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 |
|---|---|---|---|---|---|---|
Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative. | (-7,-7,18,-7,-7,-7,18,-7,-7,18,-7,-7,-7,18,-7,-7) | Suppose it is possible to have more than 16 terms in the sequence. Let $a_{1}, a_{2}, \ldots, a_{17}$ be the first 17 terms of the sequence. Consider the following array of terms in the sequence: \begin{tabular}{lllllllllll} $a_{1}$ & $a_{2}$ & $a_{3}$ & $a_{4}$ & $a_{5}$ & $a_{6}$ & $a_{7}$ & $a_{8}$ & $a_{9}$ & $a_{1... | 0 | 8,192 | -1 | 8,192 |
Given the function \( f(x) = \cos x + \log_2 x \) for \( x > 0 \), if the positive real number \( a \) satisfies \( f(a) = f(2a) \), then find the value of \( f(2a) - f(4a) \). | -1 | 0 | 8,094.125 | -1 | 8,094.125 | |
Compute $\sqrt{54}\cdot\sqrt{32}\cdot \sqrt{6}$. | 72\sqrt{2} | 1 | 3,633.6875 | 3,633.6875 | -1 | |
Given the function $f(x)=\frac{1}{2}{f'}(1){x^2}+lnx+\frac{{f(1)}}{{3x}}$, find the value of ${f'}\left(2\right)$. | \frac{33}{4} | 1 | 3,165 | 3,165 | -1 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{40} | 0 | 3,009.125 | -1 | 3,009.125 | |
The results of asking 50 students if they participate in music or sports are shown in the Venn diagram. Calculate the percentage of the 50 students who do not participate in music and do not participate in sports. | 20\% | 0.125 | 968.0625 | 3,519.5 | 603.571429 | |
Find all positive integers $k<202$ for which there exists a positive integer $n$ such that $$\left\{\frac{n}{202}\right\}+\left\{\frac{2 n}{202}\right\}+\cdots+\left\{\frac{k n}{202}\right\}=\frac{k}{2}$$ where $\{x\}$ denote the fractional part of $x$. | k \in\{1,100,101,201\} | Denote the equation in the problem statement as $\left(^{*}\right)$, and note that it is equivalent to the condition that the average of the remainders when dividing $n, 2 n, \ldots, k n$ by 202 is 101. Since $\left\{\frac{i n}{202}\right\}$ is invariant in each residue class modulo 202 for each $1 \leq i \leq k$, it s... | 0 | 7,969.8125 | -1 | 7,969.8125 |
Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$ , determine the largest possible value of $m$ .
| 64 | 0.5625 | 6,795.25 | 5,929.111111 | 7,908.857143 | |
Yesterday, Sasha cooked soup and added too little salt, requiring additional seasoning. Today, he added twice as much salt as yesterday, but still had to season the soup additionally, though with half the amount of salt he used for additional seasoning yesterday. By what factor does Sasha need to increase today's porti... | 1.5 | 0 | 7,615.5625 | -1 | 7,615.5625 | |
How many four-digit numbers are divisible by 17? | 530 | 0.9375 | 3,885.6875 | 3,598.6 | 8,192 | |
Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate. | 69 | 0.8125 | 4,828.1875 | 4,150.769231 | 7,763.666667 | |
Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
all of t... | 199 | 0.5625 | 5,957.625 | 4,611.666667 | 7,688.142857 | |
Solve for $x$ in the equation $ \frac35 \cdot \frac19 \cdot x = 6$. | 90 | 1 | 1,743.125 | 1,743.125 | -1 | |
Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected from the group to serve. Determine the number of ways to select exactly one person to serve for both days. | 60 | 0.5625 | 6,024.3125 | 4,730 | 7,688.428571 | |
Given the function $y=3\sin \left(2x-\frac{\pi }{8}\right)$, determine the horizontal shift required to transform the graph of the function $y=3\sin 2x$. | \frac{\pi}{8} | 0 | 2,497.0625 | -1 | 2,497.0625 | |
Determine the smallest positive real $K$ such that the inequality
\[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$ .
*Proposed by Fajar Yuliawan, Indonesia* | \frac{\sqrt{6}}{3} | 0 | 8,192 | -1 | 8,192 | |
If $a^{2}-4a+3=0$, find the value of $\frac{9-3a}{2a-4} \div (a+2-\frac{5}{a-2})$ . | -\frac{3}{8} | 1 | 4,601.75 | 4,601.75 | -1 | |
In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5$. The ratio of the area of triangle $DEF$ ... | 61 | By the barycentric area formula, our desired ratio is equal to \begin{align*} \begin{vmatrix} 1-p & p & 0 \\ 0 & 1-q & q \\ r & 0 & 1-r \notag \end{vmatrix} &=1-p-q-r+pq+qr+pr\\ &=1-(p+q+r)+\frac{(p+q+r)^2-(p^2+q^2+r^2)}{2}\\ &=1-\frac{2}{3}+\frac{\frac{4}{9}-\frac{2}{5}}{2}\\ &=\frac{16}{45} \end{align*}, so the answe... | 0.0625 | 8,103.0625 | 7,820 | 8,121.933333 |
Calculate
(1) Use a simplified method to calculate $2017^{2}-2016 \times 2018$;
(2) Given $a+b=7$ and $ab=-1$, find the values of $(a+b)^{2}$ and $a^{2}-3ab+b^{2}$. | 54 | 1 | 2,022.6875 | 2,022.6875 | -1 | |
Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$ . Find the expected valu... | 1866 | 0 | 8,192 | -1 | 8,192 | |
The points $(0,0)\,$, $(a,11)\,$, and $(b,37)\,$ are the vertices of an equilateral triangle. Find the value of $ab\,$.
| 315 | 0.75 | 6,000.0625 | 5,380.333333 | 7,859.25 | |
In a bucket, there are $34$ red balls, $25$ green balls, $23$ yellow balls, $18$ blue balls, $14$ white balls, and $10$ black balls. Find the minimum number of balls that must be drawn from the bucket without replacement to guarantee that at least $20$ balls of a single color are drawn. | 100 | 0.8125 | 4,276.3125 | 3,727.153846 | 6,656 | |
What is the positive difference between the median and the mode of the data given in the stem and leaf plot below? In this plot $5|8$ represents $58.$
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
1 & $2 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 5$ \\ \hline
2 & $2 \hspace{2mm} ... | 9 | 0.4375 | 1,198.6875 | 1,705.857143 | 804.222222 | |
The circle centered at $(3,-2)$ and with radius $5$ intersects the circle centered at $(3,4)$ and with radius $\sqrt{13}$ at two points $C$ and $D$. Find $(CD)^2$. | 36 | 1 | 2,811.125 | 2,811.125 | -1 | |
Let $A B C$ be a triangle with $A B=7, B C=9$, and $C A=4$. Let $D$ be the point such that $A B \| C D$ and $C A \| B D$. Let $R$ be a point within triangle $B C D$. Lines $\ell$ and $m$ going through $R$ are parallel to $C A$ and $A B$ respectively. Line $\ell$ meets $A B$ and $B C$ at $P$ and $P^{\prime}$ respectivel... | 180 | Let $R^{\prime}$ denote the intersection of the lines through $Q^{\prime}$ and $P^{\prime}$ parallel to $\ell$ and $m$ respectively. Then $\left[R P^{\prime} Q^{\prime}\right]=\left[R^{\prime} P^{\prime} Q^{\prime}\right]$. Triangles $B P P^{\prime}, R^{\prime} P^{\prime} Q^{\prime}$, and $C Q Q^{\prime}$ lie in $A B C... | 0.0625 | 8,171.8125 | 7,869 | 8,192 |
Let \(\theta\) be an angle in the second quadrant, and if \(\tan (\theta+ \frac {\pi}{3})= \frac {1}{2}\), calculate the value of \(\sin \theta+ \sqrt {3}\cos \theta\). | - \frac {2 \sqrt {5}}{5} | 0 | 7,165.625 | -1 | 7,165.625 | |
What positive integer can $n$ represent if it is known that by erasing the last three digits of the number $n^{3}$, we obtain the number $n$? | 32 | 0.25 | 7,495.0625 | 5,404.25 | 8,192 | |
Write the expression
$$
K=\frac{\frac{1}{a+b}-\frac{2}{b+c}+\frac{1}{c+a}}{\frac{1}{b-a}-\frac{2}{b+c}+\frac{1}{c-a}}+\frac{\frac{1}{b+c}-\frac{2}{c+a}+\frac{1}{a+b}}{\frac{1}{c-b}-\frac{2}{c+a}+\frac{1}{a-b}}+\frac{\frac{1}{c+a}-\frac{2}{a+b}+\frac{1}{b+c}}{\frac{1}{a-c}-\frac{2}{a+b}+\frac{1}{b-c}}
$$
in a simpler... | 0.0625 | 0 | 7,990.3125 | -1 | 7,990.3125 | |
The sides of a triangle have lengths of $15$, $20$, and $25$. Find the length of the shortest altitude. | 12 | 1 | 1,879.3125 | 1,879.3125 | -1 | |
Given the expansion of the expression $(1- \frac {1}{x})(1+x)^{7}$, find the coefficient of the term $x^{4}$. | 14 | 0.75 | 6,526.25 | 5,971 | 8,192 | |
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[ x = \sqrt{29} + \frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{x}}}}}}}}}.\] | 513 | 0.0625 | 8,113.3125 | 6,933 | 8,192 | |
On a 6 by 6 grid of points, what fraction of the larger square's area is inside the new shaded square? Place the bottom-left vertex of the square at grid point (3,3) and the square rotates 45 degrees (square's sides are diagonals of the smaller grid cells).
```
[asy]
size(6cm);
fill((3,3)--(4,4)--(5,3)--(4,2)--cycle,gr... | \frac{1}{18} | 0.5625 | 6,497 | 6,151.222222 | 6,941.571429 | |
Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \dfrac {25π}{6})$;
(2) Let $α∈(0,π)$, $f( \dfrac {α}{2})= \dfrac {1}{4}- \dfrac { \sqrt {3}}{2}$, find the value of $\sin α$. | \dfrac {1+3 \sqrt {5}}{8} | 0 | 7,655.9375 | -1 | 7,655.9375 | |
In the geometric sequence $\{a_n\}$, the common ratio $q = -2$, and $a_3a_7 = 4a_4$, find the arithmetic mean of $a_8$ and $a_{11}$. | -56 | 0.9375 | 4,920.875 | 4,860.733333 | 5,823 | |
Given a moving circle P that is internally tangent to the circle M: (x+1)²+y²=8 at the fixed point N(1,0).
(1) Find the trajectory equation of the moving circle P's center.
(2) Suppose the trajectory of the moving circle P's center is curve C. A and B are two points on curve C. The perpendicular bisector of line segmen... | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given that $\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 2$, find the value of $\frac{\sin \theta}{\cos^{3} \theta} + \frac{\cos \theta}{\sin^{3} \theta}$. | \frac{820}{27} | 0.75 | 5,842.3125 | 5,059.083333 | 8,192 | |
The seven digits in Sam's phone number and the four digits in his house number have the same sum. The four digits in his house number are distinct, and his phone number is 271-3147. What is the largest possible value of Sam's house number? | 9871 | 0.25 | 7,916.125 | 7,088.5 | 8,192 | |
Given the real numbers \( x \) and \( y \) satisfy the equations:
\[ 2^x + 4x + 12 = \log_2{(y-1)^3} + 3y + 12 = 0 \]
find the value of \( x + y \). | -2 | 0 | 8,192 | -1 | 8,192 | |
Let the reciprocals of the roots of $5x^2 + 3x + 4$ be $\alpha$ and $\beta$. Evaluate $\alpha + \beta$. | -\dfrac{3}{4} | 1 | 2,458.4375 | 2,458.4375 | -1 | |
Dylan has a \( 100 \times 100 \) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make? | 9999 | 0 | 8,135.0625 | -1 | 8,135.0625 | |
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? | 85 | An ellipse is defined as the set of points where the sum of the distances from the foci to the point is fixed. The length of major axis is equal to the sum of these distances $(2a)$. Thus if we find the sum of the distances, we get the answer. Let k be this fixed sum; then we get, by the distance formula:
$k = \sqrt{(x... | 0.3125 | 7,122.4375 | 6,276.6 | 7,506.909091 |
Let $S = {1, 2, \cdots, 100}.$ $X$ is a subset of $S$ such that no two distinct elements in $X$ multiply to an element in $X.$ Find the maximum number of elements of $X$ .
*2022 CCA Math Bonanza Individual Round #3* | 91 | 0 | 8,192 | -1 | 8,192 | |
Four cars, \( A, B, C, \) and \( D \) start simultaneously from the same point on a circular track. \( A \) and \( B \) drive clockwise, while \( C \) and \( D \) drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, \( A \) meets \( C \) for the fir... | 53 | 0.0625 | 7,797.5 | 6,967 | 7,852.866667 | |
Points $A, C$, and $B$ lie on a line in that order such that $A C=4$ and $B C=2$. Circles $\omega_{1}, \omega_{2}$, and $\omega_{3}$ have $\overline{B C}, \overline{A C}$, and $\overline{A B}$ as diameters. Circle $\Gamma$ is externally tangent to $\omega_{1}$ and $\omega_{2}$ at $D$ and $E$ respectively, and is intern... | \frac{2}{3} | Let the center of $\omega_{i}$ be $O_{i}$ for $i=1,2,3$ and let $O$ denote the center of $\Gamma$. Then $O, D$, and $O_{1}$ are collinear, as are $O, E$, and $O_{2}$. Denote by $F$ the point of tangency between $\Gamma$ and $\omega_{3}$; then $F, O$, and $O_{3}$ are collinear. Writing $r$ for the radius of $\Gamma$ we ... | 0.375 | 7,771.6875 | 7,071.166667 | 8,192 |
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k.$ | 1 | 1 | 3,719.5 | 3,719.5 | -1 | |
Given that the function $y=f(x)+\sin \frac {π}{6}x$ is an even function, and $f(\log _{ \sqrt {2}}2)= \sqrt {3}$, determine $f(\log _{2} \frac {1}{4})$. | 2 \sqrt {3} | 0 | 2,824.5 | -1 | 2,824.5 | |
Mark borrows $10$ dollars from Emily with a simple interest rate of $15\%$ everyday. What is the least integer number of days after which Mark will have to pay her back at least twice as much as he borrowed? | 7 \text{ days} | 0.9375 | 2,357.375 | 1,968.4 | 8,192 | |
Given $\sin \left(\theta -\frac{\pi }{3}\right)=\frac{3}{5}$, then $\sin \left(2\theta -\frac{\pi }{6}\right)=$_______. | \frac{7}{25} | 0.8125 | 6,003.3125 | 5,498.230769 | 8,192 | |
Given the function $y = \lg(-x^2 + x + 2)$ with domain $A$, find the range $B$ for the exponential function $y = a^x$ $(a>0$ and $a \neq 1)$ where $x \in A$.
1. If $a=2$, determine $A \cup B$;
2. If $A \cap B = (\frac{1}{2}, 2)$, find the value of $a$. | a = 2 | 0.625 | 6,342.9375 | 5,233.5 | 8,192 | |
Given that the eccentricities of a confocal ellipse and a hyperbola are \( e_1 \) and \( e_2 \), respectively, and the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, find the maximum value of \( \frac{1}{e_1} + \frac{1}{e_2} \). | 5/2 | 0.625 | 6,732.375 | 6,333.7 | 7,396.833333 | |
The coefficient of $x^3$ in the expansion of $(x^2-x-2)^4$ is __________ (fill in the answer with a number). | -40 | 0.9375 | 5,580.875 | 5,406.8 | 8,192 | |
A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number. | 727023 | 0.125 | 8,018 | 6,800 | 8,192 | |
Given the function $f(x)=|2x-9|-|x-5|$.<br/>$(1)$ Find the solution set of the inequality $f(x)\geqslant 2x-1$;<br/>$(2)$ The minimum value of the function $y=f(x)+3|x-5|$ is $m$. For positive real numbers $a$ and $b$ satisfying $\frac{1}{a}+\frac{3}{b}=m$, find the minimum value of $a+3b$. | 16 | 0.8125 | 5,219.125 | 5,230.307692 | 5,170.666667 | |
Given the function $f(x) = |\ln x|$, the solution set of the inequality $f(x) - f(x_0) \geq c(x - x_0)$ is $(0, +\infty)$, where $x_0 \in (0, +\infty)$, and $c$ is a constant. When $x_0 = 1$, the range of values for $c$ is \_\_\_\_\_\_; when $x_0 = \frac{1}{2}$, the value of $c$ is \_\_\_\_\_\_. | -2 | 0.0625 | 7,838 | 7,394 | 7,867.6 | |
The number 210 is the product of two consecutive positive integers and is also the product of three consecutive integers. What is the sum of those five integers? | 47 | 1 | 2,262.125 | 2,262.125 | -1 | |
Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a - b + c = 7$, $b - c + d = 8$, $c - d + a = 5$, and $d - a + b = 4$. What is the value of $a + b + c + d$? | 12 | 0 | 3,463.5 | -1 | 3,463.5 | |
Two numbers need to be inserted between $4$ and $16$ such that the first three numbers are in arithmetic progression and the last three numbers are in geometric progression. What is the sum of those two numbers?
A) $6\sqrt{3} + 8$
B) $10\sqrt{3} + 6$
C) $6 + 10\sqrt{3}$
D) $16\sqrt{3}$ | 6\sqrt{3} + 8 | 0 | 6,939.75 | -1 | 6,939.75 | |
Ivan Petrovich wants to save money for his retirement in 12 years. He decided to deposit 750,000 rubles in a bank account with an 8 percent annual interest rate. What will be the total amount in the account by the time Ivan Petrovich retires, assuming the interest is compounded annually using the simple interest formul... | 1470000 | 0.9375 | 2,823.0625 | 2,465.133333 | 8,192 | |
There are $52$ people in a room. what is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true? | 5 | 1. **Understanding the Problem:**
We need to determine the largest number \( n \) such that in any group of 52 people, at least \( n \) people will have their birthdays in the same month.
2. **Applying the Pigeonhole Principle:**
The Pigeonhole Principle states that if \( k \) items are put into \( n \) containe... | 1 | 2,211.6875 | 2,211.6875 | -1 |
Consider the parametric equations for a curve given by
\begin{align*}
x &= \cos t + \frac{t}{3}, \\
y &= \sin t.
\end{align*}
Determine how many times the graph intersects itself between $x = 0$ and $x = 60$. | 28 | 0 | 8,192 | -1 | 8,192 | |
Determine the number of positive integers $a$ less than $12$ such that the congruence $ax\equiv 1\pmod{12}$ has a solution in $x$. | 4 | 1 | 2,461.5625 | 2,461.5625 | -1 | |
The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each ... | 12, 15, 18 |
To solve this problem, we aim to find all integer values of \( n \) (where \( 10 < n < 20 \)) for which an \( n \times n \) floor can be completely covered using the same number of \( 2 \times 2 \) and \( 5 \times 1 \) tiles. We cannot cut the tiles and they should not overlap.
First, we calculate the total area of t... | 0 | 8,181.4375 | -1 | 8,181.4375 |
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible t... | 336 | If two of our big equilateral triangles have the same color for their center triangle and the same multiset of colors for their outer three triangles, we can carry one onto the other by a combination of rotation and reflection. Thus, to make two triangles distinct, they must differ either in their center triangle or in... | 0.5 | 5,464.9375 | 5,158.5 | 5,771.375 |
Let $A$ be a point on the parabola $y = x^2 - 4x + 4,$ and let $B$ be a point on the line $y = 2x - 3.$ Find shortest possible distance $AB.$ | \frac{2\sqrt{5}}{5} | 0 | 7,886.5 | -1 | 7,886.5 | |
What is the only integer whose square is less than its double? | 1 | 1 | 1,986.3125 | 1,986.3125 | -1 | |
In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, and $BN \perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$. | \frac{5}{2} | 1. **Identify Key Elements and Relationships**:
- $M$ is the midpoint of $BC$, so $BM = MC$.
- $AN$ bisects $\angle BAC$, making $\angle BAN = \angle NAC$.
- $BN \perp AN$, establishing $\triangle ANB$ as a right triangle.
2. **Extend $BN$ to meet $AC$ at $Q$**:
- By extending $BN$, we create a new point ... | 0.25 | 7,647 | 6,012 | 8,192 |
In triangle $XYZ$, $XY=15$, $YZ=18$, and $ZX=21$. Point $G$ is on $\overline{XY}$, $H$ is on $\overline{YZ}$, and $I$ is on $\overline{ZX}$. Let $XG = p \cdot XY$, $YH = q \cdot YZ$, and $ZI = r \cdot ZX$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of trian... | 41 | 0.25 | 7,748.3125 | 6,872.25 | 8,040.333333 | |
Using the vertices of a single rectangular solid (cuboid), how many different pyramids can be formed? | 106 | 0 | 7,400.1875 | -1 | 7,400.1875 | |
Compute $\frac{\tan ^{2}\left(20^{\circ}\right)-\sin ^{2}\left(20^{\circ}\right)}{\tan ^{2}\left(20^{\circ}\right) \sin ^{2}\left(20^{\circ}\right)}$. | 1 | If we multiply top and bottom by $\cos ^{2}\left(20^{\circ}\right)$, the numerator becomes $\sin ^{2}\left(20^{\circ}\right) \cdot(1-\cos ^{2} 20^{\circ})=\sin ^{4}\left(20^{\circ}\right)$, while the denominator becomes $\sin ^{4}\left(20^{\circ}\right)$ also. So they are equal, and the ratio is 1. | 0.9375 | 3,005.6875 | 2,659.933333 | 8,192 |
Given that the function $f(x)$ defined on $\mathbb{R}$ is an odd function and satisfies $f(1+x)=f(3+x)$. When $0\leq x\leq 1$, $f(x)=x^{3}-x$. Find $f(\frac{11}{2})+f(6)$. | \frac{3}{8} | 0.875 | 5,094.4375 | 4,651.928571 | 8,192 | |
An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. What is the difference between the largest a... | 8.875 | 0 | 6,669.375 | -1 | 6,669.375 | |
Using arithmetic operation signs, write the largest natural number using two twos. | 22 | 0.0625 | 7,790.75 | 6,330 | 7,888.133333 | |
Evaluate the expression $\sqrt{5+4\sqrt{3}} - \sqrt{5-4\sqrt{3}} + \sqrt{7 + 2\sqrt{10}} - \sqrt{7 - 2\sqrt{10}}$.
A) $4\sqrt{3}$
B) $2\sqrt{2}$
C) $6$
D) $4\sqrt{2}$
E) $2\sqrt{5}$ | 2\sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
My friend Ana likes numbers that are divisible by 8. How many different pairs of last two digits are possible in numbers that Ana likes? | 13 | 0.25 | 3,771 | 2,280.5 | 4,267.833333 | |
A magician writes the numbers 1 to 16 on 16 positions of a spinning wheel. Four audience members, A, B, C, and D, participate in the magic show. The magician closes his eyes, and then A selects a number from the wheel. B, C, and D, in that order, each choose the next number in a clockwise direction. Only A and D end up... | 120 | 0 | 7,973.4375 | -1 | 7,973.4375 | |
Given that b is an even number between 1 and 11 (inclusive) and c is any natural number, determine the number of quadratic equations x^{2} + bx + c = 0 that have two distinct real roots. | 50 | 0.9375 | 5,467.6875 | 5,286.066667 | 8,192 | |
How many integers 1-9 are divisors of the five-digit number 24,516? | 6 | 0.9375 | 2,580.75 | 2,586.8 | 2,490 | |
Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., find the value of the 8th term. | 84 | 0.875 | 3,904.5 | 3,724.5 | 5,164.5 | |
The function \( y = \cos x + \sin x + \cos x \sin x \) has a maximum value of \(\quad\). | \frac{1}{2} + \sqrt{2} | 0.5625 | 6,350.625 | 5,422.666667 | 7,543.714286 | |
In the regular tetrahedron \(ABCD\), points \(E\) and \(F\) are on edges \(AB\) and \(AC\) respectively, such that \(BE = 3\) and \(EF = 4\), and \(EF\) is parallel to face \(BCD\). What is the area of \(\triangle DEF\)? | 2\sqrt{33} | 0.0625 | 8,062.1875 | 6,752 | 8,149.533333 | |
Evaluate the absolute value of the expression $|7 - \sqrt{53}|$.
A) $7 - \sqrt{53}$
B) $\sqrt{53} - 7$
C) $0.28$
D) $\sqrt{53} + 7$
E) $-\sqrt{53} + 7$ | \sqrt{53} - 7 | 0.875 | 528.3125 | 529 | 523.5 | |
For any positive integer $n$, let \langle n\rangle denote the closest integer to \sqrt{n}. Evaluate
\[\sum_{n=1}^\infty \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}.\] | 3 | Since $(k-1/2)^2 = k^2-k+1/4$ and $(k+1/2)^2 = k^2+k+1/4$, we have that $\langle n \rangle = k$ if and only if $k^2-k+1 \leq n \leq k^2+k$. Hence
\begin{align*}
\sum_{n=1}^\infty \frac{2^{\langle n \rangle} + 2^{-\langle n \rangle}}{2^n}
&= \sum_{k=1}^\infty \sum_{n, \langle n \rangle = k}
\frac{2^{\langle n \rang... | 0.4375 | 7,746.0625 | 7,172.714286 | 8,192 |
Equilateral triangle $ABC$ has a side length of $12$. There are three distinct triangles $AD_1E_1$, $AD_2E_2$, and $AD_3E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = BD_3 = 6$. Find $\sum_{k=1}^3(CE_k)^2$. | 432 | 0 | 8,178.625 | -1 | 8,178.625 | |
Given that $\cos \alpha + \sin \alpha = \frac{2}{3}$, find the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4}) + 1}{1 + \tan \alpha}$. | -\frac{5}{9} | 0.4375 | 7,076.3125 | 5,641.857143 | 8,192 | |
In triangle \(PQR\), \(PQ = 8\), \(QR = 15\), and \(PR = 17\). If \(S\) and \(T\) are points on \(\overline{PQ}\) and \(\overline{PR}\) respectively, such that \(PS = 3\) and \(PT = 10\), find the area of triangle \(PST\). | \frac{225}{17} | 0.75 | 5,158.6875 | 4,497.833333 | 7,141.25 | |
Among the subsets of the set $\{1,2, \cdots, 100\}$, calculate the maximum number of elements a subset can have if it does not contain any pair of numbers where one number is exactly three times the other. | 67 | 0 | 8,192 | -1 | 8,192 | |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | 1 | 1 | 2,203.3125 | 2,203.3125 | -1 | |
Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$ . Determine all $n<2001$ with the property that $d_9-d_8=22$ . | 1995 | 0 | 8,192 | -1 | 8,192 | |
A quadrilateral connecting the midpoints of the sides of trapezoid $\mathrm{ABCD}$ is a rhombus. Find its area if the height of the trapezoid $\mathrm{BH}=5 \mathrm{c}$, the smaller base $\mathrm{BC}=6 \mathrm{~cm}$, and the angle $\mathrm{ABC}$ is $120^{\circ}$. | 15 | 0 | 8,192 | -1 | 8,192 | |
If the equation $\frac{m}{x-3}-\frac{1}{3-x}=2$ has a positive root with respect to $x$, then the value of $m$ is ______. | -1 | 0 | 8,020 | -1 | 8,020 | |
Define the function $g(x) = x^{-2} + \frac{x^{-2}}{1+x^{-2}}$. Determine $g(g(3))$. | \frac{72596100}{3034921} | 0 | 8,192 | -1 | 8,192 | |
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$ . Determine the measure of the angle $CBF$ . | 15 | 0.5625 | 6,566.0625 | 5,301.444444 | 8,192 | |
Let the set \( A = \{1, 2, \cdots, 2016\} \). For any 1008-element subset \( X \) of \( A \), if there exist \( x \) and \( y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good set". Find the largest positive integer \( a \) (where \( a \in A \)) such that any 1008-element subset containi... | 1008 | 0 | 8,192 | -1 | 8,192 | |
What is one-half times two-thirds times three-fourths? | \frac{1}{4} | 1 | 1,556.5625 | 1,556.5625 | -1 | |
Count how many 8-digit numbers there are that contain exactly four nines as digits. | 433755 | There are $\binom{8}{4} \cdot 9^{4}$ sequences of 8 numbers with exactly four nines. A sequence of digits of length 8 is not an 8-digit number, however, if and only if the first digit is zero. There are $\binom{7}{4} 9^{3}$ 8-digit sequences that are not 8-digit numbers. The answer is thus $\binom{8}{4} \cdot 9^{4}-\bi... | 0.0625 | 7,819.1875 | 6,232 | 7,925 |
In acute triangle $\triangle ABC$, if $\sin A = 3\sin B\sin C$, then the minimum value of $\tan A\tan B\tan C$ is \_\_\_\_\_\_. | 12 | 0.4375 | 6,596.375 | 5,165 | 7,709.666667 | |
The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$. | 13 | 0.0625 | 7,891.75 | 6,417 | 7,990.066667 | |
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$
holds for all positive integers $x, y$. | f(x) = x |
Let's analyze the problem by working with the given inequality:
\[
f(x) + y f(f(x)) \le x(1 + f(y))
\]
for all positive integers \(x, y\).
To find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) satisfying this inequality, we will first test some small values and then generalize our findings.
**Step 1: Cons... | 0.3125 | 6,712.9375 | 6,451.8 | 6,831.636364 |
Find a five-digit number that has the following property: when multiplied by 9, the result is a number represented by the same digits but in reverse order. | 10989 | 0.125 | 7,830.375 | 5,299 | 8,192 | |
Evaluate the expression: $2\log_{2}\;\sqrt {2}-\lg 2-\lg 5+ \frac{1}{ 3(\frac{27}{8})^{2} }$. | \frac{4}{9} | 0 | 4,135.4375 | -1 | 4,135.4375 |
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