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In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the basketball shooting competition. How many students participate in all three events? | 22 | 6/8 |
Find the number of pairs of integers $(x ; y)$ that satisfy the equation $6x^{2} - 7xy + y^{2} = 10^{100}$. | 19998 | 1/8 |
In a debate competition with 4 participants, the rules are as follows: each participant must choose one topic from two options, A and B. For topic A, answering correctly earns 100 points, and answering incorrectly results in a loss of 100 points. For topic B, answering correctly earns 90 points, and answering incorrectly results in a loss of 90 points. If the total score of the 4 participants is 0 points, how many different scoring situations are there for these 4 participants? | 36 | 7/8 |
Find the biggest real number $C$ , such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$ | 1010 | 1/8 |
Reading material: After studying square roots, Xiaoming found that some expressions containing square roots can be written as the square of another expression, such as: $3+2\sqrt{2}=(1+\sqrt{2})^{2}$. With his good thinking skills, Xiaoming conducted the following exploration:<br/>Let: $a+b\sqrt{2}=(m+n\sqrt{2})^2$ (where $a$, $b$, $m$, $n$ are all integers), then we have $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$.<br/>$\therefore a=m^{2}+2n^{2}$, $b=2mn$. In this way, Xiaoming found a method to convert some expressions of $a+b\sqrt{2}$ into square forms. Please follow Xiaoming's method to explore and solve the following problems:<br/>$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=(m+n\sqrt{3})^2$, express $a$, $b$ in terms of $m$, $n$, and get $a=$______, $b=$______;<br/>$(2)$ Using the conclusion obtained, find a set of positive integers $a$, $b$, $m$, $n$, fill in the blanks: ______$+\_\_\_\_\_\_=( \_\_\_\_\_\_+\_\_\_\_\_\_\sqrt{3})^{2}$;<br/>$(3)$ If $a+4\sqrt{3}=(m+n\sqrt{3})^2$, and $a$, $b$, $m$, $n$ are all positive integers, find the value of $a$. | 13 | 7/8 |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\begin{align*}
a + b + c + d &= 6, \\
a^2 + b^2 + c^2 + d^2 &= 12.
\end{align*}Let $m$ and $M$ denote minimum and maximum values of
\[4(a^3 + b^3 + c^3 + d^3) - (a^4 + b^4 + c^4 + d^4),\]respectively. Find $m + M.$ | 84 | 2/8 |
Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits, $\overline{AB}$ , such that $A$ and $B$ are both factors of $\overline{AB}$ ."
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"
Cat says, "That's cool, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!"
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
*Proposed by Andrew Wu* | 24 | 6/8 |
An triangle with coordinates $(x_1,y_1)$ , $(x_2, y_2)$ , $(x_3,y_3)$ has centroid at $(1,1)$ . The ratio between the lengths of the sides of the triangle is $3:4:5$ . Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\]
the area of the triangle can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*2021 CCA Math Bonanza Individual Round #11* | 107 | 4/8 |
Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done? | 144 | 1/8 |
Let \( S(n) \) denote the sum of the digits of a natural number \( n \). For example, \( S(123) = 1 + 2 + 3 = 6 \). If two different positive integers \( m \) and \( n \) satisfy the following conditions:
\[
\begin{cases}
m < 100 \\
n < 100 \\
m > n \\
m + S(n) = n + 2S(m)
\end{cases}
\]
then \( m \) and \( n \) form a pair \( \langle m, n \rangle \).
How many such pairs \( \langle m, n \rangle \) are there? | 99 | 1/8 |
Given two curves $y=x^{2}-1$ and $y=1-x^{3}$ have parallel tangents at point $x_{0}$, find the value of $x_{0}$. | -\dfrac{2}{3} | 6/8 |
Given integers \( m, n \) that satisfy \( m, n \in \{1, 2, \cdots, 1981\} \) and \(\left(n^{2}-mn-m^{2}\right)^{2}=1\), find the maximum value of \( m^{2}+n^{2} \). | 3524578 | 5/8 |
Given \( x \in [0, \pi] \), find the range of values for the function
$$
f(x)=2 \sin 3x + 3 \sin x + 3 \sqrt{3} \cos x
$$
| [-3\sqrt{3},8] | 7/8 |
The equation $\pi^{x-1} x^{2}+\pi^{x^{2}} x-\pi^{x^{2}}=x^{2}+x-1$ has how many solutions, where $\pi$ is the mathematical constant pi (the ratio of a circle's circumference to its diameter)? | 2 | 2/8 |
Class 3-1 of a certain school holds an evaluation activity for outstanding Young Pioneers. If a student performs excellently, they can earn a small red flower. 5 small red flowers can be exchanged for a small red flag. 4 small red flags can be exchanged for a badge. 3 badges can be exchanged for a small gold cup. To be evaluated as an outstanding Young Pioneer, a student needs to earn 2 small gold cups in one semester. How many small red flowers are needed, at least, to be evaluated as an outstanding Young Pioneer? | 120 | 7/8 |
In triangle $ABC$ with angle $A$ equal to $60^{\circ}$, the angle bisector $AD$ is drawn. The radius of the circumcircle of triangle $ADC$ with center at point $O$ is $\sqrt{3}$. Find the length of the segment $OM$, where $M$ is the intersection point of segments $AD$ and $BO$, given that $AB = 1.5$. | \frac{\sqrt{21}}{3} | 4/8 |
A sample is divided into 5 groups, with a total of 160 data points in the first, second, and third groups, and a total of 260 data points in the third, fourth, and fifth groups, and the frequency of the third group is 0.20. Calculate the frequency of the third group. | 70 | 5/8 |
Remove five out of twelve digits so that the remaining numbers sum up to 1111.
$$
\begin{array}{r}
111 \\
333 \\
+\quad 777 \\
999 \\
\hline 1111
\end{array}
$$ | 1111 | 1/8 |
Distribute 5 students into 3 groups: Group A, Group B, and Group C, with Group A having at least two people, and Groups B and C having at least one person each, and calculate the number of different distribution schemes. | 80 | 7/8 |
From an external point \(A\), a tangent \(AB\) and a secant \(ACD\) are drawn to a circle. Find the area of triangle \(CBD\), given that the ratio \(AC : AB = 2 : 3\) and the area of triangle \(ABC\) is 20. | 25 | 4/8 |
Xiaoming writes 6 numbers on three cards such that each side of each card has one number, and the sum of the numbers on each card is equal. Then, he places the cards on the table and finds that the numbers on the front sides are $28, 40, 49$. The numbers on the back sides can only be divisible by 1 and themselves. What is the average of the three numbers on the back sides?
A. 11
B. 12
C. 39
D. 40 | 12 | 1/8 |
Point $(x,y)$ is chosen randomly from the rectangular region with vertices at $(0,0)$, $(3036,0)$, $(3036,3037)$, and $(0,3037)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | \frac{506}{3037} | 7/8 |
Find the area of a triangle, given that the radius of the inscribed circle is 1, and the lengths of all three altitudes are integers. | 3\sqrt{3} | 5/8 |
Given the function $f(x)=(a+ \frac {1}{a})\ln x-x+ \frac {1}{x}$, where $a > 0$.
(I) If $f(x)$ has an extreme value point in $(0,+\infty)$, find the range of values for $a$;
(II) Let $a\in(1,e]$, when $x_{1}\in(0,1)$, $x_{2}\in(1,+\infty)$, denote the maximum value of $f(x_{2})-f(x_{1})$ as $M(a)$, does $M(a)$ have a maximum value? If it exists, find its maximum value; if not, explain why. | \frac {4}{e} | 4/8 |
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ | 9 | 5/8 |
Inside the tetrahedron \(ABCD\), points \(X\) and \(Y\) are given. The distances from point \(X\) to the faces \(ABC\), \(ABD\), \(ACD\), and \(BCD\) are \(14, 11, 29,\) and \(8\) respectively. The distances from point \(Y\) to the faces \(ABC\), \(ABD\), \(ACD\), and \(BCD\) are \(15, 13, 25,\) and \(11\) respectively. Find the radius of the inscribed sphere of the tetrahedron \(ABCD\). | 17 | 4/8 |
An influenza outbreak occurred in three areas, $A$, $B$, and $C$, where $6\%$, $5\%$, and $4\%$ of the population in each area have the flu, respectively. Assuming the population ratios in these three areas are $6:5:4$, if a person is randomly selected from these three areas, the probability that this person has the flu is ______. | \frac{77}{1500} | 3/8 |
Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\frac{1}{2}$, what is its speed? | 3 \sqrt{3} - 3 | 1/8 |
Given a positive integer \( n \geqslant 2 \), positive real numbers \( a_1, a_2, \ldots, a_n \), and non-negative real numbers \( b_1, b_2, \ldots, b_n \), which satisfy the following conditions:
(a) \( a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n = n \);
(b) \( a_1 a_2 \cdots a_n + b_1 b_2 \cdots b_n = \frac{1}{2} \).
Find the maximum value of \( a_1 a_2 \cdots a_n \left( \frac{b_1}{a_1} + \frac{b_2}{a_2} + \cdots + \frac{b_n}{a_n} \right) \). | \frac{1}{2} | 1/8 |
Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face. | 10 | 1/8 |
Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Shangmei number". Also, $f(3586)=\frac{{5-6}}{{3-8}}=\frac{1}{5}$ and $G(M)=35-86=-51$. If a "Shangmei number" $M$ has its thousands digit less than its hundreds digit, and $G(M)$ is a multiple of $7$, then the minimum value of $f(M)$ is ______. | -3 | 7/8 |
Given the circumcenter $O$ of $\triangle ABC$, and $2 \overrightarrow{OA} + 3 \overrightarrow{OB} + 4 \overrightarrow{OC} = \mathbf{0}$, find the value of $\cos \angle BAC$. | \frac{1}{4} | 5/8 |
A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence. | 10 | 1/8 |
In triangle $ABC$, $AB = 13$, $BC = 15$, and $CA = 14$. Point $D$ is on $\overline{BC}$ with $CD = 6$. Point $E$ is on $\overline{BC}$ such that $\angle BAE = \angle CAD$. Find $BE.$ | \frac{2535}{463} | 1/8 |
The cubic polynomial
\[8x^3 - 3x^2 - 3x - 1 = 0\]has a real root of the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},$ where $a,$ $b,$ and $c$ are positive integers. Find $a + b + c.$ | 98 | 3/8 |
In the polar coordinate system, the curve $C\_1$: $ρ=2\cos θ$, and the curve $C\_2$: $ρ\sin ^{2}θ=4\cos θ$. Establish a rectangular coordinate system $(xOy)$ with the pole as the coordinate origin and the polar axis as the positive semi-axis $x$. The parametric equation of the curve $C$ is $\begin{cases} x=2+ \frac {1}{2}t \ y= \frac {\sqrt {3}}{2}t\end{cases}$ ($t$ is the parameter).
(I) Find the rectangular coordinate equations of $C\_1$ and $C\_2$;
(II) The curve $C$ intersects $C\_1$ and $C\_2$ at four distinct points, arranged in order along $C$ as $P$, $Q$, $R$, and $S$. Find the value of $||PQ|-|RS||$. | \frac {11}{3} | 3/8 |
Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$ . | 17 | 3/8 |
Let \( f \) be an increasing function from \(\mathbf{R}\) to \(\mathbf{R}\) such that for any \( x \in \mathbf{R} \), \( f(x) = f^{-1}(x) \) (here \( f^{-1}(x) \) is the inverse function of \( f(x) \)). Prove that for any \( x \in \mathbf{R} \), \( f(x) = x \). | f(x)=x | 4/8 |
Given the parabola $y^{2}=4x$, a line with a slope of $\frac{\pi}{4}$ intersects the parabola at points $P$ and $Q$, and $O$ is the origin of the coordinate system. Find the area of triangle $POQ$. | 2\sqrt{2} | 1/8 |
What is the three-digit (integer) number which, when either increased or decreased by the sum of its digits, results in a number with all identical digits? | 105 | 1/8 |
Given 6 points on a plane, no three of which are collinear, prove that there always exists a triangle with vertices among these points that has an angle of at most $30^{\circ}$. | 30 | 3/8 |
On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.
(i) One may give cookies only to people adjacent to himself.
(ii) In order to give a cookie to one's neighbor, one must eat a cookie.
Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning. | 2^n | 1/8 |
There are 2013 cards with the digit 1 and 2013 cards with the digit 2. Vasya arranges these cards to form a 4026-digit number. On each move, Petya can swap any two cards and pay Vasya 1 ruble. The process ends when Petya forms a number divisible by 11. What is the maximum amount of money Vasya can earn if Petya aims to pay as little as possible? | 5 | 4/8 |
In how many ways can the numbers $1, 2, \ldots, n$ be arranged in such a way that, except for the number in the first position, each number is preceded by at least one of its (originally ordered) neighbors? | 2^{n-1} | 1/8 |
Calculate \(\int_{0}^{1} e^{-x^{2}} \, dx\) to an accuracy of 0.001. | 0.747 | 3/8 |
The height drawn to the base of an isosceles triangle is \( H \) and is twice as long as its projection on the lateral side. Find the area of the triangle. | H^2\sqrt{3} | 5/8 |
The numbers \(a, b,\) and \(c\) (not necessarily integers) are such that:
\[
a + b + c = 0 \quad \text{and} \quad \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 100
\]
What is the value of \(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\)? | -103 | 2/8 |
The Gauss family has three boys aged $7,$ a girl aged $14,$ and a boy aged $15.$ What is the mean (average) of the ages of the children? | 10 | 6/8 |
Given $$\overrightarrow {m} = (\sin \omega x + \cos \omega x, \sqrt {3} \cos \omega x)$$, $$\overrightarrow {n} = (\cos \omega x - \sin \omega x, 2\sin \omega x)$$ ($\omega > 0$), and the function $f(x) = \overrightarrow {m} \cdot \overrightarrow {n}$, if the distance between two adjacent axes of symmetry of $f(x)$ is not less than $\frac {\pi}{2}$.
(1) Find the range of values for $\omega$;
(2) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, and $a=2$, when $\omega$ is at its maximum, $f(A) = 1$, find the maximum area of $\triangle ABC$. | \sqrt {3} | 7/8 |
Find all injective functions $ f : \mathbb{N} \to \mathbb{N}$ which satisfy
\[ f(f(n)) \le\frac{n \plus{} f(n)}{2}\]
for each $ n \in \mathbb{N}$ . | f(n)=n | 4/8 |
The English college entrance examination consists of two parts: listening and speaking test with a full score of $50$ points, and the English written test with a full score of $100$ points. The English listening and speaking test is conducted twice. If a student takes both tests, the higher score of the two tests will be taken as the final score. If the student scores full marks in the first test, they will not take the second test. To prepare for the English listening and speaking test, Li Ming takes English listening and speaking mock exams every week. The table below shows his scores in $20$ English listening and speaking mock exams before the first actual test.
Assumptions:
1. The difficulty level of the mock exams is equivalent to that of the actual exam.
2. The difficulty level of the two actual listening and speaking tests is the same.
3. If Li Ming continues to practice listening and speaking after not scoring full marks in the first test, the probability of scoring full marks in the second test can reach $\frac{1}{2}$.
| $46$ | $50$ | $47$ | $48$ | $49$ | $50$ | $50$ | $47$ | $48$ | $47$ |
|------|------|------|------|------|------|------|------|------|------|
| $48$ | $49$ | $50$ | $49$ | $50$ | $50$ | $48$ | $50$ | $49$ | $50$ |
$(Ⅰ)$ Let event $A$ be "Li Ming scores full marks in the first English listening and speaking test." Estimate the probability of event $A$ using frequency.
$(Ⅱ)$ Based on the assumptions in the problem, estimate the maximum probability of Li Ming scoring full marks in the English college entrance examination listening and speaking test. | \frac{7}{10} | 3/8 |
If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$, then $r$ satisfies the equation | r^2+r+1=0 | 1/8 |
A classroom is paved with cubic bricks that have an edge length of 0.3 meters, requiring 600 bricks. If changed to cubic bricks with an edge length of 0.5 meters, how many bricks are needed? (Solve using proportions.) | 216 | 1/8 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$ and find the value of $v_3$. | 7.9 | 7/8 |
For each positive integer $n$ , determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$ | -n | 4/8 |
On the beach, there was a pile of apples belonging to 3 monkeys. The first monkey came, divided the apples into 3 equal piles with 1 apple remaining, then it threw the remaining apple into the sea and took one pile for itself. The second monkey came, divided the remaining apples into 3 equal piles with 1 apple remaining again, it also threw the remaining apple into the sea and took one pile. The third monkey did the same. How many apples were there originally at least? | 25 | 7/8 |
Given that $0 < \alpha < \frac{\pi}{2}$ and $0 < \beta < \frac{\pi}{2}$, if $\sin\left(\frac{\pi}{3}-\alpha\right) = \frac{3}{5}$ and $\cos\left(\frac{\beta}{2} - \frac{\pi}{3}\right) = \frac{2\sqrt{5}}{5}$,
(I) find the value of $\sin \alpha$;
(II) find the value of $\cos\left(\frac{\beta}{2} - \alpha\right)$. | \frac{11\sqrt{5}}{25} | 7/8 |
Color the 3 vertices of an equilateral triangle with three colors: red, blue, and green. How many distinct colorings are there? Consider the following:
1. Colorings that can be made to coincide by rotation are considered identical.
2. Colorings that can be made to coincide by rotation and reflection are considered identical. | 10 | 7/8 |
Two teams are participating in a game consisting of four contests. For each contest, each of the six judges assigns a score - an integer from 1 to 5. The computer calculates the average score for the contest and rounds it to one decimal place. The winner is determined by the sum of the four average scores calculated by the computer. Can it happen that the total of all scores given by the judges to the losing team is higher than the total for the winning team? | Yes | 1/8 |
Given that
$$
S=\left|\sqrt{x^{2}+4 x+5}-\sqrt{x^{2}+2 x+5}\right|,
$$
for real values of \(x\), find the maximum value of \(S^{4}\). | 4 | 7/8 |
Team A and Team B, each with 5 members, play a sequential knockout match in a pre-determined order. The first players of both teams compete first, and the loser is eliminated. The second player of the losing team then competes with the winner, and the loser is eliminated. This continues until all members of one team are eliminated, at which point the other team is declared the winner. Assuming each team member is equally skilled, what is the expected number of players from Team A who have not played by the time the match ends? Let $X$ represent this number. Find the mathematical expectation $E(X)$. | \frac{187}{256} | 1/8 |
The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration. | 15 | 3/8 |
Let \( n = 1 + 3 + 5 + \ldots + 31 \) and \( m = 2 + 4 + 6 \ldots + 32 \). If \( a = m - n \), find the value of \( a \).
If \(ABCD\) is a trapezium, \( AB = 4 \text{ cm}, EF = a \text{ cm}, CD = 22 \text{ cm} \) and \( FD = 8 \text{ cm} \), if the area of \( ABEF \) is \( b \text{ cm}^2 \), find the value of \( b \).
In \(\triangle ABC\), \( AB = AC = 10 \text{ cm} \) and \( \angle ABC = b^\circ - 100^\circ \). If \(\triangle ABC\) has \( c \) axis of symmetry, find the value of \( c \).
Let \( d \) be the least real root of the \( cx^{\frac{2}{3}} - 8x^{\frac{1}{3}} + 4 = 0 \), find the value of \( d \). | \frac{8}{27} | 2/8 |
If $a, b, c \in [-1, 1]$ satisfy $a + b + c + abc = 0$ , prove that $a^2 + b^2 + c^2 \ge 3(a + b + c)$ .
When does the equality hold?
| ^2+b^2+^2\ge3() | 2/8 |
During the process of choosing trial points using the 0.618 method, if the trial interval is $[3, 6]$ and the first trial point is better than the second, then the third trial point should be at _____. | 5.292 | 1/8 |
How many 0.1s are there in 1.9? How many 0.01s are there in 0.8? | 80 | 7/8 |
Out of the 200 natural numbers between 1 and 200, how many numbers must be selected to ensure that there are at least 2 numbers among them whose product equals 238? | 198 | 5/8 |
For an ordered pair $(m,n)$ of distinct positive integers, suppose, for some nonempty subset $S$ of $\mathbb R$ , that a function $f:S \rightarrow S$ satisfies the property that $f^m(x) + f^n(y) = x+y$ for all $x,y\in S$ . (Here $f^k(z)$ means the result when $f$ is applied $k$ times to $z$ ; for example, $f^1(z)=f(z)$ and $f^3(z)=f(f(f(z)))$ .) Then $f$ is called \emph{ $(m,n)$ -splendid}. Furthermore, $f$ is called \emph{ $(m,n)$ -primitive} if $f$ is $(m,n)$ -splendid and there do not exist positive integers $a\le m$ and $b\le n$ with $(a,b)\neq (m,n)$ and $a \neq b$ such that $f$ is also $(a,b)$ -splendid. Compute the number of ordered pairs $(m,n)$ of distinct positive integers less than $10000$ such that there exists a nonempty subset $S$ of $\mathbb R$ such that there exists an $(m,n)$ -primitive function $f: S \rightarrow S$ .
*Proposed by Vincent Huang* | 9998 | 1/8 |
How many positive integer multiples of $1001$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?
| 784 | 7/8 |
\( A, B, C \) are positive integers. It is known that \( A \) has 7 divisors, \( B \) has 6 divisors, \( C \) has 3 divisors, \( A \times B \) has 24 divisors, and \( B \times C \) has 10 divisors. What is the minimum value of \( A + B + C \)? | 91 | 4/8 |
In an isosceles trapezoid \(ABCD\), the angle bisectors of angles \(B\) and \(C\) intersect at the base \(AD\). Given that \(AB=50\) and \(BC=128\), find the area of the trapezoid. | 5472 | 3/8 |
There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\neq 0$, find the equation of the line. Enter your answer in the form "$y = mx + b$". | y=10x-4 | 7/8 |
By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\). | 310 | 2/8 |
Consider the integer sequence \( a_{1}, a_{2}, \cdots, a_{10} \) satisfying:
\[
a_{10} = 3a_{1}, \quad a_{2} + a_{8} = 2a_{5}
\]
and \( a_{i+1} \in \{1 + a_{i}, 2 + a_{i}\} \) for \(i = 1, 2, \cdots, 9\). Find the number of such sequences. | 80 | 3/8 |
Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube. | 7\pi - 4\sqrt{3}\pi | 1/8 |
A tribe called Mumbo-Jumbo lives by the river. One day, with urgent news, both a young warrior named Mumbo and a wise shaman named Yumbo set out simultaneously to a neighboring tribe. Mumbo ran 11 km/h to the nearest raft storage and then sailed on a raft to the neighboring tribe. Yumbo walked leisurely at 6 km/h to another raft storage and then sailed to the neighboring tribe from there. In the end, Yumbo arrived earlier than Mumbo. The river is straight, and the rafts move at the speed of the current. This speed is the same everywhere and is expressed as an integer in km/h, not less than 6. What is its maximum possible value? | 26 | 1/8 |
In triangle $ABC$, $AB=AC$ and $D$ is a point on $\overline{AC}$ so that $\overline{BD}$ bisects angle $ABC$. If $BD=BC$, what is the measure, in degrees, of angle $A$? | 36 | 7/8 |
A natural number, which does not end in zero, had one of its digits replaced with zero (if it was the leading digit, it was simply erased). As a result, the number became 9 times smaller. How many such numbers exist for which this is possible? | 7 | 1/8 |
A square is drawn on a plane with its sides horizontal and vertical. Several line segments parallel to its sides are drawn inside the square, such that no two segments lie on the same line and do not intersect at a point that is internal for both segments. The segments divide the square into rectangles, with each vertical line crossing the square and not containing any of the segments intersecting exactly $k$ rectangles, and each horizontal line crossing the square and not containing any of the segments intersecting exactly $l$ rectangles. What could be the number of rectangles in the partition? | kl | 7/8 |
In triangle \( \triangle ABC \),
\[ \tan A, \ (1+\sqrt{2}) \tan B, \ \tan C \]
form an arithmetic sequence. What is the minimum value of angle \( B \)? | \frac{\pi}{4} | 7/8 |
When \((1+x)^{38}\) is expanded in ascending powers of \(x\), \(N_{1}\) of the coefficients leave a remainder of 1 when divided by 3, while \(N_{2}\) of the coefficients leave a remainder of 2 when divided by 3. Find \(N_{1} - N_{2}\). | 4 | 6/8 |
A polynomial is called compatible if all its coefficients are equal to $0$, $1$, $2$, or $3$. Given a natural number $n$, find the number of compatible polynomials that satisfy the condition $p(2) = n$. | \lfloor\frac{n}{2}\rfloor+1 | 1/8 |
Let \( p(x) \) be a polynomial of degree at most \( 2n \), and for every integer \( k \) in the interval \([-n, n]\), it holds that \( |p(k)| \leq 1 \). Prove that for any \( x \in [-n, n] \), \( |p(x)| \leq 2^{2n} \). | 2^{2n} | 2/8 |
There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$. | 9999 | 7/8 |
The dimensions of a rectangular parallelepiped are 2, 3, and 6 cm. Find the edge length of a cube such that the ratios of their volumes are equal to the ratios of their surface areas. | 3\, | 1/8 |
Find the sum of all integer values of $h$ for which the equation $||r+h|-r|-4r=9|r-3|$ with respect to $r$ has at most one root. | -93 | 1/8 |
Lines $L_1,L_2,\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other.
All lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\{L_1,L_2,\dots,L_{100}\}$ is
$\textbf{(A) }4350\qquad \textbf{(B) }4351\qquad \textbf{(C) }4900\qquad \textbf{(D) }4901\qquad \textbf{(E) }9851$ | \textbf{(B)}4351 | 1/8 |
How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have? | 6 | 7/8 |
Two circles intersect at points \( A \) and \( B \). Chords \( A C \) and \( A D \) are drawn in each of these circles such that each chord in one circle is tangent to the other circle. Find \( A B \) if \( C B = a \) and \( D B = b \). | \sqrt{} | 1/8 |
Given points $P(\cos \alpha, \sin \alpha)$, $Q(\cos \beta, \sin \beta)$, and $R(\cos \alpha, -\sin \alpha)$ in a two-dimensional space, where $O$ is the origin, if the cosine distance between $P$ and $Q$ is $\frac{1}{3}$, and $\tan \alpha \cdot \tan \beta = \frac{1}{7}$, determine the cosine distance between $Q$ and $R$. | \frac{1}{2} | 2/8 |
A rectangular grid \(7 \times 14\) (cells) is given. What is the maximum number of three-cell corners that can be cut out from this rectangle? | 32 | 6/8 |
Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$ . For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and $7^{7^7}$ . Determine the sum of all possible values for the last two digits of $x_{2012}$ . | 75 | 4/8 |
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37}
b(a + d)\equiv b \pmod{37}
c(a + d)\equiv c \pmod{37}
bc + d^2\equiv d \pmod{37}
ad - bc\equiv 1 \pmod{37} \end{array}$ | 1 | 7/8 |
If $m$ and $b$ are real numbers and $mb>0$, then the line whose equation is $y=mx+b$ cannot contain the point
$\textbf{(A)}\ (0,1997)\qquad\textbf{(B)}\ (0,-1997)\qquad\textbf{(C)}\ (19,97)\qquad\textbf{(D)}\ (19,-97)\qquad\textbf{(E)}\ (1997,0)$ | \textbf{(E)} | 1/8 |
A secret facility is in the shape of a rectangle measuring $200 \times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths along the external perimeter (while the intruder remained in place). Three guards ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder? | 150 | 4/8 |
Find the functions \( f: \mathbb{Q}_{+}^{*} \rightarrow \mathbb{Q}_{+}^{*} \) such that for all \( x \in \mathbb{Q}_{+}^{*} \), we have \( f(x+1) = f(x) + 1 \) and \( f(1/x) = 1 / f(x) \). | f(x)=x | 2/8 |
A sequence of numbers is defined by the conditions: \(a_{1}=1\), \(a_{n+1}=a_{n}+\left\lfloor\sqrt{a_{n}}\right\rfloor\).
How many perfect squares appear among the first terms of this sequence that do not exceed 1,000,000? | 10 | 4/8 |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), prove that \(\cos \alpha + \cos \beta + \sqrt{2} \sin \alpha \sin \beta \leq \frac{3 \sqrt{2}}{2}\). | \frac{3\sqrt{2}}{2} | 7/8 |
In the diagram, \(C\) lies on \(AE\) and \(AB=BC=CD\). If \(\angle CDE=t^{\circ}, \angle DEC=(2t)^{\circ}\), and \(\angle BCA=\angle BCD=x^{\circ}\), determine the measure of \(\angle ABC\). | 60 | 2/8 |
Dots are spaced one unit part, horizontally and vertically. What is the number of square units enclosed by the polygon?
[asy]
/* AMC8 1998 #6P */
size(1inch,1inch);
pair a=(0,0), b=(10,0), c=(20,0), d=(30, 0);
pair e=(0,10), f=(10,10), g=(20,10), h=(30,10);
pair i=(0,20), j=(10,20), k=(20,20), l=(30,20);
pair m=(0,30), n=(10,30), o=(20,30), p=(30,30);
dot(a);
dot(b);
dot(c);
dot(d);
dot(e);
dot(f);
dot(g);
dot(h);
dot(i);
dot(j);
dot(k);
dot(l);
dot(m);
dot(n);
dot(o);
dot(p);
draw(a--b--g--c--d--h--l--k--o--j--i--e--a);
[/asy] | 6 | 2/8 |
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