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A regular hexagon is divided into 24 triangles. In 19 of the vertices, write distinct real numbers. Prove that among the 24 triangles formed, at least 7 of them have the numbers at their vertices in counterclockwise increasing order. | 7 | 1/8 |
On the Saturday of a weekend softball tournament, Team A plays Team B and Team C plays Team D. Then on Sunday, the two Saturday winners play for first and second places while the two Saturday losers play for third and fourth places. There are no ties. One possible ranking of the team from first place to fourth place at the end of the tournament is the sequence ACBD. What is the total number of possible four-team ranking sequences at the end of the tournament? | 16 | 7/8 |
In the plane Cartesian coordinate system $xOy$, $AB$ is a chord of the parabola $y^{2}=4x$ passing through the point $F(1,0)$, and the circumcircle of triangle $\triangle AOB$ intersects the parabola at point $P$ (different from points $O, A, B$). If $PF$ bisects the angle $\angle APB$, find all possible values of $|PF|$. | \sqrt{13}-1 | 1/8 |
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(-1));
C = extension(A,B,(0,0),(0,1));
draw(graph(parab,-2,2));
draw(A--B);
draw((0,0)--(0,4));
dot("$A$", A, E);
dot("$B$", B, SW);
dot("$(0,c)$", C, NW);
[/asy] | 4 | 6/8 |
What is the greatest integer $x$ such that $|6x^2-47x+15|$ is prime? | 8 | 7/8 |
Let $\triangle A B C$ be a scalene triangle. Let $h_{a}$ be the locus of points $P$ such that $|P B-P C|=|A B-A C|$. Let $h_{b}$ be the locus of points $P$ such that $|P C-P A|=|B C-B A|$. Let $h_{c}$ be the locus of points $P$ such that $|P A-P B|=|C A-C B|$. In how many points do all of $h_{a}, h_{b}$, and $h_{c}$ concur? | 2 | 3/8 |
If the total sum of squared deviations of a set of data is 100, and the correlation coefficient is 0.818, then the sum of squared residuals is. | 33.0876 | 2/8 |
An equiangular hexagon has side lengths 6, 7, 8, 9, 10, 11 (not necessarily in this order). If the area of the hexagon is \( k \sqrt{3} \), find the sum of all possible values of \( k \). | 213 | 1/8 |
A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth game. | 21 | 1/8 |
Convert the decimal number 2011 to a base-7 number. | 5602_7 | 1/8 |
Let \( n \) be a fixed integer, \( n \geq 2 \).
(I) Find the minimum constant \( c \) such that the inequality
\[ \sum_{1 \leq i < j \leq n} x_i x_j \left( x_i^2 + x_j^2 \right) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]
holds.
(II) For this constant \( c \), determine the necessary and sufficient conditions for which equality holds. | \frac{1}{8} | 3/8 |
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
| 320 | 7/8 |
A cuckoo clock chimes the number of times corresponding to the current hour (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it was 9:05 and started moving the minute hand forward until the clock read 7 hours later. How many times did the cuckoo chime during this period? | 43 | 7/8 |
Consider the following sequence of numbers: \( 1 ; 1 ; 2 ; 3 ; 7 ; 22 ; \ldots \). In the sequence, starting from the third element, each element is one more than the product of the two preceding elements. Prove that no element in the sequence is divisible by 4. | Noelementinthesequenceisdivisible4 | 2/8 |
Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible? | 4 | 7/8 |
Is it true that any natural number can be multiplied by one of the numbers 1, 2, 3, 4, or 5 so that the result starts with the digit 1? | True | 2/8 |
The height of a regular triangular pyramid is \( 6 \sqrt{6} \), and the lateral edge forms an angle of \( 45^\circ \) with the base plane. Find the distance from the center of the base of the pyramid to a lateral face. | \frac{6\sqrt{30}}{5} | 3/8 |
Read the material first, then answer the question.
$(1)$ Xiao Zhang encountered a problem when simplifying a quadratic radical: simplify $\sqrt{5-2\sqrt{6}}$.
After thinking about it, Xiao Zhang's process of solving this problem is as follows:
$\sqrt{5-2\sqrt{6}}=\sqrt{2-2\sqrt{2\times3}+3}$①
$=\sqrt{{(\sqrt{2})}^2}-2\sqrt{2}\times\sqrt{3}+{(\sqrt{3})}^2$②
$=\sqrt{{(\sqrt{2}-\sqrt{3})}^2}$③
$=\sqrt{2}-\sqrt{3}$④
In the above simplification process, an error occurred in step ____, and the correct result of the simplification is ____;
$(2)$ Please simplify $\sqrt{8+4\sqrt{3}}$ based on the inspiration you obtained from the above material. | \sqrt{6}+\sqrt{2} | 7/8 |
A tour group has three age categories of people, represented in a pie chart. The central angle of the sector corresponding to older people is $9^{\circ}$ larger than the central angle for children. The percentage of total people who are young adults is $5\%$ higher than the percentage of older people. Additionally, there are 9 more young adults than children. What is the total number of people in the tour group? | 120 | 7/8 |
Along the southern shore of an endless sea stretches an archipelago consisting of an infinite number of islands. The islands are connected by an infinite chain of bridges, and each island is connected to the shore by a bridge. In the event of a strong earthquake, each bridge independently has a probability \( p = 0.5 \) of being destroyed. What is the probability that after a strong earthquake, it will be possible to travel from the first island to the shore using the remaining bridges? | \frac{2}{3} | 2/8 |
The height of the pyramid $P-ABCD$ with a square base of side length $2\sqrt{2}$ is $1$. If the radius of the circumscribed sphere of the pyramid is $2\sqrt{2}$, then the distance between the center of the square $ABCD$ and the point $P$ is ______. | 2\sqrt{2} | 2/8 |
Consider $4n$ points in the plane, with no three points collinear. Using these points as vertices, we form $\binom{4n}{3}$ triangles. Show that there exists a point $X$ of the plane that belongs to the interior of at least $2n^3$ of these triangles. | 2n^3 | 1/8 |
Given vectors $\overrightarrow{a}=(\cos x, \sin x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x, 2\cos x-\sqrt{3}\sin x)$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.
$(2)$ If the maximum value of the function $g(x)=f(x-\frac{\pi}{6})+af(\frac{x}{2}-\frac{\pi}{6})-af(\frac{x}{2}+\frac{\pi}{12})$ on the interval $[0,\pi]$ is $6$, determine the value of the real number $a$. | 2\sqrt{2} | 1/8 |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 =25$ and $(x-4)^2 + 9y^2 = 81 ?$
$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 42$ | \textbf{B} | 1/8 |
In triangle \(ABC\), \(\angle B = 110^\circ\) and \(\angle C = 50^\circ\). On side \(AB\), point \(P\) is chosen such that \(\angle PCB = 30^\circ\), and on side \(AC\), point \(Q\) is chosen such that \(\angle ABQ = 40^\circ\). Find the angle \(\angle QPC\). | 40 | 1/8 |
Call a day a *perfect* day if the sum of the digits of the month plus sum of the digits of the day equals the sum of digits of the year. For example, February $28$ th, $2028$ is a perfect day because $2+2+8=2+0+2+8$ . Find the number of perfect days in $2018$ .
*2018 CCA Math Bonanza Team Round #5* | 36 | 6/8 |
Find the solutions to
\[\frac{13x - x^2}{x + 1} \left( x + \frac{13 - x}{x + 1} \right) = 42.\]Enter all the solutions, separated by commas. | 1, 6, 3 + \sqrt{2}, 3 - \sqrt{2} | 1/8 |
A number in the form of $7777 \cdots 77$ is written on the blackboard. Each time the last digit is erased, the remaining number is multiplied by 3, and then the erased digit is added back. This operation is continued repeatedly. What is the final number obtained? | 7 | 6/8 |
Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$ | |z|>\frac{1}{\sqrt[n]{n}} | 1/8 |
Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h, they will meet 16 km away from point C. What was the original speed of car A? | 30 | 7/8 |
$p$ and $q$ are primes such that the numbers $p+q$ and $p+7q$ are both squares. Find the value of $p$. | 2 | 7/8 |
Find the sum $$\frac{2^{1}}{4^{1}-1}+\frac{2^{2}}{4^{2}-1}+\frac{2^{4}}{4^{4}-1}+\frac{2^{8}}{4^{8}-1}+\cdots$$ | 1 | 6/8 |
Let \( a \) and \( b \) be natural numbers such that \( a^2 + ab + 1 \) is divisible by \( b^2 + ba + 1 \). Prove that \( a = b \). | b | 2/8 |
Let \(I\) be the center of the inscribed circle in triangle \(ABC\). We assume that \(AB = AC + CI\). Determine the value of the ratio \(\frac{\widehat{ACB}}{\widehat{ABC}}\). | 2 | 3/8 |
Along an alley, 75 trees consisting of maples and larches were planted in a single row. It is known that there are no two maples with exactly 5 trees between them. What is the maximum number of maples that could have been planted along the alley? | 39 | 6/8 |
The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}$ is a positive rational number. If $a_{n}=\frac{p_{n}}{q_{n}} (n=1,2, \cdots)$, where $p_{n}$ and $q_{n}$ are coprime positive integers, then
$$
a_{n+1}=\frac{p_{n}^{2}+2015}{p_{n} q_{n}}.
$$
Question: Does there exist an $a_{1} > 2015$ such that the sequence $\left\{a_{n}\right\}$ is a bounded sequence? Prove your conclusion. | No | 3/8 |
A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron. | 4 \sqrt{2} | 1/8 |
Solve the following equation: $\log _{2010}(2009 x)=\log _{2009}(2010 x)$. | \frac{1}{2009\cdot2010} | 1/8 |
In the quadratic equation $2x^{2}-1=6x$, the coefficient of the quadratic term is ______, the coefficient of the linear term is ______, and the constant term is ______. | -1 | 3/8 |
A regular triangular pyramid is intersected by a plane passing through its lateral edge and height. The section formed is a triangle with an angle of \(\pi / 4\) at the apex of the pyramid. Find the angle between a lateral face and the base plane of the pyramid. | \arctan(2) | 5/8 |
The administration divided the region into several districts based on the principle that the population of a large district constitutes more than 8% of the region's population and for any large district, there will be two non-large districts with a combined larger population. What is the smallest number of districts the region could have been divided into? | 8 | 1/8 |
Let triangle $ABC$ have side lengths $AB = 13$ , $BC = 14$ , $AC = 15$ . Let $I$ be the incenter of $ABC$ . The circle centered at $A$ of radius $AI$ intersects the circumcircle of $ABC$ at $H$ and $J$ . Let $L$ be a point that lies on both the incircle of $ABC$ and line $HJ$ . If the minimal possible value of $AL$ is $\sqrt{n}$ , where $n \in \mathbb{Z}$ , find $n$ . | 17 | 2/8 |
Find the legs of a right triangle if it is known that the radius of the circumcircle of the triangle is $R$ and the radius of the incircle is $r$. For what ratio $\frac{R}{r}$ does the problem have a solution? | \frac{R}{r}\ge1+\sqrt{2} | 1/8 |
The monthly pension of football fan Ivan Ivanovich is
$$
\frac{149^{6}-199^{3}}{149^{4}+199^{2}+199 \cdot 149^{2}}
$$
rubles, and the price of a ticket for the World Cup match is 22000 rubles. Will Ivan Ivanovich have enough pension for one month to buy one ticket? Justify your answer. | 1 | 1/8 |
Let $a \geq 2$ be a real number, and let $x_1$ and $x_2$ be the roots of the equation $x^{2} - a x + 1 = 0$. Define $S_{n} = x_{1}^{n} + x_{2}^{n}$ for $n = 1, 2, \cdots$.
(1) Determine the monotonicity of the sequence $\left\{\frac{S_{n}}{S_{n+1}}\right\}$ and provide a proof.
(2) Find all real numbers $a$ such that for any positive integer $n$, certain conditions are satisfied. | 2 | 2/8 |
Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\in S$, there exist $f_1, \cdots, f_k \in F$ (not necessarily different), such that $f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))$.
Find the least positive integer $m$, such that if $F$ satisfies property $P(2019)$, then it also satisfies property $P(m)$. | 595 | 1/8 |
Let \( a, b, c \) be positive numbers such that \( a + b + c = \lambda \). If the inequality
\[ \frac{1}{a(1 + \lambda b)} + \frac{1}{b(1 + \lambda c)} + \frac{1}{c(1 + \lambda a)} \geq \frac{27}{4} \]
always holds, find the range of values for \( \lambda \). | (0,1] | 7/8 |
There were 15 different non-integer numbers written on a board. For each number \( x \) among these fifteen, Vasya wrote down separately \( [x] \) and \( \frac{1}{\{x\}} \). What is the minimum number of distinct numbers that could appear in Vasya's notebook?
(Note: \( [x] \) and \( \{x\} \) represent the integer and fractional parts of the number \( x \), respectively.) | 4 | 1/8 |
Let \( n \) be a positive integer such that \( n \geq 2 \). Given distinct real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) forming a set \( S \), define \( k(S) \) as the number of distinct values in the form \( a_{i}+2^{j} \) for \( i, j = 1, 2, \ldots, n \). Determine the minimum possible value of \( k(S) \) for all sets \( S \). | \frac{n(n+1)}{2} | 1/8 |
A 1-liter carton of milk used to cost 80 rubles. Recently, in an effort to cut costs, the manufacturer reduced the carton size to 0.9 liters and increased the price to 99 rubles. By what percentage did the manufacturer's revenue increase? | 37.5 | 7/8 |
Let \( f(n) \) be a function defined on all positive integers and taking positive integer values. For all positive integers \( m \) and \( n \), it holds that \( f[f(m) + f(n)] = m + n \). Find all possible values of \( f(1988) \). (1988 Mexican Olympiad problem) | 1988 | 7/8 |
In the Metropolitan County, there are $25$ cities. From a given bar chart, the average population per city is indicated midway between $5,200$ and $5,800$. If two of these cities, due to a recent demographic survey, were found to exceed the average by double, calculate the closest total population of all these cities. | 148,500 | 1/8 |
The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3?$
$\textbf{(A)} ~-1 \qquad\textbf{(B)} ~0 \qquad\textbf{(C)} ~1 \qquad\textbf{(D)} ~2 \qquad\textbf{(E)} ~\sqrt{5}$ | (B)~0 | 1/8 |
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 | 7/8 |
The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{72}$. Find the greatest possible value of $k$. | 2\sqrt{26} | 7/8 |
In the convex quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) are drawn. It is known that \(AD = 2\), \(\angle ABD = \angle ACD = 90^\circ\), and the distance between the centers of the circles inscribed in triangles \(ABD\) and \(ACD\) is \(\sqrt{2}\). Find \(BC\). | \sqrt{3} | 1/8 |
Let the number of elements in the set \( S \) be denoted by \( |S| \), and the number of subsets of the set \( S \) be denoted by \( n(S) \). Given three non-empty finite sets \( A, B, C \) that satisfy the following conditions:
$$
\begin{array}{l}
|A| = |B| = 2019, \\
n(A) + n(B) + n(C) = n(A \cup B \cup C).
\end{array}
$$
Determine the maximum value of \( |A \cap B \cap C| \) and briefly describe the reasoning process. | 2018 | 7/8 |
Let \(a^2 + b^2 = c^2 + d^2 = 1\) and \(ac + bd = 0\) for some real numbers \(a, b, c, d\). Find all possible values of the expression \(ab + cd\). | 0 | 5/8 |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b$ . | 22 | 7/8 |
Rectangle $W X Y Z$ has $W X=4, W Z=3$, and $Z V=3$. The rectangle is curled without overlapping into a cylinder so that sides $W Z$ and $X Y$ touch each other. In other words, $W$ touches $X$ and $Z$ touches $Y$. The shortest distance from $W$ to $V$ through the inside of the cylinder can be written in the form $\sqrt{\frac{a+b \pi^{2}}{c \pi^{2}}}$ where $a, b$ and $c$ are positive integers. What is the smallest possible value of $a+b+c$? | 18 | 2/8 |
In triangle \(ABC\), side \(AB\) is \(\frac{5\sqrt{2}}{2}\), and side \(BC\) is \(\frac{5\sqrt{5}}{4}\). Point \(M\) is on side \(AB\), and point \(O\) is on side \(BC\) such that lines \(MO\) and \(AC\) are parallel. Segment \(BM\) is 1.5 times longer than segment \(AM\). The angle bisector of \(\angle BAC\) intersects line \(MO\) at point \(P\), which lies between points \(M\) and \(O\), and the radius of the circumcircle of triangle \(AMP\) is \(\sqrt{2+\sqrt{2}}\). Find the length of side \(AC\). | \frac{15}{4} | 1/8 |
Let \(\{x_1, x_2, x_3, \ldots, x_n\}\) be a set of \(n\) distinct positive integers, such that the sum of any 3 of them is a prime number. What is the maximum value of \(n\)? | 4 | 2/8 |
Given a square $ABCD$ with side length $a$, vertex $A$ lies in plane $\beta$, and the other vertices are on the same side of plane $\beta$. The distances from points $B$ and $D$ to plane $\beta$ are $1$ and $2$, respectively. If the angle between plane $ABCD$ and plane $\beta$ is $30^\circ$, find $a$. | 2\sqrt{5} | 2/8 |
In the convex quadrilateral \(ABCD\), the midpoints of sides \(BC\) and \(CD\) are \(E\) and \(F\) respectively. The segments \(AE\), \(EF\), and \(AF\) divide the quadrilateral into four triangles whose areas are four consecutive integers. What is the maximum possible area of triangle \(ABD\)? | 6 | 1/8 |
In a shop, there is one type of lollipop and one type of ice cream bar. The price of both lollipops and ice cream bars is given in whole groats.
Barborka bought three lollipops. Eliška bought four lollipops and several ice cream bars - it is known that she bought more than one and less than ten ice cream bars. Honzík bought one lollipop and one ice cream bar. Barborka paid 24 groats, and Eliška paid 109 groats.
How many groats did Honzík pay?
(L. Hozová) | 19 | 6/8 |
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color?
$\mathrm{(A)}\ 1/12\qquad\mathrm{(B)}\ 1/10\qquad\mathrm{(C)}\ 1/6\qquad\mathrm{(D)}\ 1/3\qquad\mathrm{(E)}\ 1/2$ | \mathrm{(C)}\\frac{1}{6} | 1/8 |
Given an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with a focal length of $2\sqrt{3}$, and the ellipse passes through the point $(\sqrt{3}, \frac{1}{2})$.
(Ⅰ) Find the equation of ellipse E;
(Ⅱ) Through point P$(-2, 0)$, draw two lines with slopes $k_1$ and $k_2$ respectively. These two lines intersect ellipse E at points M and N. When line MN is perpendicular to the y-axis, find the value of $k_1 \cdot k_2$. | \frac{1}{4} | 7/8 |
Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. How many minutes did Xiaoming spend on the bus that day? | 10 | 1/8 |
Define the operations:
\( a \bigcirc b = a^{\log _{7} b} \),
\( a \otimes b = a^{\frac{1}{\log ^{6} b}} \), where \( a, b \in \mathbb{R} \).
A sequence \(\{a_{n}\} (n \geqslant 4)\) is given such that:
\[ a_{3} = 3 \otimes 2, \quad a_{n} = (n \otimes (n-1)) \bigcirc a_{n-1}. \]
Then, the integer closest to \(\log _{7} a_{2019}\) is: | 11 | 2/8 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=1+t\cos\alpha}\\{y=t\sin\alpha}\end{array}}\right.$ ($t$ is the parameter, $0\leqslant \alpha\ \ \lt \pi$). Taking the origin $O$ as the pole and the non-negative $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is ${\rho^2}=\frac{{12}}{{3+{{\sin}^2}\theta}}$. <br/>$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of $C_{2}$; <br/>$(2)$ Given $F(1,0)$, the intersection points $A$ and $B$ of curve $C_{1}$ and $C_{2}$ satisfy $|BF|=2|AF|$ (point $A$ is in the first quadrant), find the value of $\cos \alpha$. | \frac{2}{3} | 6/8 |
Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\] | 3 | 7/8 |
The village council of the secret pipeline is gathering around a round table, where each arriving member can sit in any available seat. How many different seating arrangements are possible if 7 participants join the council? (Two arrangements are considered identical if the same people are sitting to the left and right of each participant, and empty seats are not considered.) | 720 | 7/8 |
Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?
[asy]
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
[/asy] | 540 | 2/8 |
Calculate the remainder when $1 + 11 + 11^2 + \cdots + 11^{1024}$ is divided by $500$. | 25 | 3/8 |
The integers $a_0, a_1, a_2, a_3,\ldots$ are defined as follows: $a_0 = 1$ , $a_1 = 3$ , and $a_{n+1} = a_n + a_{n-1}$ for all $n \ge 1$ .
Find all integers $n \ge 1$ for which $na_{n+1} + a_n$ and $na_n + a_{n-1}$ share a common factor greater than $1$ . | n\equiv3\pmod{5} | 1/8 |
A set of composite numbers from the set $\{1,2,3,4, \ldots, 2016\}$ is called good if any two numbers in this set do not have common divisors (other than 1). What is the maximum number of numbers that a good set can have? | 14 | 2/8 |
Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}-22 y-69 z+703=0 \\
y^{2}+23 x+23 z-1473=0 \\
z^{2}-63 x+66 y+2183=0
\end{array}\right.
$$ | (20,-22,23) | 1/8 |
Given points $A(-6, -1)$, $B(1, 2)$, and $C(-3, -2)$, find the coordinates of vertex $M$ of the parallelogram $ABMC$. | (4,1) | 7/8 |
Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an
integer. Find the smallest $k$ for which the set A can be partitioned into two subsets
having the same number of elements, the same sum of elements, the same sum
of the squares of elements, and the same sum of the cubes of elements. | 4 | 1/8 |
Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjacent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$? | 6 | 2/8 |
Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\). | \frac{\sqrt{5}}{5} | 7/8 |
Use $[x]$ to denote the greatest integer less than or equal to the real number $x$. Let $a_n = \sum_{k=1}^{n} \left\lfloor \frac{n}{k} \right\rfloor$. Find the number of even numbers among $a_1, a_2, \cdots, a_{2018}$. | 1028 | 3/8 |
For a positive integer $n$ , two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win? | n-1 | 1/8 |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. | \frac{\sqrt{3}}{4} | 1/8 |
Let \( f \) be a function that takes in a triple of integers and outputs a real number. Suppose that \( f \) satisfies the equations
\[
f(a, b, c) = \frac{f(a+1, b, c) + f(a-1, b, c)}{2}
\]
\[
f(a, b, c) = \frac{f(a, b+1, c) + f(a, b-1, c)}{2}
\]
\[
f(a, b, c) = \frac{f(a, b, c+1) + f(a, b, c-1)}{2}
\]
for all integers \( a, b, c \). What is the minimum number of triples at which we need to evaluate \( f \) in order to know its value everywhere? | 8 | 1/8 |
Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same:
```
4 6 10 14 22 26
6 9 15 21 33 39
10 15 25 35 55 65
16 24 40 56 88 104
18 27 45 63 99 117
20 30 50 70 110 130
``` | 648648000 | 5/8 |
Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying \begin{enumerate} \item[(a)] $f(0) > 0$, \item[(b)] $g(0) = 0$, \item[(c)] $|f'(x)| \leq |g(x)|$ for all $x$, \item[(d)] $|g'(x)| \leq |f(x)|$ for all $x$, and \item[(e)] $f(r) = 0$. \end{enumerate} | \frac{\pi}{2} | 3/8 |
Given a positive sequence $\{a_n\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \frac{5}{2}a_{n+1}a_n$, where $n \in \mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
1. If $a_2 = \frac{3}{2}$, $a_3 = x$, and $a_4 = 4$, find the range of $x$.
2. Suppose the sequence $\{a_n\}$ is a geometric sequence with a common ratio of $q$. If $\frac{1}{2}S_n < S_{n+1} < 2S_n$ for $n \in \mathbb{N}^*$, find the range of $q$.
3. If $a_1, a_2, \ldots, a_k$ ($k \geq 3$) form an arithmetic sequence, and $a_1 + a_2 + \ldots + a_k = 120$, find the minimum value of the positive integer $k$, and the corresponding sequence $a_1, a_2, \ldots, a_k$ when $k$ takes the minimum value. | 16 | 1/8 |
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
$\textbf{(A)}\ \frac{2}{5}\qquad\textbf{(B)}\ \frac{3}{7}\qquad\textbf{(C)}\ \frac{4}{7}\qquad\textbf{(D)}\ \frac{3}{5}\qquad\textbf{(E)}\ \frac{4}{5}$ | \textbf{(C)}\\frac{4}{7} | 1/8 |
Given the parabola $C$: $y^2=2px (p > 0)$ with focus $F$ and directrix $l$. A line perpendicular to $l$ at point $A$ on the parabola $C$ at $A(4,y_0)$ intersects $l$ at $A_1$. If $\angle A_1AF=\frac{2\pi}{3}$, determine the value of $p$. | 24 | 7/8 |
Given the function $f(x)=ax^{3}-4x+4$, where $a\in\mathbb{R}$, $f′(x)$ is the derivative of $f(x)$, and $f′(1)=-3$.
(1) Find the value of $a$;
(2) Find the extreme values of the function $f(x)$. | -\frac{4}{3} | 7/8 |
There are \( n \) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \( n \). | 52 | 5/8 |
The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event B occurs is $\frac{2}{3}$.
Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval
$\textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad \textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad \textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad \textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad \textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$ | \textbf{(D)}\Big[\frac{5}{12},\frac{2}{3}\Big] | 1/8 |
Jason borrowed money from his parents to buy a new surfboard. His parents have agreed to let him work off his debt by babysitting under the following conditions: his first hour of babysitting is worth $\$1$, the second hour worth $\$2$, the third hour $\$3$, the fourth hour $\$4$, the fifth hour $\$5$, the sixth hour $\$6$, the seventh hour $\$1$, the eighth hour $\$2$, etc. If he repays his debt by babysitting for 39 hours, how many dollars did he borrow? | \$132 | 6/8 |
For a natural number \( N \), if at least five of the natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "five-rule number." What is the smallest "five-rule number" greater than 2000? | 2004 | 7/8 |
Given a cube \(ABCD A_1 B_1 C_1 D_1\) with edge length \(a\), there is a segment intersecting the edge \(C_1 D_1\), with its endpoints lying on the lines \(AA_1\) and \(BC\). What is the minimum possible length of this segment? | 3a | 3/8 |
For a real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994\). | 312 | 7/8 |
Points \( A, B, C, D \) lie on a circle in that order such that \(\frac{AB}{BC} = \frac{DA}{CD}\). If \(AC = 3\) and \(BD = BC = 4\), find \(AD\). | \frac{3}{2} | 5/8 |
Simplify the expression \(\frac{2 a \sqrt{1+x^{2}}}{x+\sqrt{1+x^{2}}}\) where \(x=\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}} - \sqrt{\frac{b}{a}}\right)\) and \(a>0, b>0\). | +b | 1/8 |
A uniform cubic die with faces numbered $1, 2, 3, 4, 5, 6$ is rolled three times independently, resulting in outcomes $a_1, a_2, a_3$. Find the probability of the event "$|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_1| = 6$". | 1/4 | 7/8 |
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