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 |
|---|---|---|---|---|---|---|
For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, $x$ and $x$ are the five integers, what is the value of $x$? | 34 | 1 | 3,088.375 | 3,088.375 | -1 | |
Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between? | 21 | 0.25 | 5,251.4375 | 5,971.25 | 5,011.5 | |
A light flashes green every 3 seconds. Determine the number of times the light has flashed green after 671 seconds. | 154 | 0 | 5,124 | -1 | 5,124 | |
Suppose $\cos R = \frac{4}{9}$ in the diagram below. What is $QS$?
[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(65),0);
R = (sqrt(65),-4);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,13));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$9$",(R+S)/2,SW);
[/asy] | \sqrt{65} | 0.8125 | 5,556.625 | 5,012.384615 | 7,915 | |
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$ th day of January. What is $m$ ?
*Based on a proposal by Neelabh Deka* | 23 | 0.3125 | 7,731.5 | 6,769.4 | 8,168.818182 | |
Given that $z$ is a complex number such that $z+\frac{1}{z}=2\cos 5^\circ$, find $z^{1500}+\frac{1}{z^{1500}}$. | -\sqrt{3} | 0 | 3,847.3125 | -1 | 3,847.3125 | |
Solve the fractional equation application problem.
On the eve of Children's Day, a certain shopping mall purchased a certain electric toy for $7200. Due to good sales, after a period of time, the mall purchased the same toy again for $14800. The quantity purchased the second time was twice the quantity purchased the f... | 100 | 0.8125 | 4,053.4375 | 3,136.846154 | 8,025.333333 | |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, $f(x) = \begin{cases} \log_{2}(x+1) & \text{for } x \geq 0 \\ g(x) & \text{for } x < 0 \\ \end{cases}$. Find the value of $g\left(f(-7)\right)$. | -2 | 1 | 2,370.1875 | 2,370.1875 | -1 | |
Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way t... | 3\sqrt{3} | 0.5 | 6,608.1875 | 5,024.375 | 8,192 | |
Let \(a\), \(b\), and \(c\) be positive real numbers. Find the minimum value of
\[
\frac{5c}{a+b} + \frac{5a}{b+c} + \frac{3b}{a+c} + 1.
\] | 7.25 | 0 | 8,192 | -1 | 8,192 | |
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence? | 192 | 0 | 6,381.875 | -1 | 6,381.875 | |
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? | 380 | 1. **Identify the operation**: Ahn performs the operation $2(200 - n)$, where $n$ is a two-digit integer.
2. **Determine the range of $n$**: Since $n$ is a two-digit integer, $10 \leq n \leq 99$.
3. **Maximize the expression**: To maximize $2(200 - n)$, we need to minimize $n$ because subtracting a smaller number fro... | 1 | 2,071.375 | 2,071.375 | -1 |
What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$? | 1120 | 0.9375 | 2,884.9375 | 2,531.133333 | 8,192 | |
Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the re... | 39 |
Mattis is organizing a badminton tournament with the following setup: there are \(40\) players distributed evenly across \(20\) courts, with \(2\) players on each court. In each round of the tournament, a match is played on each court, and a winner and a loser are determined. Following the match results, the player wh... | 0.1875 | 7,898.1875 | 6,918.333333 | 8,124.307692 |
In triangle $ABC$, let vector $\vec{a} = (1, \cos B)$ and vector $\vec{b} = (\sin B, 1)$, and suppose $\vec{a}$ is perpendicular to $\vec{b}$. Find the magnitude of angle $B$. | \frac{3\pi}{4} | 0.375 | 2,219.125 | 2,344.666667 | 2,143.8 | |
Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000. | 200 | Our fraction can be written in the form $\frac{1000 - a}{a} = \frac{1000}{a} - 1.$ Thus the fraction is reducible when $a$ divides $1000.$ We also want $500 < a < 1000.$ By PIE, the total values of $a$ that make the fraction reducible is, \[249 + 99 - 49 = 299.\] By complementary counting, the answer we want is $499 - ... | 0.9375 | 5,128.875 | 4,924.666667 | 8,192 |
A solid right prism $ABCDEF$ has a height of $16,$ as shown. Also, its bases are equilateral triangles with side length $12.$ Points $X,$ $Y,$ and $Z$ are the midpoints of edges $AC,$ $BC,$ and $DC,$ respectively. A part of the prism above is sliced off with a straight cut through points $X,$ $Y,$ and $Z.$ Determine th... | 48+9\sqrt{3}+3\sqrt{91} | 0 | 7,598.1875 | -1 | 7,598.1875 | |
Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3, or 7? Express your answer as a common fraction. | \frac{107}{150} | 0.9375 | 4,074.6875 | 3,800.2 | 8,192 | |
Adam has $15$ of a certain type of rare coin and is interested in knowing how much this collection is worth. He discovers that $5$ of these coins are worth $12$ dollars in total. Assuming that the value of each coin is the same, how many dollars is his entire collection worth? | 36 | 1 | 1,190.3125 | 1,190.3125 | -1 | |
The distance from Goteborg to Jonkiping on a map is 88 cm. The scale on the map is 1 cm: 15 km. How far is it between the two city centers, in kilometers? | 1320 | 0.9375 | 1,444.375 | 1,521.666667 | 285 | |
Ten standard 6-sided dice are rolled. What is the probability that exactly one of the dice shows a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.323 | 0.3125 | 6,850.875 | 5,951.6 | 7,259.636364 | |
In the rhombus \(A B C D\), the angle at vertex \(A\) is \(60^{\circ}\). Point \(N\) divides side \(A B\) in the ratio \(A N: B N = 2: 1\). Find the tangent of angle \(D N C\). | \sqrt{\frac{243}{121}} | 0 | 5,736.8125 | -1 | 5,736.8125 | |
Let **v** be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of $\|\mathbf{v}\|$. | 10 - 2\sqrt{5} | 0.5625 | 6,627 | 5,409.777778 | 8,192 | |
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$ ,we have $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of $a_{2013}$ . | 13 | 0 | 7,854.75 | -1 | 7,854.75 | |
\( N \) is an integer whose representation in base \( b \) is 777. Find the smallest positive integer \( b \) such that \( N \) is the fourth power of a decimal integer. | 18 | 0.5625 | 6,612.625 | 5,384.222222 | 8,192 | |
Triangle $ABC$ is a right triangle with $AB = 6$, $BC = 8$, and $AC = 10$. Point $D$ is on line $\overline{BC}$ such that $\overline{AD}$ bisects angle $BAC$. The inscribed circles of $\triangle ADB$ and $\triangle ADC$ have radii $r_1$ and $r_2$, respectively. What is $r_1/r_2$?
A) $\frac{24}{35}$
B) $\frac{35}{24}$
C... | \frac{24}{35} | 0 | 8,192 | -1 | 8,192 | |
In a triangle $A B C$, points $M$ and $N$ are on sides $A B$ and $A C$, respectively, such that $M B=B C=C N$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $A B C$, respectively. Express the ratio $M N / B C$ in terms of $R$ and $r$. | \sqrt{1-\frac{2r}{R}} | Let $\omega, O$ and $I$ be the circumcircle, the circumcenter and the incenter of $A B C$, respectively. Let $D$ be the point of intersection of the line $B I$ and the circle $\omega$ such that $D \neq B$. Then $D$ is the midpoint of the arc $A C$. Hence $O D \perp C N$ and $O D=R$. We first show that triangles $M N C$... | 0 | 8,192 | -1 | 8,192 |
The product of the 9 factors $\left(1 - \frac12\right)\left(1 - \frac13\right)\left(1 - \frac14\right) \cdots \left(1 - \frac {1}{10}\right) =$ | \frac{1}{10} | 1. **Identify the Factors**: The given expression is a product of factors of the form $\left(1 - \frac{1}{n}\right)$ for $n$ ranging from $2$ to $10$. This can be rewritten as:
\[
\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\cdots\left(1 - \frac{1}{10}\right)
\]
2. **Si... | 1 | 2,148.0625 | 2,148.0625 | -1 |
Determine all integers $k\geqslant 1$ with the following property: given $k$ different colours, if each integer is coloured in one of these $k$ colours, then there must exist integers $a_1<a_2<\cdots<a_{2023}$ of the same colour such that the differences $a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}$ are all powers of $2$. | 1 \text{ and } 2 |
To solve this problem, we need to determine integers \( k \geqslant 1 \) such that when the integers are coloured with \( k \) different colours, there exist integers \( a_1 < a_2 < \cdots < a_{2023} \) all of the same colour, where the differences \( a_2-a_1, a_3-a_2, \ldots, a_{2023}-a_{2022} \) are powers of 2.
##... | 0 | 8,192 | -1 | 8,192 |
Given two-dimensional vectors $\vec{a}$, $\vec{b}$, with $|\vec{a}|=1$, $|\vec{b}|=2$, and $\vec{a} \cdot \vec{b}=1$. If $\vec{e}$ is a two-dimensional unit vector, find the maximum value of $|\vec{a} \cdot \vec{e}| + |\vec{b} \cdot \vec{e}|$. | \sqrt{7} | 0.125 | 7,879.6875 | 6,423 | 8,087.785714 | |
If $5! \cdot 3! = n!$, what is the value of $n$? | 6 | 1 | 1,480.9375 | 1,480.9375 | -1 | |
Let $\alpha$ and $\beta$ be the roots of $x^2 + px + 1 = 0,$ and let $\gamma$ and $\delta$ are the roots of $x^2 + qx + 1 = 0.$ Express
\[(\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\]in terms of $p$ and $q.$ | q^2 - p^2 | 0.4375 | 6,915.4375 | 5,274.142857 | 8,192 | |
A car is braking to a complete stop. It is known that its speed at the midpoint of the distance was 100 km/h. Determine its initial speed. | 141.4 | 0 | 7,268.1875 | -1 | 7,268.1875 | |
Let \( x, y, \) and \( z \) be real numbers such that \(\frac{4}{x} + \frac{2}{y} + \frac{1}{z} = 1\). Find the minimum of \( x + 8y + 4z \). | 64 | 0.75 | 6,226.6875 | 5,571.583333 | 8,192 | |
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? | \frac{47}{256} | 1. **Understanding the Problem:**
We need to find the probability that no two adjacent people will stand after flipping their coins around a circular table with 8 people. Each person flips a fair coin, meaning each flip has a probability of $\frac{1}{2}$ for heads (stand) and $\frac{1}{2}$ for tails (sit).
2. **Tot... | 0.25 | 8,057.375 | 7,653.5 | 8,192 |
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$. | 4018021 |
We are given the sequence \( \{a_n\}_{n \geq 0} \) which satisfies the relation:
\[
a_{m+n} + a_{m-n} - m + n - 1 = \frac{1}{2} (a_{2m} + a_{2n})
\]
for all non-negative integers \( m \) and \( n \) with \( m \ge n \). We are also given that \( a_1 = 3 \), and we need to find \( a_{2004} \).
We start by plugging sp... | 0.6875 | 6,258.6875 | 5,379.909091 | 8,192 |
The perimeter of triangle \( \mathrm{ABC} \) is 1. A circle touches side \( \mathrm{AB} \) at point \( P \) and the extension of side \( \mathrm{AC} \) at point \( Q \). A line passing through the midpoints of sides \( \mathrm{AB} \) and \( \mathrm{AC} \) intersects the circumcircle of triangle \( \mathrm{APQ} \) at po... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? | 5 | To solve this problem, we need to compare the number $2.00d5$ with $2.005$ for each possible digit $d$ (where $d$ can be any digit from $0$ to $9$).
1. **Expression of $2.00d5$:**
The number $2.00d5$ can be expressed as $2.00d5 = 2 + 0.00d + 0.0005$. Here, $0.00d$ represents the contribution of the digit $d$ in t... | 0.875 | 4,167 | 3,592 | 8,192 |
Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4.\] | -16 | 0.8125 | 5,629.75 | 5,038.461538 | 8,192 | |
A robot has 6 modules, each needing a base and a cap attached sequentially. In how many different orders can the robot attach its bases and caps, assuming that, on each module, the base must be attached before the cap?
A) $6! \cdot 2^6$
B) $(6!)^2$
C) $12!$
D) $\frac{12!}{2^6}$
E) $2^{12} \cdot 6!$ | \frac{12!}{2^6} | 0 | 5,340.6875 | -1 | 5,340.6875 | |
Calculate the sum of eight occurrences of the term $5^5$:
\[ 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 \]
A) $5^6$
B) $5^7$
C) $5^8$
D) $5^{6.29248125}$
E) $40^5$ | 5^{6.29248125} | 0 | 5,212.25 | -1 | 5,212.25 | |
The logarithm of $27\sqrt[4]{9}\sqrt[3]{9}$ to the base $3$ is: | $4\frac{1}{6}$ | 1. **Rewrite the expression using properties of exponents and logarithms:**
The given expression is $27\sqrt[4]{9}\sqrt[3]{9}$. We know that $27 = 3^3$, $\sqrt[4]{9} = 9^{1/4}$, and $\sqrt[3]{9} = 9^{1/3}$. Since $9 = 3^2$, we can rewrite these as:
\[
27 = 3^3, \quad \sqrt[4]{9} = (3^2)^{1/4} = 3^{2/4} = 3^{1/... | 0 | 2,750.9375 | -1 | 2,750.9375 |
A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probability of the arrow stopping on region $A$ is $\frac{3}{8}$, the probability of it stopping in $B$ is $\frac{1}{4}$, and the probability of it stopping in region $C$ is equal to the probability of it stopping in region $D$. What... | \frac{3}{16} | 1 | 1,257.375 | 1,257.375 | -1 | |
Given an increasing sequence $\{a_n\}$ with $2017$ terms, and all terms are non-zero, $a_{2017}=1$. If two terms $a_i$, $a_j$ are arbitrarily chosen from $\{a_n\}$, when $i < j$, $a_j-a_i$ is still a term in the sequence $\{a_n\}$. Then, the sum of all terms in the sequence $S_{2017}=$ ______. | 1009 | 0.625 | 6,466.75 | 5,431.6 | 8,192 | |
Consider a polynomial $P(x) \in \mathbb{R}[x]$ , with degree $2023$ , such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$ . If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ? | 4046 | 0.125 | 8,101.1875 | 7,610 | 8,171.357143 | |
The Chinese government actively responds to changes in gas emissions and has set a goal to reduce carbon emission intensity by 40% by 2020 compared to 2005. It is known that in 2005, China's carbon emission intensity was about 3 tons per 10,000 yuan, and each year thereafter, the carbon emission intensity decreases by ... | 2030 | 0.3125 | 6,440.5625 | 5,028.4 | 7,082.454545 | |
Ben throws four identical darts. Each hits one of four identical dartboards on the wall. After throwing the four darts, he lists the number of darts that hit each board, from greatest to least. How many different lists are possible? | 5 | 1 | 3,366.0625 | 3,366.0625 | -1 | |
In $\triangle ABC$, $AC=8$, $BC=7$, $\cos B=-\frac{1}{7}$.
(1) Find the measure of angle $A$;
(2) Find the area of $\triangle ABC$. | 6\sqrt{3} | 0.8125 | 4,499.6875 | 4,045.153846 | 6,469.333333 | |
Compute
\[\frac{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{19} \right)}{(1 + 19) \left( 1 + \dfrac{19}{2} \right) \left( 1 + \dfrac{19}{3} \right) \dotsm \left( 1 + \dfrac{19}{17} \right)}.\] | 1 | 0.0625 | 7,390.625 | 7,589 | 7,377.4 | |
In the diagram, every number beginning at 30 equals twice the sum of the two numbers to its immediate left. The value of \( c \) is:
| 10 | \(a\) | 30 | \(b\) | \(c\) |
| :--- | :--- | :--- | :--- | :--- | | 200 | 0.4375 | 749.6875 | 891.428571 | 639.444444 | |
Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \] | 20 | 0.9375 | 3,996.25 | 3,716.533333 | 8,192 | |
Find the smallest four-digit number that satisfies the following system of congruences:
\begin{align*}
3x &\equiv 6 \pmod{12} \\
5x + 20 &\equiv 25 \pmod{15} \\
3x - 2 &\equiv 2x \pmod{35}
\end{align*} | 1274 | 0 | 5,589.875 | -1 | 5,589.875 | |
The values of a function $f(x)$ are given below:
\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 3 & 4 & 5 & 6 & 7 \\ \hline $f(x)$ & 10 & 17 & 26 & 37 & 50 \\ \hline \end{tabular}Evaluate $f^{-1}\left(f^{-1}(50)\times f^{-1}(10)+f^{-1}(26)\right)$. | 5 | 1 | 1,683.125 | 1,683.125 | -1 | |
In how many ways can I choose 2 cooks on a backpacking trip with 8 people if any of the 8 people may be cooks? | 28 | 1 | 1,615.6875 | 1,615.6875 | -1 | |
The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent. | -19\% | 0.25 | 4,078.3125 | 3,992 | 4,107.083333 | |
Given the sequence 1, $\frac{1}{2}$, $\frac{2}{1}$, $\frac{1}{3}$, $\frac{2}{2}$, $\frac{3}{1}$, $\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{2}$, $\frac{4}{1}$, ..., then $\frac{3}{5}$ is the \_\_\_\_\_\_ term of this sequence. | 24 | 0.3125 | 7,070 | 4,943.6 | 8,036.545455 | |
Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,4500),\) and \((0,4500)\). What is the probability that \(x < 3y\)? Express your answer as a common fraction. | \frac{11}{18} | 0 | 6,308.6875 | -1 | 6,308.6875 | |
Alice and Bob are playing the Smallest Positive Integer Game. Alice says, "My number is 24." Bob says, "What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number."
What is the smallest possible number that Bob could have? (Remember that Bob's number has to be a p... | 6 | 0.4375 | 4,809.875 | 1,805.857143 | 7,146.333333 | |
A number $N$ is defined as follows:
\[N=2+22+202+2002+20002+\cdots+2\overbrace{00\ldots000}^{19~0\text{'s}}2\]
When the value of $N$ is simplified, what is the sum of its digits? | 42 | 0 | 8,013.25 | -1 | 8,013.25 | |
What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity? | 18 | 0.875 | 6,885.9375 | 6,699.357143 | 8,192 | |
For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance of getting them right due to my prior knowl... | 534 | 0.375 | 3,667.875 | 2,043.166667 | 4,642.7 | |
Compute the number of degrees in the smallest positive angle $x$ such that
\[8 \sin x \cos^5 x - 8 \sin^5 x \cos x = 1.\] | 7.5^\circ | 0.4375 | 5,591.8125 | 4,504.428571 | 6,437.555556 | |
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Draw a line $l$ passing through the right focus that intersects the ellipse at points $P$ and $Q$. What is the maximum area of the inscribed circle of triangle $\triangle F_{1} P Q$? | \frac{9\pi}{16} | 0 | 8,192 | -1 | 8,192 | |
Let \( a_{0}=1, a_{1}=2 \), and \( n(n+1) a_{n+1}=n(n-1) a_{n}-(n-2) a_{n-1}, n=1,2,3, \cdots \). Find the value of \( \frac{a_{0}}{a_{1}}+\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{50}}{a_{51}} \). | 1326 | 0.1875 | 7,334.4375 | 3,618.333333 | 8,192 | |
Let $f(x)$ be a polynomial such that
\[f(x^2 + 1) = x^4 + 4x^2.\]Find $f(x^2 - 1).$ | x^4 - 4 | 1 | 2,748.25 | 2,748.25 | -1 | |
In the diagram, $AB$ and $CD$ are straight lines. What is the value of $x?$ [asy]
draw((0,0)--(12,0));
draw((0,5)--(12,5));
draw((3,0)--(5,5)--(9,0));
label("$60^\circ$",(5,4.5),W);
label("$50^\circ$",(5.5,4.5),E);
label("$A$",(0,5),W);
label("$C$",(0,0),W);
label("$B$",(12,5),E);
label("$D$",(12,0),E);
label("$120^\ci... | 50 | 0 | 8,176 | -1 | 8,176 | |
Let $ABC$ be a triangle with $AB=5$ , $AC=12$ and incenter $I$ . Let $P$ be the intersection of $AI$ and $BC$ . Define $\omega_B$ and $\omega_C$ to be the circumcircles of $ABP$ and $ACP$ , respectively, with centers $O_B$ and $O_C$ . If the reflection of $BC$ over $AI$ intersects $\omega_B$ ... | \sqrt{109} | 0 | 8,192 | -1 | 8,192 | |
If the graph of the function $f(x)=a^{x-2}-2a (a > 0, a \neq 1)$ always passes through the fixed point $\left(x\_0, \frac{1}{3}\right)$, then the minimum value of the function $f(x)$ on $[0,3]$ is equal to \_\_\_\_\_\_\_\_. | -\frac{1}{3} | 0.25 | 7,245.375 | 4,405.5 | 8,192 | |
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2]. | 10 | 0.1875 | 7,048.375 | 4,589 | 7,615.923077 | |
In a certain country, the airline system is arranged in such a way that any city is connected by airlines with no more than three other cities, and from any city to any other, one can travel with no more than one stopover. What is the maximum number of cities that can be in this country? | 10 | 0.8125 | 4,434.25 | 3,567.076923 | 8,192 | |
Suppose that $x$ and $y$ satisfy $\frac{x-y}{x+y}=9$ and $\frac{xy}{x+y}=-60$. What is the value of $(x+y)+(x-y)+xy$? | -150 | The first equation $\frac{x-y}{x+y}=9$ gives $x-y=9x+9y$ and so $-8x=10y$ or $-4x=5y$. The second equation $\frac{xy}{x+y}=-60$ gives $xy=-60x-60y$. Multiplying this equation by 5 gives $5xy=-300x-300y$ or $x(5y)=-300x-60(5y)$. Since $5y=-4x$, then $x(-4x)=-300x-60(-4x)$ or $-4x^{2}=-60x$. Rearranging, we obtain $4x^{2... | 1 | 2,606.3125 | 2,606.3125 | -1 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases}x=a\cos \varphi \\ y=b\sin \varphi\end{cases} (a > b > 0,$ $\varphi$ is the parameter), and the point $M(2, \sqrt{3})$ on curve $C1$ corresponds to the parameter $\varphi= \frac{\pi}{3}$. Using $O$ as the pole and the po... | \frac{5}{16} | 0.0625 | 7,840.375 | 7,203 | 7,882.866667 | |
For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary. | 0.79 | 0 | 8,192 | -1 | 8,192 | |
The real numbers $x$ and $y$ satisfy the equation $2\cos ^{2}(x+y-1)= \frac {(x+1)^{2}+(y-1)^{2}-2xy}{x-y+1}$. Find the minimum value of $xy$. | \frac{1}{4} | 0.625 | 6,615.5 | 5,743.4 | 8,069 | |
An urn contains nine balls, numbered from 1 to 9. José and Maria each simultaneously draw one ball from the urn. They form a two-digit number with the number on José's ball as the tens digit and the number on Maria's ball as the units digit. What is the probability that this number is even? | 4/9 | 0.9375 | 2,433 | 2,049.066667 | 8,192 | |
On a $10000 order a merchant has a choice between three successive discounts of 20%, 20%, and 10% and three successive discounts of 40%, 5%, and 5%. By choosing the better offer, he can save: | $345 | To determine which discount option provides the greater savings, we need to calculate the final price for each option and then compare them.
#### Option 1: Three successive discounts of 20%, 20%, and 10%
1. **First discount of 20%**:
\[
\$10,000 \times (1 - 0.20) = \$10,000 \times 0.80 = \$8,000
\]
2. **Secon... | 0 | 2,171.25 | -1 | 2,171.25 |
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find... | 507 | Let $A$ be the origin, so $B=(16,0)$ and $C=(20,0).$ Using equilateral triangle properties tells us that $D=(8,8\sqrt3)$ and $E=(18,2\sqrt3)$ as well. Therefore, $M=(9,\sqrt3)$ and $N=(14,4\sqrt3).$ Applying the Shoelace Theorem to triangle $BMN$ gives
\[x=\dfrac 1 2 |16\sqrt3+36\sqrt3+0-(0+14\sqrt3+64\sqrt3)| =13\sqrt... | 0.875 | 3,532.5 | 3,129.214286 | 6,355.5 |
A robot invented a cipher for encoding words: it replaced certain letters of the alphabet with one-digit or two-digit numbers, using only the digits 1, 2, and 3 (different letters were replaced with different numbers). Initially, it encoded itself: ROBOT = 3112131233. After encoding the words CROCODILE and HIPPOPOTAMUS... | 2232331122323323132 | 0 | 8,192 | -1 | 8,192 | |
Given in the tetrahedron P-ABC, PA is perpendicular to the plane ABC, AB=AC=PA=2, and in triangle ABC, ∠BAC=120°, then the volume of the circumscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_. | \frac{20\sqrt{5}\pi}{3} | 0 | 5,959.9375 | -1 | 5,959.9375 | |
Determine the length of side $x$ in the following diagram:
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(2))--cycle);
draw((0,0)--(-2,0)--(0,2*sqrt(2))--cycle);
label("10",(-1, sqrt(2)),NW);
label("$x$",(sqrt(2),sqrt(2)),NE);
draw("$30^{\circ}$",(2.2,0),NW);
draw("$45^{\circ}$",(-1.8,0),NE);
draw(rightanglemar... | \frac{20\sqrt{3}}{3} | 0 | 7,832.375 | -1 | 7,832.375 | |
Given an equilateral triangle with one vertex at the origin and the other two vertices on the parabola $y^2 = 2\sqrt{3}x$, find the length of the side of this equilateral triangle. | 12 | 0.625 | 7,047.9375 | 6,361.5 | 8,192 | |
Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$, where angles are in radians. What is $abc$? | \frac{1}{32} | 1. **Identify the roots and express them in terms of complex exponentials:**
Let $z = e^{\frac{2\pi i}{7}}$. The roots of the polynomial $P(x) = x^3 + ax^2 + bx + c$ are given as $\cos \frac{2\pi}{7}, \cos \frac{4\pi}{7},$ and $\cos \frac{6\pi}{7}$. These can be expressed using Euler's formula:
\[
\cos \theta ... | 0.0625 | 8,144.1875 | 7,427 | 8,192 |
The base of isosceles $\triangle XYZ$ is 30 units and its area is 60 square units. | \sqrt{241} | 0.8125 | 3,056.5 | 2,849.538462 | 3,953.333333 | |
Find the number of four-digit numbers, composed of the digits 1, 2, 3, 4, 5, 6, 7 (each digit can be used no more than once), that are divisible by 15. | 36 | 0.4375 | 7,265.875 | 6,075.142857 | 8,192 | |
Consider all polynomials of the form
\[x^7 + b_6 x^6 + b_5 x^5 + \dots + b_2 x^2 + b_1 x + b_0,\]
where \( b_i \in \{0,1\} \) for all \( 0 \le i \le 6 \). Find the number of such polynomials that have exactly two different integer roots, -1 and 0. | 15 | 0.125 | 8,010.25 | 7,263.5 | 8,116.928571 | |
A system of inequalities defines a region on a coordinate plane as follows:
$$
\begin{cases}
x+y \leq 5 \\
3x+2y \geq 3 \\
x \geq 1 \\
y \geq 1
\end{cases}
$$
Determine the number of units in the length of the longest side of the quadrilateral formed by the region satisfying all these conditions. Express your answer in... | 3\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Let $F(0)=0, F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2} F(n-1)-F(n-2)$ for $n \geq 2$. Determine whether or not $\sum_{n=0}^{\infty} \frac{1}{F\left(2^{n}\right)}$ is a rational number. | 1 | The characteristic equation of our linear recurrence is $x^{2}-\frac{5}{2} x+1=0$, with roots $x_{1}=2$ and $x_{2}=\frac{1}{2}$. So $F(n)=a \cdot 2^{n}+b \cdot\left(\frac{1}{2}\right)^{n}$ with some constants $a, b$. By $F(0)=0$ and $F(1)=\frac{3}{2}$, these constants satisfy $a+b=0$ and $2 a+\frac{b}{2}=\frac{3}{2}$. ... | 0.625 | 6,692.1875 | 5,792.3 | 8,192 |
The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are | 63,65 | To solve the problem, we need to find the factors of $2^{48} - 1$ that lie between $60$ and $70$. We start by factoring $2^{48} - 1$ using the difference of squares repeatedly.
1. **Factorization using difference of squares:**
\[
2^{48} - 1 = (2^{24} - 1)(2^{24} + 1)
\]
We can further factor $2^{24} - 1$ u... | 0 | 7,428.75 | -1 | 7,428.75 |
A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food? | 10/11 | 0.875 | 5,634.8125 | 5,269.5 | 8,192 | |
When the length of a rectangle is increased by $20\%$ and the width increased by $10\%$, by what percent is the area increased? | 32 \% | 1 | 1,785.6875 | 1,785.6875 | -1 | |
In how many ways can a bamboo trunk (a non-uniform natural material) of length 4 meters be cut into three parts, the lengths of which are multiples of 1 decimeter, and from which a triangle can be formed? | 171 | 0 | 8,158.75 | -1 | 8,158.75 | |
What is the sum of the whole-number factors of 24? | 60 | 1 | 2,048.625 | 2,048.625 | -1 | |
Given that $M(m,n)$ is any point on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.
(I) Find the maximum and minimum values of $|MQ|$;
(II) Find the maximum and minimum values of $\frac{n-3}{m+2}$. | 2-\sqrt{3} | 0.4375 | 7,371.6875 | 6,317 | 8,192 | |
How many natural numbers are there whose square and cube together require 10 digits to describe? | 53 | 0.375 | 7,730.6875 | 7,079 | 8,121.7 | |
Given the function $f(x)=2\ln(3x)+8x$, calculate the value of $\lim_{\triangle x \to 0} \frac{f(1-2\triangle x)-f(1)}{\triangle x}$. | -20 | 0.8125 | 4,140.375 | 3,205.384615 | 8,192 | |
In $\triangle ABC$, $\angle C = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle? | 12 + 12\sqrt{2} | 1. **Identify the center of the circle**: Let $O$ be the midpoint of $AB$. Since $ABXY$ and $ACWZ$ are squares, the diagonals of these squares bisect each other at right angles and have equal lengths. Therefore, the perpendicular bisectors of $XY$ and $WZ$ will meet at $O$, making $O$ the center of the circle on which ... | 0.0625 | 8,142.375 | 7,398 | 8,192 |
You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the seco... | 2021 | 0.0625 | 8,149.875 | 8,192 | 8,147.066667 | |
How many positive integers $N$ less than $1000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$? | 412 | For a positive integer $k$, we find the number of positive integers $N$ such that $x^{\lfloor x\rfloor}=N$ has a solution with ${\lfloor x\rfloor}=k$. Then $x=\sqrt[k]{N}$, and because $k \le x < k+1$, we have $k^k \le x^k < (k+1)^k$, and because $(k+1)^k$ is an integer, we get $k^k \le x^k \le (k+1)^k-1$. The number o... | 0.3125 | 7,852.6875 | 7,106.2 | 8,192 |
A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane,... | 810 | Go to 2020 AMC 10A #19, and connect all of the centers of the faces on the dodecahedron to get the icosahedron. The answer is $\boxed{810}$. | 0 | 8,192 | -1 | 8,192 |
Find the smallest integer \( n > 1 \) such that \(\frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n}\) is a square. | 337 | 0.1875 | 8,052.375 | 7,447.333333 | 8,192 |
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