatoes peeled by Homer alone**: Homer peels potatoes at a rate of 3 potatoes per minute. He peels alone for 4 minutes. Therefore, the number of potatoes peeled by Homer in these 4 minutes is:
\[
4 \text{ minutes} \times 3 \frac{\text{potatoes}}{\text{minute}} = 12 \text{ potatoes}
... | 1 | 1,173.0625 | 1,173.0625 | -1 |
For any positive integer \( n \), define
\( g(n) =\left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right. \)
What is \( \sum_{n = 1}^{1023}{g(n)} \)?
**A** \( \frac{40}{2} \)
**B** \( \frac{42}{2} \)
**C** \( \frac{45}{2} \)
**D** \( \frac{48}{2} \)... | \frac{45}{2} | 0 | 5,057.875 | -1 | 5,057.875 | |
In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay is the income minus the tax on that income.) | 50000 | 0.8125 | 4,268.5625 | 3,677.538462 | 6,829.666667 | |
Calculate $52103_{8} - 1452_{9}$ in base 10. | 20471 | 0.75 | 3,853.3125 | 3,478.5 | 4,977.75 | |
If $\angle A = 60^\circ$, $\angle E = 40^\circ$ and $\angle C = 30^\circ$, then $\angle BDC =$ | 50^\circ | 1. **Identify the Known Angles**: Given $\angle A = 60^\circ$, $\angle E = 40^\circ$, and $\angle C = 30^\circ$. We assume that points $A$, $B$, $C$, $D$, and $E$ are positioned such that these angles are part of triangles or other geometric figures, but the exact configuration is not specified. We need to make logical... | 0 | 6,215.875 | -1 | 6,215.875 |
One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube. | 64 | 0.1875 | 8,128.9375 | 7,855.666667 | 8,192 | |
Narsa buys a package of 45 cookies on Monday morning. How many cookies are left in the package after Friday? | 15 | On Monday, Narsa ate 4 cookies. On Tuesday, Narsa ate 12 cookies. On Wednesday, Narsa ate 8 cookies. On Thursday, Narsa ate 0 cookies. On Friday, Narsa ate 6 cookies. This means that Narsa ate $4+12+8+0+6=30$ cookies. Since the package started with 45 cookies, there are $45-30=15$ cookies left in the package after Frid... | 0 | 372.3125 | -1 | 372.3125 |
The smallest 9-digit integer that can be divided by 11 is ____. | 100000010 | 0.0625 | 3,000.4375 | 4,781 | 2,881.733333 | |
Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $4:30$ PM. If she pedals at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arrive home at $5:00$ PM? | 12 | 0.4375 | 5,624.5 | 4,763.428571 | 6,294.222222 | |
Solve the equations:
(1) $(x-2)^2=25$;
(2) $x^2+4x+3=0$;
(3) $2x^2+4x-1=0$. | \frac{-2-\sqrt{6}}{2} | 0 | 3,085.125 | -1 | 3,085.125 | |
Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$. | 30^\circ |
To solve for \(\angle XAY\), we first establish the geometry of the problem. We have a regular hexagon \(ABCDEF\) with side length 1. Since it is regular, each interior angle of the hexagon is \(120^\circ\).
Points \(X\) and \(Y\) are located on sides \(CD\) and \(DE\), respectively, with the condition that the perim... | 0.1875 | 8,028.375 | 7,319.333333 | 8,192 |
If $z=3+4i$, find $z^2$. (Note that $i^2 = -1.$) | -7+24i | 0.9375 | 2,647.5625 | 2,277.933333 | 8,192 | |
Point $Q$ lies on the diagonal $AC$ of square $EFGH$ with $EQ > GQ$. Let $R_{1}$ and $R_{2}$ be the circumcenters of triangles $EFQ$ and $GHQ$ respectively. Given that $EF = 8$ and $\angle R_{1}QR_{2} = 90^{\circ}$, find the length $EQ$ in the form $\sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c... | 40 | 0 | 8,068 | -1 | 8,068 | |
Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find the perimeter of triangle APR. | 48 | 0.375 | 6,337.875 | 4,045.166667 | 7,713.5 | |
In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 1:2$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 24, determine the area of $\triangle ABC$. | 432 | 0.5625 | 6,282.6875 | 5,029.555556 | 7,893.857143 | |
If \( 10^{x} \cdot 10^{5}=100^{4} \), what is the value of \( x \)? | 3 | Since \( 100=10^{2} \), then \( 100^{4}=(10^{2})^{4}=10^{8} \). Therefore, we must solve the equation \( 10^{x} \cdot 10^{5}=10^{8} \), which is equivalent to \( 10^{x+5}=10^{8} \). Thus, \( x+5=8 \) or \( x=3 \). | 1 | 820.5625 | 820.5625 | -1 |
In the figure below, $ABCD$ is a square piece of paper 6 cm on each side. Corner $C$ is folded over so that it coincides with $E$, the midpoint of $\overline{AD}$. If $\overline{GF}$ represents the crease created by the fold such that $F$ is on $CD,$ what is the length of $\overline{FD}$? Express your answer as a commo... | \frac94 | 0.625 | 6,288.625 | 5,287.2 | 7,957.666667 | |
If $f(x)=\dfrac{x-3}{x-4}$, then for what value of $x$ is $f^{-1}(x)$ undefined? | 1 | 1 | 2,408.6875 | 2,408.6875 | -1 | |
The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes? | 216 | 0 | 5,704.125 | -1 | 5,704.125 | |
Given that the odd function $f(x)$ is a monotonically increasing function defined on $\mathbb{R}$ and the sequence $\{x_n\}$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, find the value of $x_{2012}$. | 4005 | 0.9375 | 4,076.125 | 3,801.733333 | 8,192 | |
How many four-digit positive integers are multiples of 7? | 1286 | 0.9375 | 3,170.8125 | 2,836.066667 | 8,192 | |
Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $-\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then when $n\in N^*$, the sum of the maximum and minimum values of $S_n - \frac{1}{S_n}$ is ______. | \frac{1}{4} | 0.5 | 7,326.0625 | 6,535.875 | 8,116.25 | |
Calculate:<br/>$(1)-1^{2023}-\sqrt{2\frac{1}{4}}+\sqrt[3]{-1}+\frac{1}{2}$;<br/>$(2)2\sqrt{3}+|1-\sqrt{3}|-\left(-1\right)^{2022}+2$. | 3\sqrt{3} | 0.8125 | 1,524.125 | 1,644.153846 | 1,004 | |
There are a total of $400$ machine parts. If person A works alone for $1$ day, and then person A and person B work together for $2$ days, there will still be $60$ parts unfinished. If both work together for $3$ days, they can produce $20$ parts more than needed. How many parts can each person make per day? | 80 | 1 | 1,823.625 | 1,823.625 | -1 | |
The volume of a given sphere is \( 72\pi \) cubic inches. Calculate the surface area of the sphere in terms of \( \pi \). | 36\pi 2^{2/3} | 0 | 2,895.8125 | -1 | 2,895.8125 | |
Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two... | \frac{2^{n}-1}{8^{n}} | At any point during this competition, we shall say that the situation is living if both players 1 and $2^{n}$ are still in the running. A living situation is far if those two players are diametrically opposite each other, and near otherwise, in which case (as one can check inductively) they must be just one person shy ... | 0 | 8,192 | -1 | 8,192 |
The symbol $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the solution to the equation $\left[3 x - 4 \frac{5}{6}\right] - 2 x - 1 = 0$. | 6.5 | 0.1875 | 7,187.625 | 7,424 | 7,133.076923 | |
Two isosceles triangles each have at least one angle that measures $70^{\circ}$. In the first triangle, the measure in degrees of each of the remaining two angles is even. In the second triangle, the measure in degrees of each of the remaining two angles is odd. Let $S$ be the sum of the equal angles in the first trian... | 250 | 1 | 3,546.4375 | 3,546.4375 | -1 | |
In the isosceles triangle \( ABC \) (\( AB = BC \)), medians \( AD \) and \( EC \) intersect at point \( O \). The ratio of the radius of the circle inscribed in triangle \( AOC \) to the radius of the circle inscribed in quadrilateral \( ODBE \) is \(\frac{2}{3}\). Find the ratio \(\frac{AC}{BC}\). | 20/17 | 0.0625 | 8,157.9375 | 7,647 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\cos A= \frac {c}{a}\cos C$, $b+c=2+ \sqrt {2}$, and $\cos B= \frac {3}{4}$, find the area of $\triangle ABC$. | \frac { \sqrt {7}}{2} | 0 | 6,635.125 | -1 | 6,635.125 | |
Yura has unusual clocks with several minute hands, moving in different directions. Yura counted that in one hour, the minute hands coincided exactly 54 times in pairs. What is the maximum number of minute hands that Yura's clock can have? | 28 | 0 | 7,644.5625 | -1 | 7,644.5625 | |
Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______. | -\sqrt{3} | 0.1875 | 7,974.75 | 7,033.333333 | 8,192 | |
Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$ ? | 333 | 0.0625 | 8,190.5 | 8,168 | 8,192 | |
How many integers from 1 to 2001 have a digit sum that is divisible by 5? | 399 | 0.375 | 7,718.125 | 6,928.333333 | 8,192 | |
Find all real numbers \( p \) such that the cubic equation \( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \) has three roots, all of which are positive integers. | 76 | 0 | 8,192 | -1 | 8,192 | |
Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits. | 32768 | 0.3125 | 7,574.6875 | 6,216.6 | 8,192 | |
Given an inverted cone with a base radius of $15 \mathrm{cm}$ and a height of $15 \mathrm{cm}$, and a cylinder with a horizontal base radius of $18 \mathrm{cm}$, determine the height in centimeters of the water in the cylinder after $10\%$ of the water is lost from the cone. | 3.125 | 0.3125 | 4,015 | 2,833.6 | 4,552 | |
Consider the function $g(x) = \frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are nonzero real numbers. Assume $g(5) = 5$ and $g(25) = 25$, and it also satisfies $g(g(x)) = x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $g$. | 15 | 0.9375 | 5,018.3125 | 4,806.733333 | 8,192 | |
If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is: | \sqrt{50} | 1. **Identify the relationship between the perimeter and the sides of the rectangle:**
Given that the perimeter of rectangle $ABCD$ is $20$ inches, we can express this in terms of the length $l$ and width $w$ of the rectangle:
\[
2l + 2w = 20
\]
Simplifying, we get:
\[
l + w = 10
\]
2. **Expres... | 0 | 2,239.5625 | -1 | 2,239.5625 |
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$ . | \[
7
\] | Note that $A_0 = 2^0 + 3^2 + 5^2 = 35$ , so the GCD must be a factor of 35. The prime factorization of $35$ is $5 \cdot 7$ , so we need to check if $5$ and $7$ are factors of the rest of the numbers.
Note that $A_1 = 2^3 + 3^8 + 5^8$ . Taking both sides modulo 5 yields $A_1 \equiv 2^3 + 3^8 \equiv 4 \pmod{5}$ , and ... | 0 | 6,161.375 | -1 | 6,161.375 |
If $2x+1=8$, then $4x+1=$ | 15 | 1. **Start from the given equation:**
Given that $2x + 1 = 8$.
2. **Isolate $x$:**
Subtract $1$ from both sides to simplify the equation:
\[
2x + 1 - 1 = 8 - 1 \implies 2x = 7.
\]
Divide both sides by $2$ to solve for $x$:
\[
x = \frac{7}{2}.
\]
3. **Substitute $x$ into the new expressi... | 1 | 2,063.625 | 2,063.625 | -1 |
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1$. Determine how many complex numbers $z$ exist such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute values at most $15$ and $\text{Re}(f(z)) = \text{Im}(f(z))$. | 31 | 0 | 8,192 | -1 | 8,192 | |
A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to i... | 14 | 1 | 2,206.875 | 2,206.875 | -1 | |
The function $f(x)$ takes positive real numbers to real numbers, such that
\[xf(y) - yf(x) = f \left( \frac{x}{y} \right)\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas. | 0 | 0.875 | 6,174.625 | 5,886.428571 | 8,192 | |
Given the function $f(x)=1-2\sin ^{2}(x+ \frac {π}{8})+2\sin (x+ \frac {π}{8})\cos (x+ \frac {π}{8})$.
(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[- \frac {π}{4}, \frac {3π}{8}]$. | -1 | 0.8125 | 4,873.375 | 4,740 | 5,451.333333 | |
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform nu... | 11 | Let's denote the uniform numbers of Ashley, Bethany, and Caitlin as $a$, $b$, and $c$ respectively. According to the problem, all these numbers are 2-digit primes. The smallest 2-digit primes are $11, 13, 17, 19, 23, \ldots$.
From the conversation:
- Bethany's birthday earlier this month corresponds to the sum of Ashl... | 0.125 | 7,762.4375 | 7,968 | 7,733.071429 |
Estimate the population of Nisos in the year 2050. | 2000 | 1. **Identify the growth pattern**: The problem states that the population of Nisos triples every 25 years. This means that if the population is $P$ in a certain year, it will be $3P$ 25 years later.
2. **Calculate the population in 2050**:
- The population in 2000 is given as 200.
- From 2000 to 2025 (25 years... | 0 | 5,583.1875 | -1 | 5,583.1875 |
Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 3$? | 9 | 0.4375 | 7,475.3125 | 6,988.428571 | 7,854 | |
Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$ .
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
| 23 | 0 | 8,192 | -1 | 8,192 | |
Determine the number of subsets $S$ of $\{1,2,3, \ldots, 10\}$ with the following property: there exist integers $a<b<c$ with $a \in S, b \notin S, c \in S$. | 968 | 968. There are $2^{10}=1024$ subsets of $\{1,2, \ldots, 10\}$ altogether. Any subset without the specified property must be either the empty set or a block of consecutive integers. To specify a block of consecutive integers, we either have just one element (10 choices) or a pair of distinct endpoints $\left(\binom{10}{... | 0.0625 | 8,037.8125 | 5,762 | 8,189.533333 |
Determine the sum of all integer values $n$ for which $\binom{25}{n} + \binom{25}{12} = \binom{26}{13}$. | 25 | 0.9375 | 2,924.125 | 2,774.333333 | 5,171 | |
A 5-dimensional ant starts at one vertex of a 5-dimensional hypercube of side length 1. A move is when the ant travels from one vertex to another vertex at a distance of $\sqrt{2}$ away. How many ways can the ant make 5 moves and end up on the same vertex it started at? | 6240 | We let the cube lie in $\mathbb{R}^{5}$ with each corner with coordinates 1 or 0. Assume the ant starts at $(0,0,0,0,0)$. Every move the ant adds or subtracts 1 to two of the places. Note that this means the ant can only land on a vertex with the sum of its coordinates an even number. Every move the ant has $\binom{5}{... | 0 | 6,866.875 | -1 | 6,866.875 |
What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$ ? | 20 | 1 | 4,806.3125 | 4,806.3125 | -1 | |
What is the greatest integer value of $x$ such that $\frac{x^2 + 2x + 5}{x-3}$ is an integer? | 23 | 1 | 2,452.375 | 2,452.375 | -1 | |
What is the value of $525^2 - 475^2$? | 50000 | 0.9375 | 3,064.875 | 2,723.066667 | 8,192 | |
When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
| 26 | 1 | 2,269.5 | 2,269.5 | -1 |
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