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Isosceles triangles \(ABC\) (\(AB = BC\)) and \(A_1B_1C_1\) (\(A_1B_1 = B_1C_1\)) are similar, and \(AC : A_1C_1 = 5 : \sqrt{3}\).
Vertices \(A_1\) and \(B_1\) are located on sides \(AC\) and \(BC\), respectively, and vertex \(C_1\) is on the extension of side \(AB\) beyond point \(B\), with \(A_1B_1 \perp BC\). Find the angle \(B\). | 120 | 1/8 |
An electronic watch displays the time as $09:15:12$ at 9:15:12 AM and 13:11:29 at 1:11:29 PM. How many times in a 24-hour day does the six digits of the time form a symmetric sequence (i.e., the time reads the same forwards and backwards, such as 01:33:10)? | 96 | 3/8 |
The function \( f(x) \) satisfies for all real numbers \( x \):
\[ f(2-x) = f(2+x) \]
\[ f(5+x) = f(5-x) \]
and \( f(0) = 0 \).
Determine the minimum number of zeros of \( f(x) \) on the interval \([-21, 21]\). | 14 | 4/8 |
Binbin's height is 1.46 meters, his father is 0.32 meters taller than Binbin, and his mother's height is 1.5 meters.
(1) How tall is Binbin's father?
(2) How much shorter is Binbin's mother than his father? | 0.28 | 3/8 |
What fraction of the Earth's volume lies above the 45 degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane. | \frac{8-5\sqrt{2}}{16} | 4/8 |
A boy named Vasya looked at a clock and saw a time where the number of hours was greater than zero. He read the digits as a three- or four-digit number (for example, 5:07 would become 507). Then he calculated the number of minutes that had passed since midnight. Could it be that the doubled second number he got is divisible by the first number? | No | 2/8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a,c)$ and $\vec{n}=(\cos C,\cos A)$.
1. If $\vec{m}\parallel \vec{n}$ and $a= \sqrt {3}c$, find angle $A$;
2. If $\vec{m}\cdot \vec{n}=3b\sin B$ and $\cos A= \frac {3}{5}$, find the value of $\cos C$. | \frac {4-6 \sqrt {2}}{15} | 6/8 |
Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$? | $a+4$ | 5/8 |
Let \( a \in \mathbf{R}_{+} \). If the function
\[
f(x)=\frac{a}{x-1}+\frac{1}{x-2}+\frac{1}{x-6} \quad (3 < x < 5)
\]
achieves its maximum value at \( x=4 \), find the value of \( a \). | -\frac{9}{2} | 1/8 |
Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. | 441 | 2/8 |
If five geometric means are inserted between $8$ and $5832$, the fifth term in the geometric series:
$\textbf{(A)}\ 648\qquad\textbf{(B)}\ 832\qquad\textbf{(C)}\ 1168\qquad\textbf{(D)}\ 1944\qquad\textbf{(E)}\ \text{None of these}$ | \textbf{(A)}\648 | 1/8 |
The integer part and decimal part of $(\sqrt{10}+3)^{2n+1}$ for $n \in \mathbf{N}$ are denoted as $I$ and $F$, respectively. What is the value of $P(I + F)$? | 1 | 1/8 |
Given real numbers \( x, y, z \) satisfying
\[
\frac{y}{x-y}=\frac{x}{y+z}, \quad z^{2}=x(y+z)-y(x-y)
\]
find the value of
\[
\frac{y^{2}+z^{2}-x^{2}}{2 y z}.
\] | \frac{1}{2} | 3/8 |
Find all positive integers $k$ satisfying: there is only a finite number of positive integers $n$ , such that the positive integer solution $x$ of $xn+1\mid n^2+kn+1$ is not unique. | k\ne2 | 1/8 |
Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers? | 28 | 1/8 |
Find all 10-digit numbers \( a_0 a_1 \ldots a_9 \) such that for each \( k \), \( a_k \) is the number of times that the digit \( k \) appears in the number. | 6210001000 | 1/8 |
Let QR = x, PR = y, and PQ = z. Given that the area of the square on side QR is 144 = x^2 and the area of the square on side PR is 169 = y^2, find the area of the square on side PQ. | 25 | 7/8 |
A rhombus \(ABCD\) and a triangle \(ABC\), containing its longer diagonal, are inscribed with circles. Find the ratio of the radii of these circles, if the acute angle of the rhombus is \(\alpha\). | 1+\cos\frac{\alpha}{2} | 7/8 |
Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer. | 252 | 5/8 |
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 12$ and $CP = 4.$ If $\tan \angle APD = 2,$ then find $AB.$ | 12 | 5/8 |
On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.
At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet.
After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls.
Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls? | n+1 | 2/8 |
If \( a, b, c \) are the three real roots of the equation
\[ x^{3} - x^{2} - x + m = 0, \]
then the minimum value of \( m \) is _____. | -\frac{5}{27} | 6/8 |
If P and Q are points on the line y = 1 - x and the curve y = -e^x, respectively, find the minimum value of |PQ|. | \sqrt{2} | 7/8 |
A certain pharmaceutical company has developed a new drug to treat a certain disease, with a cure rate of $p$. The drug is now used to treat $10$ patients, and the number of patients cured is denoted as $X$.
$(1)$ If $X=8$, two patients are randomly selected from these $10$ people for drug interviews. Find the distribution of the number of patients cured, denoted as $Y$, among the selected patients.
$(2)$ Given that $p\in \left(0.75,0.85\right)$, let $A=\{k\left|\right.$ probability $P\left(X=k\right)$ is maximum$\}$, and $A$ contains only two elements. Find $E\left(X\right)$. | \frac{90}{11} | 7/8 |
The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid. | 104 | 1/8 |
Determine by how many times the number \((2014)^{2^{2014}} - 1\) is greater than the number written in the following form:
\[
\left(\left((2014)^{2^0} + 1\right) \cdot \left((2014)^{2^1} + 1\right) \cdot \left((2014)^{2^2} + 1\right) \ldots \cdot \left((2014)^{2^{2013}} + 1\right)\right) + 1.
\] | 2013 | 3/8 |
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 26$ | \textbf{(C)}\13 | 1/8 |
Given the function \( f(x) = \sqrt{x+2} + k \), and that there exist \( a, b \) (\(a < b\)) such that the range of \( f(x) \) on \([a, b]\) is \([a, b]\), find the range of values for the real number \( k \). | (-\frac{9}{4},-2] | 7/8 |
A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52 | 38 | 1/8 |
Let \( a_{1}, a_{2}, \ldots, a_{2018} \) be 2018 real numbers such that:
\[ \sum_{i=1}^{2018} a_{i} = 0 \]
\[ \sum_{i=1}^{2018} a_{i}^{2} = 2018 \]
Determine the maximum possible minimum value of the product of any two of these 2018 real numbers. | -1 | 2/8 |
Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that in any \( n \)-element subset of \( S \), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a number in one group that is coprime with the other four numbers in the same group, and a number in the other group that is not coprime with the other four numbers in that group. | 50 | 2/8 |
In the sequence \(1^{2}, 2^{2}, 3^{2}, \cdots, 2005^{2}\), add a "+" or "-" sign before each number to make the algebraic sum the smallest non-negative number. Write the resulting expression. | 1 | 4/8 |
Let \( P \in \mathbb{R}[X] \) be a non-zero polynomial such that \( P \mid P(X^2 + X + 1) \). Show that the degree of \( P \) is even.
Hint: Show that \( P \) has no real roots. | TheofPiseven. | 1/8 |
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\] | (\frac{1}{3},\frac{1}{3},\frac{1}{3}) | 1/8 |
The numbers \(a, b,\) and \(c\) are such that \((a + b)(b + c)(c + a) = abc\) and \(\left(a^3 + b^3\right)\left(b^3 + c^3\right)\left(c^3 + a^3\right) = a^3b^3c^3\). Prove that \(abc = 0\). | abc=0 | 1/8 |
Given that $0 < a \leqslant \frac{5}{4}$, find the range of real number $b$ such that all real numbers $x$ satisfying the inequality $|x - a| < b$ also satisfy the inequality $|x - a^2| < \frac{1}{2}$. | \frac{3}{16} | 1/8 |
Among all the roots of
\[z^8 - z^6 + z^4 - z^2 + 1 = 0,\]the maximum imaginary part of a root can be expressed as $\sin \theta,$ where $-90^\circ \le \theta \le 90^\circ.$ Find $\theta.$ | 54^\circ | 5/8 |
Find the smallest positive period of the function \( f(x) = \cos(\sqrt{2} x) + \sin\left(\frac{3}{8} \sqrt{2} x\right) \). | 8\sqrt{2}\pi | 6/8 |
Let \( f(x) = \frac{x}{x+1} \).
Given:
\[ a_1 = \frac{1}{2}, \, a_2 = \frac{3}{4} \]
\[ a_{n+2} = f(a_n) + f(a_{n+1}) \quad \text{for} \, n = 1, 2, \ldots \]
Prove: For any positive integer \( n \), the following inequality holds:
\[ f\left(3 \times 2^{n-1}\right) \leq a_{2n} \leq f\left(3 \times 2^{2n-2}\right) \] | f(3\times2^{n-1})\lea_{2n}\lef(3\times2^{2n-2}) | 1/8 |
In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo).
What is the maximal possible number of edges in such a graph? | 42550 | 3/8 |
One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), it could be that she is in the cafe (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has a one in two chance of not finding her. If she is on one of the courts, there is another one in three chance of missing her, but if she went to the cafe, François will definitely find her: he knows which cafe Béatrice usually enjoys her ice cream. François visited all three possible meeting places but still did not find Béatrice.
What is the probability that Béatrice was on the beach, assuming she did not change locations while François was searching for her? | \frac{3}{5} | 3/8 |
The opposite sides of a quadrilateral inscribed in a circle intersect at points \( P \) and \( Q \). Find the length of the segment \( |PQ| \), given that the tangents to the circle drawn from \( P \) and \( Q \) are \( a \) and \( b \) respectively. | \sqrt{^2+b^2} | 5/8 |
A certain school conducted physical fitness tests on the freshmen to understand their physical health conditions. Now, $20$ students are randomly selected from both male and female students as samples. Their test data is organized in the table below. It is defined that data $\geqslant 60$ indicates a qualified physical health condition.
| Level | Data Range | Number of Male Students | Number of Female Students |
|---------|--------------|-------------------------|---------------------------|
| Excellent | $[90,100]$ | $4$ | $6$ |
| Good | $[80,90)$ | $6$ | $6$ |
| Pass | $[60,80)$ | $7$ | $6$ |
| Fail | Below $60$ | $3$ | $2$ |
$(Ⅰ)$ Estimate the probability that the physical health level of the freshmen in this school is qualified.
$(Ⅱ)$ From the students in the sample with an excellent level, $3$ students are randomly selected for retesting. Let the number of female students selected be $X$. Find the distribution table and the expected value of $X$.
$(Ⅲ)$ Randomly select $2$ male students and $1$ female student from all male and female students in the school, respectively. Estimate the probability that exactly $2$ of these $3$ students have an excellent health level. | \frac{31}{250} | 2/8 |
Little Tiger places chess pieces on the grid points of a 19 × 19 Go board, forming a solid rectangular dot matrix. Then, by adding 45 more chess pieces, he transforms it into a larger solid rectangular dot matrix with one side unchanged. What is the maximum number of chess pieces that Little Tiger originally used? | 285 | 1/8 |
Does there exist a positive integer \( m \) such that the equation \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = \frac{m}{a+b+c}\) has infinitely many solutions in positive integers \( (a, b, c) \)? | 12 | 1/8 |
What is the least possible number of cells that can be marked on an \( n \times n \) board such that for each \( m > \frac{n}{2} \), both diagonals of any \( m \times m \) sub-board contain a marked cell? | n | 1/8 |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | \frac{9}{16} | 1/8 |
In triangle $ABC$, if $a=2$, $c=2\sqrt{3}$, and $\angle A=30^\circ$, then the area of $\triangle ABC$ is equal to __________. | \sqrt{3} | 1/8 |
The area of triangle $\triangle OFA$, where line $l$ has an inclination angle of $60^\circ$ and passes through the focus $F$ of the parabola $y^2=4x$, and intersects with the part of the parabola that lies on the x-axis at point $A$, is equal to $\frac{1}{2}\cdot OA \cdot\frac{1}{2} \cdot OF \cdot \sin \theta$. Determine the value of this expression. | \sqrt {3} | 1/8 |
By how many times is the number \( A \) greater or smaller than the number \( B \), if
\[
A = \underbrace{1+\ldots+1}_{2022 \text{ times}}+\underbrace{2+\ldots+2}_{2021 \text{ times}}+\ldots+2021+2021+2022,
\]
\[
B = \underbrace{2023+\ldots+2023}_{2022 \text{ times}}+\underbrace{2022+\ldots+2022}_{2021 \text{ times}}+\ldots+3+3+2
\] | 2 | 2/8 |
Arrange the sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cycle: $(3)$, $(5, 7)$, $(9, 11, 13)$, $(15, 17, 19, 21)$, $(23)$, $(25, 27)$, $(29, 31, 33)$, $(35, 37, 39, 41)$, $(43)$, ..., then 2013 is the number in the $\boxed{\text{nth}}$ bracket. | 403 | 4/8 |
In the Cartesian coordinate system, the parametric equations of the line $C_{1}$ are $\left\{\begin{array}{l}x=1+t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ (where $t$ is the parameter). Using the origin $O$ as the pole and the positive x-axis as the polar axis, the polar equation of the curve $C_{2}$ is ${\rho}^{2}=\frac{4}{3-\cos2\theta}$.
$(1)$ Find the Cartesian equation of the curve $C_{2}$.
$(2)$ If the line $C_{1}$ intersects the curve $C_{2}$ at points $A$ and $B$, and $P(1,0)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$. | 2\sqrt{2} | 7/8 |
A whole number representing the number of meters of sold fabric in a record book was covered in ink. The total revenue couldn't be read either, but the end of this record showed: 7 p. 28 k., and it is known that this amount does not exceed 500 p. The price of 1 meter of fabric is 4 p. 36 k. Help the auditor restore this record. | 98 | 1/8 |
In a pasture where the grass grows evenly every day, the pasture can feed 10 sheep for 20 days, or 14 sheep for 12 days. How many days of grass growth each day is enough to feed 2 sheep? | 2 | 1/8 |
In a polar coordinate system with the pole at point $O$, the curve $C\_1$: $ρ=6\sin θ$ intersects with the curve $C\_2$: $ρ\sin (θ+ \frac {π}{4})= \sqrt {2}$. Determine the maximum distance from a point on curve $C\_1$ to curve $C\_2$. | 3+\frac{\sqrt{2}}{2} | 5/8 |
At a mathematics competition, three problems were given: $A$, $B$, and $C$. There were 25 students who each solved at least one problem. Among the students who did not solve problem $A$, twice as many solved $B$ as solved $C$. One more student solved only problem $A$ than the number of those who also solved problem $A$. Half of the students who solved only one problem did not solve $A$. How many students solved only problem $B$? | 6 | 4/8 |
Define \( x \star y = \frac{\sqrt{x^2 + 3xy + y^2 - 2x - 2y + 4}}{xy + 4} \). Compute
\[ ((\cdots((2007 \star 2006) \star 2005) \star \cdots) \star 1) . \] | \frac{\sqrt{15}}{9} | 1/8 |
Given a point P is 9 units away from the center of a circle with a radius of 15 units, find the number of chords passing through point P that have integer lengths. | 12 | 7/8 |
Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write ten Cs: CcCcCc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are different colors? | 222480 | 4/8 |
Let $PA$, $PB$, and $PC$ be three non-coplanar rays originating from point $P$, with each pair of rays forming a $60^\circ$ angle. A sphere with a radius of 1 is tangent to each of these three rays. Find the distance from the center of the sphere $O$ to point $P$. | \sqrt{3} | 4/8 |
Given positive integers \(a\), \(b\), \(c\), and \(d\) satisfying the equations
\[
a^{2}=c(d+29) \quad \text{and} \quad b^{2}=c(d-29),
\]
what is the value of \(d\)? | 421 | 5/8 |
A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets the other two sides at points X and Y. What is the maximum length of XY if the triangle has a perimeter \( p \)? | \frac{p}{8} | 7/8 |
The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A,$ respectively. In this process, card number $(n+1)$ becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original positions. Find the number of cards in the magical stack in which card number 131 retains its original position.
| 392 | 3/8 |
Four mathematicians, two physicists, one chemist, and one biologist take part in a table tennis tournament. The eight players are to compete in four pairs by drawing lots. What is the probability that no two mathematicians play against each other? | \frac{8}{35} | 7/8 |
Show that a rectangle is the only convex polygon in which all angles are right angles. | 4 | 1/8 |
A circle with its center at the intersection point of the diagonals $K M$ and $L N$ of an isosceles trapezoid $K L M N$ touches the shorter base $L M$ and the leg $M N$. Find the perimeter of trapezoid $K L M N$ given that its height is 36 and the radius of the circle is 11. | 129 | 3/8 |
Given that the perimeter of each of the nine small equilateral triangles is $6 \mathrm{cm}$, calculate the perimeter of $\triangle A B C$. | 18 | 1/8 |
Two cyclists depart simultaneously from point \( A \) with different speeds and travel to point \( B \). Upon reaching \( B \), they immediately turn back. The first cyclist, who is faster than the second, meets the second cyclist on the return journey at a distance of \( a \) km from \( B \). Then, after reaching \( A \), the first cyclist travels again towards \( B \), and after covering \(\boldsymbol{k}\)-th part of the distance \( AB \), meets the second cyclist, who is returning from \( B \). Find the distance from \( \boldsymbol{A} \) to \( \boldsymbol{B} \). | 2ak | 1/8 |
For every positive integer $n$ , let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$ ; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$ . If we define $f(n)=(-1)^{s(n)}$ , prove that the limit
\[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} \]
exists and determine its value. | \frac{20}{21} | 3/8 |
There are 49 children, each wearing a unique number from 1 to 49 on their chest. Select several children and arrange them in a circle such that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? | 18 | 1/8 |
Prove that for any scalene triangle, \( l_{1}^{2} > \sqrt{3} S > l_{2}^{2} \), where \( l_{1} \) and \( l_{2} \) are the lengths of the longest and shortest angle bisectors of the triangle, respectively, and \( S \) is its area. | l_1^2>\sqrt{3}S>l_2^2 | 1/8 |
Let \( S \) be the smallest subset of the integers with the property that \( 0 \in S \) and for any \( x \in S \), we have \( 3x \in S \) and \( 3x + 1 \in S \). Determine the number of non-negative integers in \( S \) less than 2008. | 128 | 3/8 |
Find the area of the region defined by the inequality: \( |y - |x - 2| + |x|| \leq 4 \). | 32 | 1/8 |
A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coordinates of the vertices of $Q_3$. | 44 | 1/8 |
There are 200 candies. What is the minimum number of schoolchildren that these candies can be distributed to so that, no matter how the candies are distributed, there are always at least two schoolchildren who receive the same number of candies (possibly none)? | 21 | 7/8 |
Find the sum of the infinite series $1+2\left(\dfrac{1}{1998}\right)+3\left(\dfrac{1}{1998}\right)^2+4\left(\dfrac{1}{1998}\right)^3+\cdots$. | \frac{3992004}{3988009} | 1/8 |
What is the maximum value that the area of the projection of a regular tetrahedron with an edge length of 1 can take? | \frac{1}{2} | 1/8 |
In the set of positive integers less than 10,000, how many integers \( x \) are there such that \( 2^x - x^2 \) is divisible by 7? | 2857 | 5/8 |
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\sin A + \sin B = 2\sin C\cos B$ and the area of $\triangle ABC$ is $S = \frac{\sqrt{3}}{2}c$, find the minimum value of $ab$. | 12 | 7/8 |
Find the positive integer $n\,$ for which
\[\lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994\]
(For real $x\,$, $\lfloor x\rfloor\,$ is the greatest integer $\le x.\,$) | 312 | 6/8 |
A circle passes through vertex $B$ of triangle $ABC$, touches side $AC$ at its midpoint $D$, and intersects sides $AB$ and $BC$ at points $M$ and $N$ respectively. Given that $AB:BC=3:2$, find the ratio of the area of triangle $AMD$ to the area of triangle $DNC$. | 4:9 | 1/8 |
Use the five digits $0$, $1$, $2$, $3$, $4$ to form integers that satisfy the following conditions:
(I) All four-digit integers;
(II) Five-digit integers without repetition that are greater than $21000$. | 66 | 7/8 |
For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always | -6[-W] | 5/8 |
Given a unit cube $A B C D - A_{1} B_{1} C_{1} D_{1}$ with the midpoints of its edges $A B$, $A_{1} D_{1}$, $A_{1} B_{1}$, and $B C$ being $L$, $M$, $N$, and $K$ respectively, find the radius of the inscribed sphere of the tetrahedron $L M N K$. | \frac{\sqrt{3}-\sqrt{2}}{2} | 6/8 |
How many integers from 1 to 2001 have a digit sum that is divisible by 5? | 399 | 2/8 |
In a plane, two rectangular coordinate systems \( O x y \) and \( O' x' y' \) are chosen such that the points \( O \) and \( O' \) do not coincide, the axis \( O x \) is not parallel to the axis \( O' x' \), and the units of length in these systems are different.
Does there exist a point in the plane where both coordinates are the same in each of the coordinate systems? | Yes | 4/8 |
If the solution set of the system of linear inequalities in one variable $x$, $\left\{{\begin{array}{l}{x-1≥2x+1}\\{2x-1＜a}\end{array}}\right.$, is $x\leqslant -2$, and the solution of the fractional equation in variable $y$, $\frac{{y-1}}{{y+1}}=\frac{a}{{y+1}}-2$, is negative, then the sum of all integers $a$ that satisfy the conditions is ______. | -8 | 3/8 |
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$? | 15 | 7/8 |
Elsa wrote the numbers $1, 2, \ldots, 8$ on the board. She then allows herself operations of the following form: she selects two numbers $a$ and $b$ such that $a+2 \leq b$, if such numbers exist, then she erases them, and writes the numbers $a+1$ and $b-1$ in their place.
(a) Demonstrate that Elsa must stop after a finite number of operations.
(b) What is the maximum number of operations Elsa can perform? | 20 | 2/8 |
Given \( x, y, z \in \mathbf{R}_{+} \) and \( x + y + z = 1 \). Find the maximum value of \( x + \sqrt{2xy} + 3 \sqrt[3]{xyz} \). | 2 | 1/8 |
Suppose $\angle A=75^{\circ}$ and $\angle C=45^{\circ}$. Then $\angle B=60^{\circ}$. Points $M$ and $N$ lie on the circle with diameter $AC$. Triangle $BMN$ is similar to triangle $BCA$:
$$MNA = \angle MCA \quad (\text{they subtend the same arc}), \quad \angle BNM = 90^{\circ} - \angle MNA = 90^{\circ} - \angle MCA = \angle A = 75^{\circ}$$
$$NMC = \angle NAC (\text{they subtend the same arc}), \quad \angle BMN = 90^{\circ} - \angle NMC = 90^{\circ} - \angle NAC = \angle C = 45^{\circ}$$
The similarity coefficient $k_{1}$ of triangles $BMN$ and $BCA$ is the ratio of corresponding sides:
$$
k_{1}=\frac{BN}{BA}=\cos \angle B = \frac{1}{2}.
$$
Similarly, triangle $MAP$ is similar to triangle $ABC$ with similarity coefficient $k_{2}=\cos \angle A = \cos 75^{\circ}$, and triangle $CNP$ is similar to triangle $ABC$ with similarity coefficient:
$$
k_{3}=\cos \angle C = \frac{\sqrt{2}}{2}.
$$
The areas of similar triangles are proportional to the squares of their similarity coefficients:
$$
S_{BMN} = k_{1}^{2} \cdot S_{ABC} = \frac{1}{4} S_{ABC}, \quad S_{AMP} = k_{2}^{2} \cdot S_{ABC}, \quad S_{CNP} = k_{3}^{2} \cdot S_{ABC} = \frac{1}{2} S_{ABC}.
$$
Then:
$$
S_{MNP} = \left(1 - \frac{1}{4} - \frac{1}{2} - \cos^{2} 75^{\circ}\right) \cdot S_{ABC} \rightarrow S_{MNP} : S_{ABC} = \left(\frac{1}{4} - \frac{1 + \cos 150^{\circ}}{2}\right) = \frac{\sqrt{3} - 1}{4}.
$$ | (\sqrt{3}-1):4 | 1/8 |
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? | 173 | 1/8 |
Given that the central angle of a sector is $\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \_\_\_\_\_\_ cm, and the area of the sector is \_\_\_\_\_\_ cm<sup>2</sup>. | 27 | 6/8 |
Let \( n \) be a positive integer. Given \( n \) positive numbers \( x_{1}, x_{2}, \cdots, x_{n} \) whose product is 1, prove that:
$$
\sum_{i=1}^{n}\left(x_{i} \sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{i}^{2}}\right) \geqslant \frac{(n+1) \sqrt{n}}{2}.
$$ | \frac{(n+1)\sqrt{n}}{2} | 1/8 |
In $\triangle PQR,$ $PQ=PR=30$ and $QR=28.$ Points $M, N,$ and $O$ are located on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{MN}$ and $\overline{NO}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PMNO$? | 60 | 4/8 |
Define the digitlength of a positive integer to be the total number of letters used in spelling its digits. For example, since "two zero one one" has a total of 13 letters, the digitlength of 2011 is 13. We begin at any positive integer and repeatedly take the digitlength. Show that after some number of steps, we must arrive at the number 4. | 4 | 7/8 |
An express train overtakes a freight train. The speed of the express train is as many times greater than the speed of the freight train as the time spent passing by each other when traveling in the same direction is greater than the time they would spend traveling past each other in opposite directions. What is this ratio? | 1+\sqrt{2} | 7/8 |
Let $f:[0,1]\to[0,\infty)$ be an arbitrary function satisfying $$ \frac{f(x)+f(y)}2\le f\left(\frac{x+y}2\right)+1 $$ for all pairs $x,y\in[0,1]$ . Prove that for all $0\le u<v<w\le1$ , $$ \frac{w-v}{w-u}f(u)+\frac{v-u}{w-u}f(w)\le f(v)+2. $$ | \frac{w-v}{w-u}f(u)+\frac{v-u}{w-u}f(w)\lef(v)+2 | 1/8 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seven-letter good words are there? | 192 | 7/8 |
How many positive integers $n$ satisfy the inequality
\[
\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ?
\]
Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$ . | 15049 | 1/8 |
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