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 |
|---|---|---|---|---|---|---|
In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table? | 256 | 0.1875 | 7,793.6875 | 6,067.666667 | 8,192 | |
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and p... | 84 | We will analyze the problem by considering different cases based on the distribution of crops in the sections. The key restrictions are that corn cannot be adjacent to wheat, and soybeans cannot be adjacent to potatoes.
#### Case 1: All sections have the same crop.
- There are 4 types of crops (corn, wheat, soybeans, ... | 0 | 8,192 | -1 | 8,192 |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 0.3125 | 5,984.3125 | 4,160.8 | 6,813.181818 | |
Given that $3^8 \cdot 5^2 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$. | 407 | 1. **Express the given equation in terms of $a$ and $b$:**
Given the equation $3^8 \cdot 5^2 = a^b$, we need to find $a$ and $b$ such that $a^b$ equals $3^8 \cdot 5^2$.
2. **Simplify the expression:**
\[
3^8 \cdot 5^2 = (3^4)^2 \cdot 5^2 = (81)^2 \cdot 25 = 81 \cdot 81 \cdot 25
\]
We can further simplif... | 0.875 | 5,471.0625 | 5,082.357143 | 8,192 |
Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money? | 4 | 1. **Convert Etienne's euros to dollars**: Given that 1 euro is equivalent to 1.3 dollars, we can calculate the dollar value of Etienne's 400 euros as follows:
\[
400 \text{ euros} \times 1.3 \frac{\text{dollars}}{\text{euro}} = 520 \text{ dollars}
\]
2. **Compare Etienne's dollar amount to Diana's dollar amo... | 1 | 1,290.875 | 1,290.875 | -1 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that fits inside the cylinder? | 2\sqrt{61} | 0.875 | 3,534.1875 | 2,868.785714 | 8,192 | |
Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\] | \frac{3}{9\sqrt{5} + 4\sqrt{7} - 10} | 0 | 6,798.5625 | -1 | 6,798.5625 | |
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and K... | 163 | Consider the unit cicle of radius 50. Assume that they start at points $(-50,100)$ and $(-50,-100).$ Then at time $t$, they end up at points $(-50+t,100)$ and $(-50+3t,-100).$ The equation of the line connecting these points and the equation of the circle are \begin{align}y&=-\frac{100}{t}x+200-\frac{5000}{t}\\50^2&=x^... | 0 | 8,192 | -1 | 8,192 |
The expression $\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: | \frac{15}{8} | 1. **Simplify the Numerator**:
The expression $1^{4y-1}$ simplifies to $1$ for any real number $y$, because any power of $1$ is $1$. Thus, the expression becomes:
\[
\frac{1^{4y-1}}{5^{-1} + 3^{-1}} = \frac{1}{5^{-1} + 3^{-1}}
\]
2. **Simplify the Denominator**:
We rewrite the terms in the denominator ... | 1 | 1,286.125 | 1,286.125 | -1 |
Numbers from 1 to 9 are arranged in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal is 7, and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
| 25 | 0 | 7,678.75 | -1 | 7,678.75 | |
Given vectors $\overrightarrow {a}=(\sin(2x+ \frac {\pi}{6}), 1)$, $\overrightarrow {b}=( \sqrt {3}, \cos(2x+ \frac {\pi}{6}))$, and the function $f(x)= \overrightarrow {a} \cdot \overrightarrow {b}$.
(Ⅰ) Find the interval where the function $f(x)$ is monotonically decreasing;
(Ⅱ) In $\triangle ABC$, where $A$, $B$,... | \sqrt {3}+2 \sqrt {2} | 0 | 6,303.375 | -1 | 6,303.375 | |
Find the minimum value of the function \( f(x) = 3^x - 9^x \) for real numbers \( x \). | \frac{1}{4} | 0.0625 | 8,127.1875 | 7,831 | 8,146.933333 | |
If $\frac{x}{x-1} = \frac{y^2 + 2y - 1}{y^2 + 2y - 2},$ then $x$ equals | y^2 + 2y - 1 | 1. Start with the given equation:
\[
\frac{x}{x-1} = \frac{y^2 + 2y - 1}{y^2 + 2y - 2}
\]
2. Cross multiply to eliminate the fractions:
\[
x(y^2 + 2y - 2) = (x-1)(y^2 + 2y - 1)
\]
3. Expand both sides of the equation:
\[
xy^2 + 2xy - 2x = xy^2 + 2xy - y^2 - 2y - x + 1
\]
4. Rearrange the e... | 1 | 2,640.4375 | 2,640.4375 | -1 |
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying \( f(m+n) \geq f(m) + f(f(n)) - 1 \) for all \( m, n \in \mathbb{N} \). What values can \( f(2019) \) take? | 2019 | 0 | 8,192 | -1 | 8,192 | |
Use \((a, b)\) to represent the greatest common divisor of \(a\) and \(b\). Let \(n\) be an integer greater than 2021, and \((63, n+120) = 21\) and \((n+63, 120) = 60\). What is the sum of the digits of the smallest \(n\) that satisfies the above conditions? | 15 | 0 | 7,527.875 | -1 | 7,527.875 | |
Let \( S = \{1, 2, 3, 4, \ldots, 16\} \). Each of the following subsets of \( S \):
\[ \{6\},\{1, 2, 3\}, \{5, 7, 9, 10, 11, 12\}, \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \]
has the property that the sum of all its elements is a multiple of 3. Find the total number of non-empty subsets \( A \) of \( S \) such that the sum of all... | 21855 | 0.125 | 7,943 | 6,200 | 8,192 | |
The domain of the function \( f(x) \) is \( D \). If for any \( x_{1}, x_{2} \in D \), when \( x_{1} < x_{2} \), it holds that \( f(x_{1}) \leq f(x_{2}) \), then \( f(x) \) is called a non-decreasing function on \( D \). Suppose that the function \( f(x) \) is non-decreasing on \( [0,1] \) and satisfies the following t... | \frac{3}{4} | 0.125 | 7,990.4375 | 6,579.5 | 8,192 | |
From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes. | 60 | 0.75 | 3,532.8125 | 2,758.583333 | 5,855.5 | |
Given triangle \( ABC \) with \( AB = 12 \), \( BC = 10 \), and \( \angle ABC = 120^\circ \), find \( R^2 \), where \( R \) is the radius of the smallest circle that can contain this triangle. | 91 | 0 | 5,348.625 | -1 | 5,348.625 | |
40 pikes were released into a pond. A pike is considered well-fed if it has eaten three other pikes (whether well-fed or hungry). What is the maximum number of pikes that can be well-fed? | 13 | 0.125 | 7,877.125 | 6,659.5 | 8,051.071429 | |
Mr. Canada chooses a positive real $a$ uniformly at random from $(0,1]$, chooses a positive real $b$ uniformly at random from $(0,1]$, and then sets $c=a /(a+b)$. What is the probability that $c$ lies between $1 / 4$ and $3 / 4$ ? | 2 / 3 | From $c \geq 1 / 4$ we get $$ \frac{a}{a+b} \geq \frac{1}{4} \Longleftrightarrow b \leq 3 a $$ and similarly $c \leq 3 / 4$ gives $$ \frac{a}{a+b} \leq \frac{3}{4} \Longleftrightarrow a \leq 3 b $$ Choosing $a$ and $b$ randomly from $[0,1]$ is equivalent to choosing a single point uniformly and randomly from the unit s... | 0.3125 | 7,831.1875 | 7,037.4 | 8,192 |
In the encoded equality $A B + A B + A B + A B + A B + A B + A B + A B + A B = A A B$, digits are replaced with letters: the same digits with the same letter, and different digits with different letters. Find all possible decipherings. (I. Rubanov) | 25 | 0 | 4,898.0625 | -1 | 4,898.0625 | |
If $\tan (\alpha+\beta)= \frac {3}{4}$ and $\tan (\alpha- \frac {\pi}{4})= \frac {1}{2}$, find the value of $\tan (\beta+ \frac {\pi}{4})$. | \frac {2}{11} | 0.9375 | 3,130.6875 | 2,793.266667 | 8,192 | |
The symbol \( R_{k} \) represents an integer whose decimal representation consists of \( k \) consecutive 1s. For example, \( R_{3} = 111 \), \( R_{5} = 11111 \), and so on. If \( R_{4} \) divides \( R_{24} \), the quotient \( Q = \frac{R_{24}}{R_{4}} \) is an integer, and its decimal representation contains only the d... | 15 | 0.1875 | 7,630.375 | 5,454 | 8,132.615385 | |
What is the value of the expression $ rac{3}{10}+ rac{3}{100}+ rac{3}{1000}$? | 0.333 | Evaluating, $ rac{3}{10}+ rac{3}{100}+ rac{3}{1000}=0.3+0.03+0.003=0.333$. | 0.125 | 523.875 | 396.5 | 542.071429 |
In the diagram, $\triangle PQR$ is right-angled at $P$ and has $PQ=2$ and $PR=2\sqrt{3}$. Altitude $PL$ intersects median $RM$ at $F$. What is the length of $PF$? [asy]
draw((0,0)--(10,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*sqrt(3))--(5,0));
draw((6.68,3.86)--(7.17,3.01)--(7.99,3.49));
label(... | \frac{4\sqrt{3}}{7} | 0 | 4,090.0625 | -1 | 4,090.0625 | |
Consider a circle with radius $4$, and there are numerous line segments of length $6$ that are tangent to the circle at their midpoints. Compute the area of the region consisting of all such line segments.
A) $8\pi$
B) $7\pi$
C) $9\pi$
D) $10\pi$ | 9\pi | 0 | 8,192 | -1 | 8,192 | |
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area 3 and triangle $ACD$ has area 4, find the area of triangle $ABC$. | \frac{37}{7} | A rotation by $120^{\circ}$ about the center of the circle will take $ABE$ to $BCD$, so $BCD$ has area 3. Let $AD=x, BD=y$, and observe that $\angle ADC=\angle CDB=60^{\circ}$. By Ptolemy's Theorem, $CD=x+y$. We have $$4=[ACD]=\frac{1}{2} AD \cdot CD \cdot \sin 60^{\circ}=\frac{\sqrt{3}}{4} x(x+y)$$ $$3=[BCD]=\frac{1}{... | 0 | 8,192 | -1 | 8,192 |
The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$? | 39 | To solve this problem, we need to understand the symmetry in the sum of the numbers on the faces of the dice when rolled. The minimum sum occurs when all dice show a $1$, which is $7 \times 1 = 7$. The maximum sum occurs when all dice show a $6$, which is $7 \times 6 = 42$.
The sums are symmetric around the average v... | 1 | 2,161.8125 | 2,161.8125 | -1 |
When measuring a part, random errors occur that follow a normal distribution with a parameter $\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15$ mm. | 0.8664 | 0.5625 | 4,504.1875 | 4,185.222222 | 4,914.285714 | |
Given a triangular pyramid $D-ABC$ with all four vertices lying on the surface of a sphere $O$, if $DC\bot $ plane $ABC$, $\angle ACB=60^{\circ}$, $AB=3\sqrt{2}$, and $DC=2\sqrt{3}$, calculate the surface area of sphere $O$. | 36\pi | 0.4375 | 7,130.125 | 5,764.857143 | 8,192 | |
Suppose that $f(x)=4x+5$. What is $f^{-1}(f^{-1}(9))$? | -1 | 1 | 2,018.625 | 2,018.625 | -1 | |
There are \( n \) players participating in a round-robin chess tournament, where each player competes exactly once against every other player. The winner of a match earns 3 points, a draw gives each player 1 point, and the loser earns 0 points. If the total sum of all players' points is 120, what is the maximum number ... | 11 | 0.5625 | 6,576.8125 | 5,320.555556 | 8,192 | |
Define a sequence by \( a_0 = \frac{1}{3} \) and \( a_n = 1 + (a_{n-1} - 1)^3 \). Compute the infinite product \( a_0 a_1 a_2 \dotsm \). | \frac{3}{5} | 0.0625 | 7,953.6875 | 4,379 | 8,192 | |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | 1 - 2x | 0.9375 | 3,830.125 | 3,796 | 4,342 | |
Given the function $$f(x)=\cos\omega x\cdot \sin(\omega x- \frac {\pi}{3})+ \sqrt {3}\cos^{2}\omega x- \frac { \sqrt {3}}{4}(\omega＞0，x\in\mathbb{R})$$, and the distance from a center of symmetry of the graph of $y=f(x)$ to the nearest axis of symmetry is $$\frac {\pi}{4}$$.
(Ⅰ) Find the value of $\omega$ and the equat... | \frac {3+2 \sqrt {6}}{3} | 0 | 8,148.4375 | -1 | 8,148.4375 | |
A spider is on the edge of a ceiling of a circular room with a radius of 65 feet. The spider moves straight across the ceiling to the opposite edge, passing through the circle's center. It then moves directly to another point on the edge of the circle, not passing through the center. The final segment of the journey is... | 220 + 20\sqrt{22} | 0.25 | 6,716.375 | 5,633.75 | 7,077.25 | |
Find $1_6 + 2_6 + 3_6 + \cdots + 45_6$. Express your answer in base $6$. | 2003 | 0 | 5,838.3125 | -1 | 5,838.3125 | |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $2acosC + ccosA = b$.
(I) Find the measure of angle $C$;
(II) Find the maximum value of $sinAcosB + sinB$. | \frac{5}{4} | 0.9375 | 3,521.625 | 3,210.266667 | 8,192 | |
Given an ellipse $C$ with one focus at $F_{1}(2,0)$ and the corresponding directrix $x=8$, and eccentricity $e=\frac{1}{2}$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Find the length of the chord cut from the ellipse $C$ by a line passing through the other focus and having a slope of $45^{\circ}$. | \frac{48}{7} | 0.5 | 6,669.5 | 5,516.875 | 7,822.125 | |
Given the function $f(x)= \sqrt {3}\cos ( \frac {π}{2}+x)\cdot \cos x+\sin ^{2}x$, where $x\in R$.
(I) Find the interval where $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, angles $A$, $B$, and $C$ have corresponding opposite sides $a$, $b$, and $c$. If $B= \frac {π}{4}$, $a=2$, and angle $A$ satisfies $... | \frac {3+ \sqrt {3}}{3} | 0 | 7,640.1875 | -1 | 7,640.1875 | |
Solve for $n$, if $9^n\cdot9^n\cdot9^n\cdot9^n=81^4$. | 2 | 1 | 1,731.375 | 1,731.375 | -1 | |
In a group of five friends, Amy is taller than Carla. Dan is shorter than Eric but taller than Bob. Eric is shorter than Carla. Who is the shortest? | Bob | We use $A, B, C, D, E$ to represent Amy, Bob, Carla, Dan, and Eric, respectively. We use the greater than symbol $(>)$ to represent 'is taller than' and the less than symbol $(<)$ to represent 'is shorter than'. From the first bullet, $A > C$. From the second bullet, $D < E$ and $D > B$ so $E > D > B$. From the third b... | 0.5 | 363.75 | 338 | 389.5 |
Determine the value of \(\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot \ln \left(1+\frac{1}{2 n}\right) \cdot \ln \left(1+\frac{1}{2 n+1}\right)\). | \frac{1}{3} \ln ^{3}(2) | Define \(f(n)=\ln \left(\frac{n+1}{n}\right)\) for \(n \geq 1\), and observe that \(f(2 n)+f(2 n+1)=f(n)\). The well-known inequality \(\ln (1+x) \leq x\) implies \(f(n) \leq 1 / n\). Furthermore introduce \(g(n)=\sum_{k=n}^{2 n-1} f^{3}(k)<n f^{3}(n) \leq 1 / n^{2}\). Then \(g(n)-g(n+1) =f^{3}(n)-f^{3}(2 n)-f^{3}(2 n+... | 0 | 8,178.3125 | -1 | 8,178.3125 |
A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$. | 44 | 0 | 7,901.4375 | -1 | 7,901.4375 | |
There are 2008 red cards and 2008 white cards. 2008 players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.
$ (*)$ If you have more than one red car... | 1004 | 0.0625 | 7,508.9375 | 6,482 | 7,577.4 | |
Given the quadratic equation $ax^{2}+bx+c=0$ with $a > 0$ and $b, c \in \mathbb{R}$, and the roots of the equation lying in the interval $(0, 2)$, determine the minimum value of the real number $a$ given that $25a+10b+4c \geqslant 4$ for $c \geqslant 1$. | \frac{16}{25} | 0 | 8,164.5625 | -1 | 8,164.5625 | |
Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number. | 426 | 0.8125 | 6,257.625 | 5,958 | 7,556 | |
There are three types of snacks for the kitten. It eats a stick of cat food every 1 day, an egg yolk every 2 days, and nutritional cream every 3 days. The kitten ate cat stick and nutritional cream on March 23, and ate cat stick and egg yolk on March 25. Which day in March does the kitten eat all three types of snacks ... | 29 | 0.375 | 5,958.8125 | 4,827.333333 | 6,637.7 | |
How many possible sequences of the experiment are there, given that 6 procedures need to be implemented in sequence, procedure A can only appear in the first or last step, and procedures B and C must be adjacent when implemented? | 96 | 0.8125 | 5,701.5 | 5,126.769231 | 8,192 | |
How many students chose Greek food if 200 students were asked to choose between pizza, Thai food, or Greek food, and the circle graph shows the results? | 100 | Of the 200 students, $50 \%$ (or one-half) of the students chose Greek food. Since one-half of 200 is 100, then 100 students chose Greek food. | 0 | 5,429.75 | -1 | 5,429.75 |
The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is: | \frac{1}{5} | To solve the expression $[2 - 3(2 - 3)^{-1}]^{-1}$, we start by simplifying the innermost expression first.
1. **Simplify $(2 - 3)^{-1}$:**
\[
(2 - 3)^{-1} = (-1)^{-1} = -1
\]
2. **Substitute back into the expression:**
\[
2 - 3(-1) = 2 + 3 = 5
\]
3. **Now, simplify the outer inverse:**
\[
[5... | 1 | 2,436.5 | 2,436.5 | -1 |
Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$. | 15 | 0 | 7,953.4375 | -1 | 7,953.4375 | |
Two particles move along the edges of a square $ABCD$ with \[A \Rightarrow B \Rightarrow C \Rightarrow D \Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at vertex $A$, and the other starts at the midpoint of side $CD$. The midpoint of the line segment joining the two particles traces o... | \frac{1}{4} | 0 | 8,192 | -1 | 8,192 | |
Square \(ABCD\) is inscribed in circle \(\omega\) with radius 10. Four additional squares are drawn inside \(\omega\) but outside \(ABCD\) such that the lengths of their diagonals are as large as possible. A sixth square is drawn by connecting the centers of the four aforementioned small squares. Find the area of the s... | 144 | Let \(DEGF\) denote the small square that shares a side with \(AB\), where \(D\) and \(E\) lie on \(AB\). Let \(O\) denote the center of \(\omega, K\) denote the midpoint of \(FG\), and \(H\) denote the center of \(DEGF\). The area of the sixth square is \(2 \cdot \mathrm{OH}^{2}\). Let \(KF=x\). Since \(KF^{2}+OK^{2}=... | 0 | 8,192 | -1 | 8,192 |
Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$ , inclusive, such that, if $$ q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}}, $$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$ .
*Proposed by Ankit Bisain* | 132 | 0.125 | 8,008.9375 | 7,880 | 8,027.357143 | |
The maximum value of the function \( y = \frac{\sin x \cos x}{1 + \sin x + \cos x} \) is $\quad$ . | \frac{\sqrt{2} - 1}{2} | 0 | 7,747.375 | -1 | 7,747.375 | |
Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $- \frac{1}{2}$, the sum of the first $n$ terms is $S_n$. If for any $n \in N^*$, it holds that $S_n - \frac{1}{S_n} \in [s, t]$, then the minimum value of $t-s$ is \_\_\_\_\_\_. | \frac{17}{12} | 0.4375 | 7,815.25 | 7,330.857143 | 8,192 | |
Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the ... | 840 | 0.0625 | 7,586.4375 | 4,107 | 7,818.4 | |
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$ th time, for any nonnegative integer $n$ , he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$ . Find $a + b$ .
*Proposed by Isaac Ch... | 201 | 0 | 7,931.125 | -1 | 7,931.125 | |
There are three two-digit numbers $A$, $B$, and $C$.
- $A$ is a perfect square, and each of its digits is also a perfect square.
- $B$ is a prime number, and each of its digits is also a prime number, and their sum is also a prime number.
- $C$ is a composite number, and each of its digits is also a composite number... | 120 | 1 | 4,059.25 | 4,059.25 | -1 | |
Find the value of \(\cos ^{5} \frac{\pi}{9}+\cos ^{5} \frac{5 \pi}{9}+\cos ^{5} \frac{7 \pi}{9}\). | \frac{15}{32} | 0 | 8,192 | -1 | 8,192 | |
The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true? | 1 | 1 | 1,682.25 | 1,682.25 | -1 | |
Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$. | 12 | 1 | 1,541.8125 | 1,541.8125 | -1 | |
Given that $O$ is the circumcenter of $\triangle ABC$, and $D$ is the midpoint of $BC$, if $\overrightarrow{AO} \cdot \overrightarrow{AD} = 4$ and $BC = 2\sqrt{6}$, find the length of $AD$. | \sqrt{2} | 0.6875 | 6,925.9375 | 6,350.454545 | 8,192 | |
Kelvin and 15 other frogs are in a meeting, for a total of 16 frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is cool if for each of the 16 frogs, the number of friends they made during the meeting is a multiple of 4. S... | 1167 | Consider the multivariate polynomial $$\prod_{1 \leq i<j \leq 16}\left(1+x_{i} x_{j}\right)$$ We're going to filter this by summing over all $4^{16} 16$-tuples $\left(x_{1}, x_{2}, \ldots, x_{16}\right)$ such that $x_{j}= \pm 1, \pm i$. Most of these evaluate to 0 because $i^{2}=(-i)^{2}=-1$, and $1 \cdot-1=-1$. If you... | 0 | 7,852.375 | -1 | 7,852.375 |
In a larger geoboard grid of $7 \times 7$, points are evenly spaced vertically and horizontally. Points $A$ and $B$ are at $(3,3)$ and $(5,3)$ respectively. How many of the remaining points in the grid will allow for triangle $ABC$ to be isosceles? | 10 | 0 | 7,769.125 | -1 | 7,769.125 | |
Six bags of marbles contain $18, 19, 21, 23, 25$ and $34$ marbles, respectively. One bag contains chipped marbles only. The other $5$ bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, h... | 23 | 1. **Identify the total number of marbles:**
The total number of marbles in all bags is $18 + 19 + 21 + 23 + 25 + 34 = 140$.
2. **Set up the relationship between Jane's and George's marbles:**
Let the number of marbles George has be $x$. Then, Jane has $2x$ marbles. Since one bag remains, the total number of... | 0.8125 | 4,281.0625 | 3,378.538462 | 8,192 |
Given the sequence $\{a_n\}$ that satisfies $a_1=1$, $a_2=2$, and $2na_n=(n-1)a_{n-1}+(n+1)a_{n+1}$ for $n \geq 2$ and $n \in \mathbb{N}^*$, find the value of $a_{18}$. | \frac{26}{9} | 0.1875 | 7,818.1875 | 6,198.333333 | 8,192 | |
Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have? | 1950 | 1. **Define Variables:**
Let $P$ represent the number of students in Portia's high school and $L$ represent the number of students in Lara's high school.
2. **Set Up Equations:**
From the problem, we know:
\[ P = 3L \]
\[ P + L = 2600 \]
3. **Substitute and Solve for $L$:**
Substitute $P = 3L$ into the... | 1 | 1,645.25 | 1,645.25 | -1 |
William is taking the 25-question, multiple choice American Mathematics Competition. Each question has five answer choices. William guesses random answers for the last four questions. What is the probability that he will get at least one of these final four questions right? | \frac{369}{625} | 0.8125 | 3,444.1875 | 2,656.230769 | 6,858.666667 | |
The [Fibonacci sequence](https://artofproblemsolving.com/wiki/index.php/Fibonacci_sequence) $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten [digits](https://artofproblemsolving.com/wiki/index.php/Digit) is the last to appear in the units p... | 6 | To solve this problem, we need to determine the first occurrence of each digit from 0 to 9 in the units position of the Fibonacci sequence. We can simplify the problem by considering the Fibonacci sequence modulo 10, which will directly give us the units digit of each Fibonacci number.
The Fibonacci sequence is define... | 0.125 | 6,923.3125 | 5,652 | 7,104.928571 |
What is the diameter of the circle inscribed in triangle $ABC$ if $AB = 11,$ $AC=6,$ and $BC=7$? Express your answer in simplest radical form. | \sqrt{10} | 1 | 2,934.125 | 2,934.125 | -1 | |
Given the expansion of $(1+x){(x-\frac{2}{x})}^{3}$, calculate the coefficient of $x$. | -6 | 0.8125 | 5,903.1875 | 5,375 | 8,192 | |
Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $R$. Meanwhile, a secant from $Q$ intersects $C$ at points $D$ and $E$, such that $QD < QE$. If $QD = 4$ and $QR = ED - QD$, then what is $QE$? | 16 | 1 | 2,008.4375 | 2,008.4375 | -1 | |
Triangle \(\triangle A B C\) has \(A B=21, B C=55\), and \(C A=56\). There are two points \(P\) in the plane of \(\triangle A B C\) for which \(\angle B A P=\angle C A P\) and \(\angle B P C=90^{\circ}\). Find the distance between them. | \frac{5}{2} \sqrt{409} | Let \(P_{1}\) and \(P_{2}\) be the two possible points \(P\), with \(A P_{1}<A P_{2}\). Both lie on the \(\angle A\)-bisector and the circle \(\gamma\) with diameter \(B C\). Let \(D\) be the point where the \(\angle A\)-bisector intersects \(B C\), let \(M\) be the midpoint of \(B C\), and let \(X\) be the foot of the... | 0 | 7,984.1875 | -1 | 7,984.1875 |
The polynomial $f(x)=x^4+ax^3+bx^2+cx+d$ has real coefficients, and $f(2i)=f(2+i)=0$. What is $a+b+c+d$? | 9 | 0.875 | 4,242.4375 | 3,678.214286 | 8,192 | |
A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is found that if the price o... | 120 | 0.3125 | 7,338.1875 | 6,245.4 | 7,834.909091 | |
Given that
$$
\begin{array}{l}
a + b + c = 5, \\
a^2 + b^2 + c^2 = 15, \\
a^3 + b^3 + c^3 = 47.
\end{array}
$$
Find the value of \((a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)\). | 625 | 0.5625 | 6,569.5625 | 5,682.222222 | 7,710.428571 | |
In \(\triangle PMO\), \(PM = 6\sqrt{3}\), \(PO = 12\sqrt{3}\), and \(S\) is a point on \(MO\) such that \(PS\) is the angle bisector of \(\angle MPO\). Let \(T\) be the reflection of \(S\) across \(PM\). If \(PO\) is parallel to \(MT\), find the length of \(OT\). | 2\sqrt{183} | 0.875 | 6,675.0625 | 6,458.357143 | 8,192 | |
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction. | \frac{7}{144} | 1 | 1,681.625 | 1,681.625 | -1 | |
Given $f(x) = x^{3} + 3xf''(2)$, then $f(2) = \_\_\_\_\_\_$. | -28 | 0 | 4,144.5625 | -1 | 4,144.5625 | |
Let \(a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of all positive integers that are relatively prime to 75, where \(a_{1}<a_{2}<a_{3}<\cdots\). (The first five terms of the sequence are: \(a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8\).) Find the value of \(a_{2008}\). | 3764 | 0.1875 | 7,052.375 | 6,057.333333 | 7,282 | |
Evgeny is laying tiles on the floor of his living room, which measures 12 by 16 meters. He plans to place square tiles of size 1 m × 1 m along the border of the room and to lay the rest of the floor with square tiles of size 2 m × 2 m. How many tiles will he need in total? | 87 | 0.625 | 932.875 | 954.6 | 896.666667 | |
Estimate $N=\prod_{n=1}^{\infty} n^{n^{-1.25}}$. An estimate of $E>0$ will receive $\lfloor 22 \min (N / E, E / N)\rfloor$ points. | 9000000 | We approximate $\ln N=\sum_{n=1}^{\infty} \frac{\ln n}{n^{5 / 4}}$ with an integral as $\int_{1}^{\infty} \frac{\ln x}{x^{5 / 4}} d x =\left.\left(-4 x^{-1 / 4} \ln x-16 x^{-1 / 4}\right)\right|_{1} ^{\infty} =16$. Therefore $e^{16}$ is a good approximation. We can estimate $e^{16}$ by repeated squaring: $e \approx 2.7... | 0 | 8,192 | -1 | 8,192 |
There exists a constant $c,$ so that among all chords $\overline{AB}$ of the parabola $y = x^2$ passing through $C = (0,c),$
\[t = \frac{1}{AC} + \frac{1}{BC}\]is a fixed constant. Find the constant $t.$
[asy]
unitsize(1 cm);
real parab (real x) {
return(x^2);
}
pair A, B, C;
A = (1.7,parab(1.7));
B = (-1,parab(... | 4 | 0.25 | 7,501.125 | 5,428.5 | 8,192 | |
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime. | n=1,2 |
We are tasked with finding all natural numbers \( n \) such that every natural number with \( n - 1 \) digits being \( 1 \) and one digit being \( 7 \) is prime. To explore this, consider the structure of such numbers.
For a given \( n \), any number of this kind can be represented as \( 111\ldots17111\ldots1 \), whe... | 0 | 7,837.8125 | -1 | 7,837.8125 |
Given the regular octagon $ABCDEFGH$ with its center at $J$, and each of the vertices and the center associated with the digits 1 through 9, with each digit used once, such that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal, determine the number of ways in which this can be done. | 1152 | 0.25 | 7,564.6875 | 7,355.5 | 7,634.416667 | |
You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime p... | 67 | 0.5 | 5,887.5625 | 5,255.625 | 6,519.5 | |
The graph of the equation $x + 2y + 3 = 0$ is perpendicular to the graph of the equation $ax + 2y + 3 = 0$. What is the value of $a$? | -4 | 1 | 2,052.5 | 2,052.5 | -1 | |
The graph of $r = \sin \theta$ is also seen in a circular form but pertains to a vertical alignment. Find the smallest value of $t$ so that when $r = \sin \theta$ is plotted for $0 \le \theta \le t,$ the resulting graph forms a complete circle. | 2\pi | 0.375 | 7,835.4375 | 7,456 | 8,063.1 | |
Find all real numbers $x$ such that \[\frac{x^3+2x^2}{x^2+3x+2} + x = -6.\]Enter all the solutions, separated by commas. | -\tfrac32 | 1 | 3,692 | 3,692 | -1 | |
In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters? | 89 | 0 | 8,011.0625 | -1 | 8,011.0625 | |
What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal. | 1.25 | 1 | 2,821.5 | 2,821.5 | -1 | |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | \frac{\sqrt{15}}{3} | 0 | 4,469.875 | -1 | 4,469.875 | |
Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction. | \frac{1}{44} | 0.75 | 3,564.3125 | 2,621.166667 | 6,393.75 | |
The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$. | 2y | 1 | 2,394 | 2,394 | -1 | |
Given a rectangle with dimensions \(100 \times 101\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal. | 200 | 0.6875 | 6,970.75 | 6,415.636364 | 8,192 | |
In triangle $ABC$ the medians $\overline{AD}$ and $\overline{CE}$ have lengths $18$ and $27$, respectively, and $AB=24$. Extend $\overline{CE}$ to intersect the circumcircle of $ABC$ at $F$. The area of triangle $AFB$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any ... | 63 | Use the same diagram as in Solution 1. Call the centroid $P$. It should be clear that $PE=9$, and likewise $AP=12$, $AE=12$. Then, $\sin \angle AEP = \frac{\sqrt{55}}{8}$. Power of a Point on $E$ gives $FE=\frac{16}{3}$, and the area of $AFB$ is $AE * EF* \sin \angle AEP$, which is twice the area of $AEF$ or $FEB$ (the... | 0 | 8,192 | -1 | 8,192 |
Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base. | \frac{1}{4} | 0.75 | 6,104.8125 | 5,409.083333 | 8,192 |
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