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 |
|---|---|---|---|---|---|---|
The curve $y = \sin x$ cuts the line whose equation is $y = \sin 70^\circ$ into segments having the successive ratios
\[\dots p : q : p : q \dots\]with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p,q).$ | (1,8) | 0.6875 | 6,000.8125 | 5,004.818182 | 8,192 | |
Given the ellipse $C$: $mx^{2}+3my^{2}=1$ ($m > 0$) with a major axis length of $2\sqrt{6}$, and $O$ is the origin.
$(1)$ Find the equation of the ellipse $C$.
$(2)$ Let point $A(3,0)$, point $B$ be on the $y$-axis, and point $P$ be on the ellipse $C$ and to the right of the $y$-axis. If $BA=BP$, find the minimum val... | 3\sqrt{3} | 0.375 | 7,928.8125 | 7,788.666667 | 8,012.9 | |
A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from end to end, counting two halves as one whole... | 23 | 0 | 8,192 | -1 | 8,192 | |
If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 10 | 0 | 8,192 | -1 | 8,192 | |
The diagram below shows a $4\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j)); [/asy]
Define a growing path to be a sequence of distinc... | 240 | 0 | 8,173.625 | -1 | 8,173.625 | |
How many one-fourths are there in $\frac{7}{2}$? | 14 | 1 | 1,506.4375 | 1,506.4375 | -1 | |
During the New Year, Xiaoming's family bought many bottles of juice. On New Year's Eve, they drank half of the total amount minus 1 bottle. On the first day of the New Year, they drank half of the remaining amount again. On the second day of the New Year, they drank half of the remaining amount plus 1 bottle, leaving t... | 22 | 0.4375 | 1,094.875 | 1,214.857143 | 1,001.555556 | |
If triangle $PQR$ has sides of length $PQ = 7$, $PR = 8$, and $QR = 6$, then calculate
\[
\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.
\] | \frac{16}{7} | 0.5625 | 5,302.0625 | 3,840.333333 | 7,181.428571 | |
The inverse of $f(x) = \frac{2x-1}{x+5}$ may be written in the form $f^{-1}(x)=\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are real numbers. Find $a/c$. | -5 | 0.9375 | 3,711 | 3,412.266667 | 8,192 | |
Given that $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $3\pi < \alpha < \frac{7}{2}\pi$, find $\cos \alpha + \sin \alpha$. | -\sqrt{2} | 0.4375 | 6,592.25 | 6,697.428571 | 6,510.444444 | |
In triangle $ABC$, $AB = 16$, $AC = 24$, $BC = 19$, and $AD$ is an angle bisector. Find the ratio of the area of triangle $ABD$ to the area of triangle $ACD$. (Express your answer as a fraction in lowest terms.) | \frac{2}{3} | 1 | 2,417.0625 | 2,417.0625 | -1 | |
In a large box of ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{6}$ are orange, and the remaining 40 are black. How many of the ribbons are orange? | 27 | 0 | 7,644.25 | -1 | 7,644.25 | |
A herd of 183 elephants could drink the lake in 1 day, and a herd of 37 elephants could do it in 5 days.
In how many days will one elephant drink the lake? | 365 | 0 | 675.6875 | -1 | 675.6875 | |
Given that in △ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively, and $b^{2}=c^{2}+a^{2}- \sqrt {2}ac$.
(I) Find the value of angle B;
(II) If $a= \sqrt {2}$ and $cosA= \frac {4}{5}$, find the area of △ABC. | \frac {7}{6} | 0.75 | 6,672.25 | 6,165.666667 | 8,192 | |
Given that in $\triangle ABC$, $C = 2A$, $\cos A = \frac{3}{4}$, and $2 \overrightarrow{BA} \cdot \overrightarrow{CB} = -27$.
(I) Find the value of $\cos B$;
(II) Find the perimeter of $\triangle ABC$. | 15 | 0.375 | 7,195.875 | 5,966 | 7,933.8 | |
Given two lines $l_{1}: 3mx+(m+2)y+1=0$ and $l_{2}: (m-2)x+(m+2)y+2=0$, and $l_{1} \parallel l_{2}$, determine the possible values of $m$. | -2 | 0.375 | 5,388.875 | 4,979.166667 | 5,634.7 | |
Regions I, II, and III are bounded by shapes. The perimeter of region I is 16 units and the perimeter of region II is 36 units. Region III is a triangle with a perimeter equal to the average of the perimeters of regions I and II. What is the ratio of the area of region I to the area of region III? Express your answer a... | \frac{144}{169\sqrt{3}} | 0 | 6,251.125 | -1 | 6,251.125 | |
Let $x,$ $y,$ and $z$ be angles such that
\begin{align*}
\cos x &= \tan y, \\
\cos y &= \tan z, \\
\cos z &= \tan x.
\end{align*}Find the largest possible value of $\sin x.$ | \frac{\sqrt{5} - 1}{2} | 0 | 6,791.75 | -1 | 6,791.75 | |
Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______. | \dfrac{\sqrt{5}}{2} | 0 | 5,686.125 | -1 | 5,686.125 | |
How many 3-element subsets of the set $\{1,2,3, \ldots, 19\}$ have sum of elements divisible by 4? | 244 | Consider the elements of the sets mod 4. Then we would need to have sets of the form $\{0,0,0\}$, $\{0,2,2\},\{0,1,3\},\{1,1,2\}$, or $\{2,3,3\}$. In the set $\{1,2, \ldots, 19\}$ there are four elements divisible by 4 and 5 elements congruent to each of $1,2,3 \bmod 4$. Hence the desired number is given by $$\binom{4}... | 0 | 8,030.3125 | -1 | 8,030.3125 |
$(81)^{\frac12}=3^m$. Find $m$. | 2 | 1 | 1,559.8125 | 1,559.8125 | -1 | |
Given that
\[2^{-\frac{3}{2} + 2 \cos \theta} + 1 = 2^{\frac{1}{4} + \cos \theta},\]compute $\cos 2 \theta.$ | \frac{1}{8} | 1 | 3,209.5 | 3,209.5 | -1 | |
Sets \(A, B\), and \(C\) satisfy \(|A| = 92\), \(|B| = 35\), \(|C| = 63\), \(|A \cap B| = 16\), \(|A \cap C| = 51\), and \(|B \cap C| = 19\). Compute the number of possible values of \(|A \cap B \cap C|\). | 10 | 0.625 | 6,229.8125 | 5,052.5 | 8,192 | |
Write $\sqrt{\frac{16}{25}+\frac{9}{4}}$ as a common fraction. | \frac{17}{10} | 1 | 1,521.5625 | 1,521.5625 | -1 | |
Let $m$ be the largest real solution to the equation
\[\dfrac{3}{x-3} + \dfrac{5}{x-5} + \dfrac{17}{x-17} + \dfrac{19}{x-19} = x^2 - 11x - 4\]There are positive integers $a, b,$ and $c$ such that $m = a + \sqrt{b + \sqrt{c}}$. Find $a+b+c$. | 263 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Rationalize the denominator of $\frac{5}{2+\sqrt{6}}$. The answer can be written as $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. If the greatest common divisor of $A$, $C$, and $D$ is 1, find $A+B+C+D$. | 3 | 0.9375 | 3,440.5 | 3,123.733333 | 8,192 | |
Evaluate the argument $\theta$ of the complex number
\[
e^{11\pi i/60} + e^{31\pi i/60} + e^{51 \pi i/60} + e^{71\pi i /60} + e^{91 \pi i /60}
\]
expressed in the form $r e^{i \theta}$ with $0 \leq \theta < 2\pi$. | \frac{17\pi}{20} | 0.0625 | 8,004.9375 | 5,199 | 8,192 | |
Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\cdots+2006(x-2006)^2+2007(x-2007)$.
| 2005 | 0.8125 | 4,106.375 | 3,798.846154 | 5,439 | |
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y,$ r... | 033 | Notice that line $\overline{PQ}$ is the radical axis of circles $\omega_1$ and $\omega_2$. By the radical axis theorem, we know that the tangents of any point on line $\overline{PQ}$ to circles $\omega_1$ and $\omega_2$ are equal. Therefore, line $\overline{PQ}$ must pass through the midpoint of $\overline{AB}$, call t... | 0 | 8,192 | -1 | 8,192 |
In triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of triangle ABC is given by $\frac{\sqrt{3}}{6}b(b + c - a\cos C)$.
1. Find angle A.
2. If b = 1 and c = 3, find the value of $\cos(2C - \frac{\pi}{6})$. | -\frac{4\sqrt{3}}{7} | 0 | 6,521.5 | -1 | 6,521.5 | |
The real number $x$ satisfies $x^2 - 5x + 6 < 0.$ Find all possible values of $x^2 + 5x + 6.$ | (20,30) | 0.9375 | 3,512.125 | 3,200.133333 | 8,192 | |
Given the parabola $y=\frac{1}{4}x^2$ and the circle $C: (x-1)^2+(y-2)^2=r^2$ $(r > 0)$ share a common point $P$. If the tangent line to the parabola at point $P$ also touches circle $C$, find the value of $r$. | r = \sqrt{2} | 0.375 | 7,510.6875 | 6,375.166667 | 8,192 | |
In acute triangle $ABC$ , points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$ . Compute $EC - EB$ . | 54 | 0.1875 | 7,584.6875 | 4,953 | 8,192 | |
In the diagram, $\angle PQR = 90^\circ$. What is the value of $x$?
[asy]
size(100);
draw((0,1)--(0,0)--(1,0));
draw((0,0)--(.9,.47));
draw((0,.1)--(.1,.1)--(.1,0));
label("$P$",(0,1),N); label("$Q$",(0,0),SW); label("$R$",(1,0),E); label("$S$",(.9,.47),NE);
label("$2x^\circ$",(.15,.2)); label("$x^\circ$",(.32,-.02),... | 30 | 0.5625 | 5,872.4375 | 4,068.333333 | 8,192 | |
Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$. | 8 | If a factor of $10^{n}$ has a $2$ and a $5$ in its prime factorization, then that factor will end in a $0$. Therefore, we have left to consider the case when the two factors have the $2$s and the $5$s separated, so we need to find the first power of 2 or 5 that contains a 0.
For $n = 1:$ \[2^1 = 2 , 5^1 = 5\] $n = 2:$... | 0.0625 | 8,139.5625 | 7,353 | 8,192 |
How many distinct four-digit positive integers are there such that the product of their digits equals 18? | 48 | 0 | 7,935.75 | -1 | 7,935.75 | |
In how many ways can one select five books from a row of twelve books so that no two adjacent books are chosen? | 56 | 0.75 | 5,568.125 | 4,693.5 | 8,192 | |
Given the function $f(x)= \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2}$, where $a \in \mathbb{R}$, and the curve $y=f(x)$ has a tangent at the point $(1,f(1))$ which is perpendicular to the line $y=\frac{1}{2}x$.
(i) Find the value of $a$;
(ii) Determine the intervals of monotonicity and the extreme values for t... | -\ln 5 | 0.875 | 3,449.5 | 3,389.642857 | 3,868.5 | |
The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) is equal to: | 0.333 | 0.0625 | 2,909.3125 | 2,440 | 2,940.6 | |
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights... | 1009 | 0 | 8,192 | -1 | 8,192 | |
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are t... | 148 | It is obvious that $N=a\cdot 2^4 \cdot 3\cdot 5\cdot 7$ and so the only mod $3$ number of students are $9, 11, 13$. Therefore, $N=1287\cdot k+3$. Try some approaches and you will see that this one is one of the few successful ones:
Start by setting the two $N$ equations together, then we get $1680a=1287k+3$. Divide by... | 0.0625 | 7,947.25 | 6,991 | 8,011 |
Gretchen has eight socks, two of each color: magenta, cyan, black, and white. She randomly draws four socks. What is the probability that she has exactly one pair of socks with the same color? | \frac{24}{35} | 0.875 | 5,737.75 | 5,387.142857 | 8,192 | |
On the board, two sums are written:
$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$
and
$9+98+987+9876+98765+987654+9876543+$
$+98765432+987654321$
Determine which of them is greater (or if they are equal). | 1097393685 | 0 | 8,192 | -1 | 8,192 | |
Marcelle and Jaclyn each think of a polynomial. Each of their polynomials is monic, has degree 4, and has the same positive constant term and the same coefficient of $z$. The product of their polynomials is \[z^8 +3z^7 +z^6 +3z^5 +4z^4 +6z^3 +2z^2 +4.\]What is the constant term of Jaclyn's polynomial? | 2 | 0.0625 | 8,128.25 | 7,172 | 8,192 | |
Given the hyperbola $C$: $\frac{x^{2}}{4}-y^{2}=1$ with left and right foci $F\_1$ and $F\_2$ respectively, find the area of the quadrilateral $P\_1P\_2P\_3P\_4$ for a point $P$ on the hyperbola that satisfies $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=0$. | \frac{8\sqrt{6}}{5} | 0 | 7,559.8125 | -1 | 7,559.8125 | |
Suppose that there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ such that $AB = AC = AD,$ as shown in the following diagram. If $\angle BAC = 20^\circ,$ then what is $\angle BDC$? [asy]
pair pA, pB, pC, pD;
pA = (0, 0);
pB = pA + dir(240);
pC = pA + dir(260);
pD = pA + dir(280);
draw(pA--pB--pC--pA);... | 10^\circ | 0.5 | 6,266.0625 | 5,272.25 | 7,259.875 | |
Let \( S = \{1,2, \cdots, 15\} \). From \( S \), extract \( n \) subsets \( A_{1}, A_{2}, \cdots, A_{n} \), satisfying the following conditions:
(i) \(\left|A_{i}\right|=7, i=1,2, \cdots, n\);
(ii) \(\left|A_{i} \cap A_{j}\right| \leqslant 3,1 \leqslant i<j \leqslant n\);
(iii) For any 3-element subset \( M \) of \( S ... | 15 | 0 | 8,157.375 | -1 | 8,157.375 | |
What is the value of $\left(\sqrt{4!\cdot 3!}\right)^2$? | 144 | 1 | 1,928.3125 | 1,928.3125 | -1 | |
A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest piece to the volume of the largest piece. | \frac{1}{27} | 0 | 6,516.3125 | -1 | 6,516.3125 | |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | 15 | 0.1875 | 7,647.9375 | 5,290.333333 | 8,192 | |
How many different routes can Samantha take by biking on streets to the southwest corner of City Park, then taking a diagonal path through the park to the northeast corner, and then biking on streets to school? | 400 | 0 | 5,810.4375 | -1 | 5,810.4375 | |
Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a... | 7500 | 1 | 1,791.375 | 1,791.375 | -1 | |
Express $\sqrt{x} \div\sqrt{y}$ as a common fraction, given:
$\frac{ {\left( \frac{1}{2} \right)}^2 + {\left( \frac{1}{3} \right)}^2 }{ {\left( \frac{1}{4} \right)}^2 + {\left( \frac{1}{5} \right)}^2} = \frac{13x}{41y} $ | \frac{10}{3} | 0.875 | 3,581.875 | 3,519 | 4,022 | |
A rectangle has its length increased by $30\%$ and its width increased by $15\%$. What is the percentage increase in the area of the rectangle? | 49.5\% | 1 | 2,329 | 2,329 | -1 | |
Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.
\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24
\hline
25 & 26 & 27 & 28 & 2... | 1056 | 0.625 | 6,811.5625 | 5,983.3 | 8,192 | |
How many integers between $200$ and $300$ have three different digits in increasing order? | 21 | 1 | 3,057.875 | 3,057.875 | -1 | |
For some positive integers $c$ and $d$, the product \[\log_c(c+1) \cdot \log_{c+1} (c+2) \dotsm \log_{d-2} (d-1) \cdot\log_{d-1} d\]contains exactly $930$ terms, and its value is $3.$ Compute $c+d.$ | 1010 | 0 | 8,192 | -1 | 8,192 | |
Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \perp XY.$ Meanwhile, $R$ is a point on line $PY$ such that $XR \perp PY.$ If $XR = 3$, $PQ = 6$, and $XY = 7$, then what is the length of $PY?$ | 14 | 0.5625 | 6,162.3125 | 4,583.666667 | 8,192 | |
Using the six digits 0, 1, 2, 3, 4, 5,
(1) How many distinct three-digit numbers can be formed?
(2) How many distinct three-digit odd numbers can be formed? | 48 | 0.25 | 5,275.5625 | 5,219.25 | 5,294.333333 | |
In the rectangular coordinate system, point \( O(0,0) \), \( A(0,6) \), \( B(-3,2) \), \( C(-2,9) \), and \( P \) is any point on line segment \( OA \) (including endpoints). Find the minimum value of \( PB + PC \). | 5 + \sqrt{13} | 0.125 | 8,135.5625 | 7,740.5 | 8,192 | |
Let $A$ be the vertex of the graph of the equation $y=x^2 - 2x + 3 $. Let $B$ be the vertex of the graph of the equation $y=x^2 + 4x + 10 $. What is the distance between $A$ and $B$? | 5 | 1 | 1,540.8125 | 1,540.8125 | -1 | |
Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}.$ For how many such quadruples is it true that $a \cdot d-b \cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0 \cdot 1-3 \cdot 1 = -3$ is odd.) | 96 |
To solve the problem, we need to count the number of ordered quadruples $(a, b, c, d)$ such that $a\cdot d - b\cdot c$ is odd. We will use parity analysis to determine the conditions under which the expression is odd.
#### Step 1: Understanding Parity of Products
The product of two integers is odd if and only if both... | 0.5625 | 6,897.25 | 5,890.222222 | 8,192 |
The lengths of the edges of a rectangular parallelepiped extending from one vertex are 8, 8, and 27. Divide the parallelepiped into four parts that can be assembled into a cube. | 12 | 0 | 8,192 | -1 | 8,192 | |
Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $|AZ|$ ? | 10 | 0.125 | 8,037.625 | 6,957 | 8,192 | |
Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\square$ is unclearly printed.<br/>$(1)$ She guessed $\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find the value of the expression when $x=-2$;<br/... | -2 | 0.875 | 2,700.3125 | 2,317.357143 | 5,381 | |
Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals? | 27.55 | 0 | 6,004.8125 | -1 | 6,004.8125 | |
What is the area enclosed by the graph of $|x| + |3y| = 9$? | 54 | 0.9375 | 3,448.8125 | 3,132.6 | 8,192 | |
Let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\lfloor x \rfloor +51=0$ is | 4 | 1. **Rewrite the equation**: Start by rewriting the given equation:
\[
4x^2 - 40\lfloor x \rfloor + 51 = 0
\]
Rearrange this to isolate $4x^2$:
\[
4x^2 = 40\lfloor x \rfloor - 51
\]
Let $n = 40\lfloor x \rfloor - 51$, which implies $4x^2 = n$. Since $n$ must be an integer (as $\lfloor x \rfloor$... | 0.1875 | 7,619.625 | 6,235.333333 | 7,939.076923 |
Compute $17^9 \div 17^7$. | 289 | 0.75 | 3,985.4375 | 2,583.25 | 8,192 | |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-6x+5|$ and $y=\frac{29}{4}-x$. | \frac{17}{2} | 0.625 | 6,018.25 | 4,714 | 8,192 | |
Compute the positive integer less than 1000 which has exactly 29 positive proper divisors. | 720 | Recall that the number $N=p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{k}^{e_{k}}$ (where the $p_{i}$ are distinct primes) has exactly $(e_{1}+1)(e_{2}+1) \cdots(e_{k}+1)$ positive integer divisors including itself. We seek $N<1000$ such that this expression is 30. Since $30=2 \cdot 3 \cdot 5$, we take $e_{1}=1, e_{2}=2, e_{3... | 0.625 | 6,922.0625 | 6,160.1 | 8,192 |
How many factors of 2 are in the prime factorization of 1984!? | 1979 | 1 | 3,262.3125 | 3,262.3125 | -1 | |
A line passing through the intersection point of the medians of triangle \(ABC\) intersects the sides \(BA\) and \(BC\) at points \(A^{\prime}\) and \(C_1\) respectively. Given that:
\(BA^{\prime} < BA = 3\), \(BC = 2\), and \(BA^{\prime} \cdot BC_1 = 3\). Find \(BA^{\prime}\). | \frac{3}{2} | 0.6875 | 6,258.875 | 5,380.181818 | 8,192 | |
Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy]unitsize(0.2 cm); pair A, B, C, D, E, F; A = (0,13); B = (13,13); C = (13,0); D = (0,0); E = A + (12*12/13,5*12/13); F = D + (5*5/13,-5*12/13); draw(A--B--C--D--cycle); draw(A--E... | 578 | Drawing $EF$, it clearly passes through the center of $ABCD$. Letting this point be $P$, we note that $AEBP$ and $CFDP$ are congruent cyclic quadrilaterals, and that $AP=BP=CP=DP=\frac{13}{\sqrt{2}}.$ Now, from Ptolemy's, $13\cdot EP=\frac{13}{\sqrt{2}}(12+5)\implies EP+\frac{17\sqrt{2}}{2}$. Since $EF=EP+FP=2\cdot EP$... | 0.875 | 5,368.75 | 4,965.428571 | 8,192 |
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer. | 132 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$, $A=30^{\circ}$, $AB=\sqrt {3}$, $BC=1$, find the area of $\triangle ABC$. | \frac {\sqrt {3}}{4} | 0 | 7,416.4375 | -1 | 7,416.4375 | |
Round to the nearest tenth: 36.89753 | 36.9 | 1 | 360.0625 | 360.0625 | -1 | |
The 5-digit number $52\,28\square$ is a multiple of 6. Which digit is represented by $\square$? | 4 | 1 | 1,714.9375 | 1,714.9375 | -1 | |
Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = 3,$ $\|\mathbf{v}\| = 4,$ and $\|\mathbf{w}\| = 5,$ and
\[\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}.\]Compute $\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{w}.$ | -25 | 1 | 2,639 | 2,639 | -1 | |
We are given some similar triangles. Their areas are $1^{2}, 3^{2}, 5^{2} \ldots$, and $49^{2}$. If the smallest triangle has a perimeter of 4, what is the sum of all the triangles' perimeters? | 2500 | Because the triangles are all similar, they all have the same ratio of perimeter squared to area, or, equivalently, the same ratio of perimeter to the square root of area. Because the latter ratio is 4 for the smallest triangle, it is 4 for all the triangles, and thus their perimeters are $4 \cdot 1,4 \cdot 3,4 \cdot 5... | 0.875 | 3,182.625 | 2,809.214286 | 5,796.5 |
A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? | 18 | 0.125 | 7,461.75 | 5,669.5 | 7,717.785714 | |
Let $M=\{1,2, \cdots, 2005\}$. Subset $A$ of $M$ satisfies the condition: if $x \in A$, then $15x \notin A$. What is the maximum number of elements in $A$? | 1880 | 0 | 8,192 | -1 | 8,192 | |
A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximiz... | \frac{5}{16} |
1. **Form of the Polynomial:**
The disrespectful quadratic polynomial $p(x)$ with leading coefficient $1$ can be written as:
\[
p(x) = (x-r)(x-s) = x^2 - (r+s)x + rs
\]
where $r$ and $s$ are the roots of the polynomial.
2. **Condition for $p(p(x)) = 0$:**
We substitute $p(x)$ into itself:
\[
p... | 0.25 | 7,965.0625 | 7,284.25 | 8,192 |
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them? | \frac{\sqrt{6} + 2}{2} | 0 | 5,785.0625 | -1 | 5,785.0625 | |
What is the sum of the positive divisors of 1184? | 2394 | We start by finding the prime factors of 1184: $1184=2 \cdot 592=2^{2} \cdot 296=2^{3} \cdot 148=2^{4} \cdot 74=2^{5} \cdot 37$. The positive divisors of 1184 cannot contain prime factors other than 2 and 37, and cannot contain more than 5 factors of 2 or 1 factor of 37. Thus, the positive divisors are $1,2,4,8,16,32,3... | 1 | 2,043.125 | 2,043.125 | -1 |
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, if ${a_1}=\frac{5}{2}$, and ${a_{n+1}}({2-{a_n}})=2$ for $n\in\mathbb{N}^*$, then $S_{22}=$____. | -\frac{4}{3} | 0.6875 | 5,974.0625 | 4,965.909091 | 8,192 | |
Find the minimum value of
\[x^2 + xy + y^2\]over all real numbers $x$ and $y.$ | 0 | 1 | 4,953.5 | 4,953.5 | -1 | |
Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition:
For any three elements in $M$ , there exist two of them $a$ and $b$ such that $a|b$ or $b|a$ .
Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$ | 18 | 0 | 8,192 | -1 | 8,192 | |
In a plane Cartesian coordinate system, points where both the vertical and horizontal coordinates are integers are called lattice points. The number of lattice points $(x, y)$ satisfying the inequality $(|x|-1)^{2}+(|y|-1)^{2}<2$ is: | 16 | 0.3125 | 7,666.0625 | 6,509 | 8,192 | |
On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could have answered correctly? | 29 | Let's define the variables:
- Let $a$ be the number of questions Jesse answered correctly.
- Let $b$ be the number of questions Jesse left blank.
- Let $c$ be the number of questions Jesse answered incorrectly.
Given the total number of questions is $50$, we have the equation:
\[ a + b + c = 50 \]
Jesse's scoring rul... | 0.9375 | 3,775.6875 | 3,481.266667 | 8,192 |
An economist and a statistician play a game on a calculator which does only one
operation. The calculator displays only positive integers and it is used in the following
way: Denote by $n$ an integer that is shown on the calculator. A person types an integer,
$m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the fir... | 951 |
To solve this problem, we need to understand the specific condition under which the current displayed number \( n \) on the calculator can be transformed to another integer through the operation described, where \( m \) is chosen from the set \(\{1, 2, \ldots, 99\}\).
The process involves finding \( m\% \) of \( n \)... | 0 | 8,192 | -1 | 8,192 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(3-\frac{2}{\cos x}\right)^{\operatorname{cosec}^{2} x}
$$ | e^{-1} | 0 | 7,756.6875 | -1 | 7,756.6875 | |
Given the positive integer \( A = \overline{a_{n} a_{n-1} \cdots a_{1} a_{0}} \), where \( a_{n}, a_{n-1}, \ldots, a_{0} \) are all non-zero and not all equal (with \( n \) being a positive integer), consider the following cyclic permutations of \( A \):
$$
\begin{array}{l}
A_{1}=\overline{a_{n-1} \cdots a_{1} a_{0} a_... | 142857 | 0.1875 | 8,029.75 | 7,326.666667 | 8,192 | |
A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ De... | 5 + \frac{15\sqrt{2}}{2} | 0 | 6,974.25 | -1 | 6,974.25 | |
With the popularity of cars, the "driver's license" has become one of the essential documents for modern people. If someone signs up for a driver's license exam, they need to pass four subjects to successfully obtain the license, with subject two being the field test. In each registration, each student has 5 chances to... | \frac{1}{9} | 0 | 6,735 | -1 | 6,735 | |
Given the standard equation of the hyperbola $M$ as $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$. Find the length of the real axis, the length of the imaginary axis, the focal distance, and the eccentricity of the hyperbola $M$. | \frac{\sqrt{6}}{2} | 0 | 1,824.8125 | -1 | 1,824.8125 | |
Given the parameterized equation of a line is $$\begin{cases} x=1+ \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{cases}$$ (where $t$ is the parameter), determine the angle of inclination of the line. | \frac{\pi}{3} | 0.1875 | 2,298.25 | 2,676 | 2,211.076923 | |
Little Pang, Little Dingding, Little Ya, and Little Qiao's four families, totaling 8 parents and 4 children, went to the amusement park together. The ticket prices are as follows: adult tickets are 100 yuan per person; children's tickets are 50 yuan per person; if there are 10 or more people, they can buy group tickets... | 800 | 0.0625 | 6,491.9375 | 8,192 | 6,378.6 | |
Find the maximum value of the expression \((\sin 3x + \sin 2y + \sin z)(\cos 3x + \cos 2y + \cos z)\).
(Possible points: 15) | 4.5 | 0 | 8,192 | -1 | 8,192 | |
To determine the minimum time required for the concentration of the drug in the air to drop below 0.25 milligrams per cubic meter, solve the equation y = 0.25 for t using the function y=\begin{cases} 10t & (0 \leqslant t \leqslant 0.1), \\ {\left( \frac{1}{16} \right)}^{t- \frac{1}{10}} & (t > 0.1) \end{cases}. | 0.6 | 0.375 | 5,204.25 | 4,224 | 5,792.4 |
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