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A rectangular sheet of cardboard was cut along its diagonal. In one of the obtained pieces, 2 cuts were made parallel to the 2 shorter sides and through the midpoints of those sides. In the end, a rectangle with a perimeter of \(129 \, \text{cm}\) remained. What was the perimeter of the sheet before the cut? | 258\, | 1/8 |
The analysis of bank accounts revealed that the balances on each of them are more than 10 rubles. Additionally, there was a group of clients, each having the same amount of money on their account. This amount is a number consisting solely of ones. If the total amount of money on the accounts of this group of clients is calculated, this sum will also be a number consisting solely of ones. Find the minimum number of clients in the group for which this is possible, given that there are more than one person in the group. | 101 | 7/8 |
Given the ellipse \( C: \frac{x^{2}}{9} + \frac{y^{2}}{8} = 1 \) with left and right foci \( F_{1} \) and \( F_{2} \), and left and right vertices \( A \) and \( B \), respectively, a line \( l: x = my + 1 \) passing through the right focus \( F_{2} \) intersects the ellipse \( C \) at points \( M(x_{1}, y_{1}) \) and \( N(x_{2}, y_{2}) \) where \( y_{1} > 0 \) and \( y_{2} < 0 \). If \( MA \perp NF_{1} \), find the real number \( m \). | \frac{\sqrt{3}}{12} | 2/8 |
What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$? | 23 | 2/8 |
There are 100 distinct cards with numbers \(2, 5, 2^{2}, 5^{2}, \ldots, 2^{50}, 5^{50}\) (each card has exactly one number, and each number occurs exactly once). In how many ways can you choose 2 cards such that the product of the numbers on the chosen cards is a cube of an integer? | 1074 | 7/8 |
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$ . Determine the area of $\vartriangle DEF$ . | \frac{9\sqrt{3}}{16} | 3/8 |
Three fair dice are tossed at random (i.e., all faces have the same probability of coming up).
What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?
$\textbf{(A) }\frac{1}{6}\qquad \textbf{(B) }\frac{1}{9}\qquad \textbf{(C) }\frac{1}{27}\qquad \textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$ | (B)\frac{1}{9} | 1/8 |
Given vectors $\overrightarrow{a} = (3, 4)$ and $\overrightarrow{b} = (t, -6)$, and $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear, the projection of vector $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ is \_\_\_\_\_. | -5 | 7/8 |
Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway. | 193 | 7/8 |
In a group of seven friends, the mean (average) age of three of the friends is 12 years and 3 months, and the mean age of the remaining four friends is 13 years and 5 months. In months, what is the mean age of all seven friends? | 155 | 7/8 |
The two figures shown are made of unit squares. What is the positive difference of the perimeters, in units?
[asy]
draw((0,0)--(0,1)--(5,1)--(5,0)--cycle,linewidth(1));
draw((1,0)--(1,2)--(4,2)--(4,0),linewidth(1));
draw((2,-1)--(2,3)--(3,3)--(3,-1)--cycle,linewidth(1));
draw((7,0)--(7,2)--(12,2)--(12,0)--cycle,linewidth(1));
draw((7,1)--(12,1),linewidth(1));
draw((8,0)--(8,2),linewidth(1));
draw((9,0)--(9,2),linewidth(1));
draw((10,0)--(10,2),linewidth(1));
draw((11,0)--(11,2),linewidth(1));
[/asy] | 4 | 1/8 |
Distribute 10 volunteer positions among 4 schools, with the requirement that each school receives at least one position. How many different ways can the positions be distributed? (Answer with a number.) | 84 | 7/8 |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
draw((-5,i)--(5,i),gray(0.7));
}
draw((-5,0)--(5,0),Arrows(6));
draw((0,-5)--(0,5),Arrows(6));
label("$x$", (5,0), E);
label("$y$", (0,5), N);
draw(graph(func,-3,3),red);
label("$y = f(x)$", (3,-2), UnFill);
[/asy]
Which is the graph of $y = f(x + 2)$?
[asy]
unitsize(0.5 cm);
picture[] graf;
int i, n;
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
real funcc(real x) {
return(func(x - 2));
}
real funce(real x) {
return(func(x + 2));
}
for (n = 1; n <= 5; ++n) {
graf[n] = new picture;
for (i = -5; i <= 5; ++i) {
draw(graf[n],(i,-5)--(i,5),gray(0.7));
draw(graf[n],(-5,i)--(5,i),gray(0.7));
}
draw(graf[n],(-5,0)--(5,0),Arrows(6));
draw(graf[n],(0,-5)--(0,5),Arrows(6));
label(graf[n],"$x$", (5,0), E);
label(graf[n],"$y$", (0,5), N);
}
draw(graf[1],shift((2,-1))*graph(func,-3,3),red);
draw(graf[2],shift((2,1))*graph(func,-3,3),red);
draw(graf[3],graph(funcc,-1,5),red);
draw(graf[4],shift((-2,-1))*graph(func,-3,3),red);
draw(graf[5],graph(funce,-5,1),red);
label(graf[1], "A", (0,-6));
label(graf[2], "B", (0,-6));
label(graf[3], "C", (0,-6));
label(graf[4], "D", (0,-6));
label(graf[5], "E", (0,-6));
add(graf[1]);
add(shift((12,0))*(graf[2]));
add(shift((24,0))*(graf[3]));
add(shift((6,-12))*(graf[4]));
add(shift((18,-12))*(graf[5]));
[/asy]
Enter the letter of the graph of $y = f(x + 2).$ | \text{E} | 7/8 |
Given \(a, b, c, d \in \mathbf{R}\) such that \(a^{2} + b^{2} + c^{2} + d^{2} = 1\), find the maximum value of \(ab + bc + cd + da + ac + 4bd\). | \frac{5}{2} | 1/8 |
In an experiment, a certain constant \( c \) is measured to be 2.43865 with an error range of \(\pm 0.00312\). The experimenter wants to publish the value of \( c \), with each digit being significant. This means that regardless of how large \( c \) is, the announced value of \( c \) (with \( n \) digits) must match the first \( n \) digits of the true value of \( c \). What is the most precise value of \( c \) that the experimenter can publish? | 2.44 | 2/8 |
A bottle of cola costs 2 yuan, and two empty bottles can be exchanged for one more bottle of cola. With 30 yuan, what is the maximum number of bottles of cola that you can drink? | 29 | 7/8 |
A cubic polynomial $p(x)$ satisfies
\[p(n) = \frac{1}{n^2}\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$ | -\frac{5}{12} | 2/8 |
Given the ellipse $C$: $mx^{2}+3my^{2}=1$ ($m > 0$) with a major axis length of $2\sqrt{6}$, and $O$ is the origin.
$(1)$ Find the equation of the ellipse $C$.
$(2)$ Let point $A(3,0)$, point $B$ be on the $y$-axis, and point $P$ be on the ellipse $C$ and to the right of the $y$-axis. If $BA=BP$, find the minimum value of the area of quadrilateral $OPAB$. | 3\sqrt{3} | 7/8 |
On the sides $BC$ and $AC$ of the isosceles triangle $ABC$ ($AB = AC$), points $D$ and $E$ were found respectively such that $AE = AD$ and $\angle EDC = 18^\circ$. Find the measure of the angle $\angle BAD$. | 36 | 3/8 |
Two consecutive sides of a quadrilateral measure $10 \mathrm{~cm}$ and $15 \mathrm{~cm}$. If each diagonal divides the quadrilateral into two regions of equal area, calculate its perimeter. | 50\, | 1/8 |
As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces 1 counterfeit coin for every 99 real ones. Alex isn't very good at this, so none of the counterfeit coins are the right weight. Since the mint is not perfect, each coin is weighed before leaving. If the coin is not the right weight, then it is sent to a lab for testing. The scale is accurate $95 \%$ of the time, $5 \%$ of all the coins minted are sent to the lab, and the lab's test is accurate $90 \%$ of the time. If the lab says a coin is counterfeit, what is the probability that it really is? | \frac{19}{28} | 1/8 |
Determine the largest constant $K\geq 0$ such that $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .
*Proposed by Orif Ibrogimov (Czech Technical University of Prague).* | 18 | 3/8 |
A right circular cone has a base with radius $600$ and height $200\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}.$ Find the least distance that the fly could have crawled.
| 625 | 7/8 |
Square $XYZW$ has area $144$. Point $P$ lies on side $\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\overline{ZP}$ and $\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$. | 12 | 1/8 |
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgina, one card to each, and asked them to figure out which trigonometric function (sin, cos, or tan) produced their cards. Even after sharing the values on their cards with each other, only Malvina was able to surely identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Malvina's card. | \frac{1 + \sqrt{5}}{2} | 5/8 |
Given the set $\{a, \frac{b}{a}, 1\} = \{a^2, a+b, 0\}$, find the value of $a^{2015} + b^{2016}$. | -1 | 6/8 |
Let \( f \) be a function from the set of non-negative integers to itself, satisfying for all \( n \geqslant 0 \):
(1) \( [f(2n+1)]^2 - [f(2n)]^2 = 6 f(n) + 1 \);
(2) \( f(2n) \geqslant f(n) \).
How many numbers less than 2008 are in the range of \( f \)? | 128 | 2/8 |
Calculate: \(\frac{1}{2 \cos \frac{2 \pi}{7}}+\frac{1}{2 \cos \frac{4 \pi}{7}}+\frac{1}{2 \cos \frac{6 \pi}{7}}\). | -2 | 2/8 |
Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Which of the following expressions describes the number of cans of soda that can be purchased for $D$ dollars, where $1$ dollar is worth $4$ quarters?
$\textbf{(A) } \frac{4DQ}{S} \qquad \textbf{(B) } \frac{4DS}{Q} \qquad \textbf{(C) } \frac{4Q}{DS} \qquad \textbf{(D) } \frac{DQ}{4S} \qquad \textbf{(E) } \frac{DS}{4Q}$ | \textbf{(B)}\frac{4DS}{Q} | 1/8 |
In a regular tetrahedron $\mathrm{ABCD}$ with edge length 2, the angle $\theta$ between edge $\mathrm{AD}$ and plane $\alpha$ satisfies $\theta \in \left[\frac{\pi}{3}, \frac{\pi}{2}\right]$. Vertex $\mathrm{A}$ lies in plane $\alpha$, while vertices $\mathrm{B}$, $\mathrm{C}$, and $\mathrm{D}$ are outside $\alpha$. Determine the range of distances from the midpoint $\mathrm{E}$ of edge $\mathrm{BC}$ to plane $\alpha$. | [\frac{\sqrt{3}-\sqrt{2}}{2},\frac{\sqrt{3}+\sqrt{2}}{2}] | 1/8 |
Convex quadrilateral \(ABCD\) has sides \(AB = BC = 7\), \(CD = 5\), and \(AD = 3\). Given additionally that \( \angle ABC = 60^{\circ} \), find \(BD\). | 8 | 3/8 |
Florián is deciding on a bouquet for Mother's Day. According to the price list at the florist, he calculated that whether he buys 5 classic gerberas or 7 mini gerberas, the bouquet, after adding a decorative ribbon, will cost the same, specifically 295 korunas. However, if he buys only 2 mini gerberas and 1 classic gerbera without any additional decorations, he will pay 102 korunas.
How much does one ribbon cost? | 85 | 7/8 |
For integers $a, b, c, d$, let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \in \{1,2,3,4,5\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5. Find the sum of all possible values of $f(a, b, c, d)$. | 31 | 6/8 |
Given a plane $\alpha$ and a triangle $ABC$ whose plane is not parallel to the plane $\alpha$, find a point $S$ in space such that the lines $SA, SB, SC$ intersecting the plane $\alpha$ produce points $A', B', C'$ such that triangle $A'B'C'$ is congruent to another given triangle $MNP$. | S | 5/8 |
In a number matrix as shown, the three numbers in each row are in arithmetic progression, and the three numbers in each column are also in arithmetic progression. Given that \( a_{22} = 2 \), find the sum of all 9 numbers in the matrix. | 18 | 5/8 |
The equation with integer coefficients \( x^{4} + a x^{3} + b x^{2} + c x + d = 0 \) has four positive roots considering their multiplicities.
Find the smallest possible value of the coefficient \( b \) given these conditions. | 6 | 6/8 |
Four visually identical coins weigh 1, 2, 3, and 4 grams. Is it possible to determine the exact weight of each coin using a balance scale in four weighings without using any weights? | 4 | 1/8 |
A said: "I am 10 years old, 2 years younger than B, and 1 year older than C."
B said: "I am not the youngest, C and I have a 3-year difference, and C is 13 years old."
C said: "I am younger than A, A is 11 years old, and B is 3 years older than A."
Among the three statements made by each person, one of them is incorrect. Please determine A's age. | 11 | 6/8 |
Let $ P_1$ be a regular $ r$ -gon and $ P_2$ be a regular $ s$ -gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$ . What's the largest possible value of $ s$ ? | 117 | 7/8 |
To monitor the skier's movements, the coach divided the track into three sections of equal length. It was found that the skier's average speeds on these three separate sections were different. The time required for the skier to cover the first and second sections together was 40.5 minutes, and for the second and third sections - 37.5 minutes. Additionally, it was determined that the skier's average speed on the second section was the same as the average speed for the first and third sections combined. How long did it take the skier to reach the finish? | 58.5 | 5/8 |
\(AB\) is the diameter of a circle; \(BC\) is a tangent; \(D\) is the point where line \(AC\) intersects the circle. It is known that \(AD = 32\) and \(DC = 18\). Find the radius of the circle. | 20 | 5/8 |
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$ . Find $a_{100}$ . | 53 | 2/8 |
Of the following statements, the one that is incorrect is:
$\textbf{(A)}\ \text{Doubling the base of a given rectangle doubles the area.}$
$\textbf{(B)}\ \text{Doubling the altitude of a triangle doubles the area.}$
$\textbf{(C)}\ \text{Doubling the radius of a given circle doubles the area.}$
$\textbf{(D)}\ \text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$
$\textbf{(E)}\ \text{Doubling a given quantity may make it less than it originally was.}$ | \textbf{(C)} | 1/8 |
Ship A and Ship B travel downstream at 8 kilometers per hour and upstream at 4 kilometers per hour. If Ship A travels downstream and then returns, and Ship B travels upstream and then returns, both starting at the same time and returning to their respective starting points after 3 hours, how much time do Ship A and Ship B travel in the same direction during these 3 hours? | 1 | 7/8 |
A circle is inscribed in a triangle with a perimeter of \(2p\). A tangent parallel to a side of the triangle is drawn to this circle. Find the maximum possible length of the segment of this tangent that lies inside the triangle. | \frac{p}{4} | 7/8 |
Let \( a, b, c \) be positive real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if \( a = b = c = 1 \). | \frac{1}{(1+b)}+\frac{1}{b(1+)}+\frac{1}{(1+)}\ge\frac{3}{1+abc} | 1/8 |
How many functions $f:\{0,1\}^{3} \rightarrow\{0,1\}$ satisfy the property that, for all ordered triples \left(a_{1}, a_{2}, a_{3}\right) and \left(b_{1}, b_{2}, b_{3}\right) such that $a_{i} \geq b_{i}$ for all $i, f\left(a_{1}, a_{2}, a_{3}\right) \geq f\left(b_{1}, b_{2}, b_{3}\right)$? | 20 | 1/8 |
The line passing through the point $(0,-2)$ intersects the parabola $y^{2}=16x$ at two points $A(x_1,y_1)$ and $B(x_2,y_2)$, with $y_1^2-y_2^2=1$. Calculate the area of the triangle $\triangle OAB$, where $O$ is the origin. | \frac{1}{16} | 2/8 |
Find an integer $n$, where $100 \leq n \leq 1997$, such that
\[ \frac{2^n+2}{n} \]
is also an integer. | 946 | 6/8 |
Given that the three lateral edges \( PA, PB, PC \) of a triangular prism \( P-ABC \) are mutually perpendicular, let the dihedral angles between the planes \( P-AB \) and \( BC, P-BC \) and \( CA, P-CA \) and \( AB \) be \( \alpha, \beta, \gamma \), respectively. Then, \(\frac{\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma}{\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma} = \) ______. | 2 | 6/8 |
Let $f(x) = \frac{x + 6}{x}.$ The sequence $(f_n)$ of functions is defined by $f_1 = f$ and
\[f_n = f \circ f_{n - 1}\]for all $n \ge 2.$ For example,
\[f_2(x) = f(f(x)) = \frac{\frac{x + 6}{x} + 6}{\frac{x + 6}{x}} = \frac{7x + 6}{x + 6}\]and
\[f_3(x) = f(f_2(x)) = \frac{\frac{7x + 6}{x + 6} + 6}{\frac{7x + 6}{x + 6}} = \frac{13x + 42}{7x + 6}.\]Let $S$ be the set of all real numbers $x$ such that
\[f_n(x) = x\]for some positive integer $n.$ Find the number of elements in $S.$ | 2 | 6/8 |
Regular hexagon $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ has side length 2. For $1 \leq i \leq 6$, let $C_{i}$ be a unit circle centered at $P_{i}$ and $\ell_{i}$ be one of the internal common tangents of $C_{i}$ and $C_{i+2}$, where $C_{7}=C_{1}$ and $C_{8}=C_{2}$. Assume that the lines $\{\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6}\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. | 1603 | 1/8 |
Given the sequence \( T_0 = 2 \), \( T_1 = 3 \), \( T_2 = 6 \), and for \( n \geq 3 \),
\[ T_n = (n+4) T_{n-1} - 4n T_{n-2} + (4n - 8) T_{n-3}, \]
the first \( n \) terms of this sequence are \( 2, 3, 6, 14, 40, 152, 784, 5186, 40576, \ldots \). Find the general term formula for \( T_n \) which is of the form \( T_n = A_n + B_n \), where \( \{A_n\} \) and \( \{B_n\} \) are well-known sequences, and provide a proof. | T_n=n!+2^n | 1/8 |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\mbox{th}}$ term of this sequence.
| 981 | 7/8 |
The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage? | 400 | 7/8 |
Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence is an integer. | \frac{3}{4} | 6/8 |
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square. | 3 | 5/8 |
Let \( P \) be the number of triangles whose side lengths are integers less than or equal to 9. Find the value of \( P \). | 95 | 5/8 |
If the sum of the digits of a natural number is the same as the sum of the digits of three times that number, but different from the sum of the digits of twice that number, we call such a number a "wonder number." Find the smallest "wonder number." | 144 | 3/8 |
In the right triangle \(ABC\), the altitude \(BH\) is drawn to the hypotenuse \(AC\). Points \(X\) and \(Y\) are the centers of the circles inscribed in triangles \(ABH\) and \(CBH\) respectively. The line \(XY\) intersects the legs \(AB\) and \(BC\) at points \(P\) and \(Q\). Find the area of triangle \(BPQ\), given that \(BH = h\). | \frac{^2}{2} | 1/8 |
Find an axis of symmetry for the function $f(x) = \cos(2x + \frac{\pi}{6})$. | \frac{5\pi}{12} | 1/8 |
In triangle \(ABC\), the altitudes \(AH\) and \(CP\) are drawn. Find the measure of angle \(B\) if it is known that \(|AC| = 2|PH|\). | 60 | 4/8 |
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble? | 6480/2401 | 7/8 |
Vasya has \( n \) candies of several types, where \( n \geq 145 \). It is known that if any group of at least 145 candies is chosen from these \( n \) candies (in particular, one can choose the group of all \( n \) candies), there exists a type of candy such that the chosen group contains exactly 10 candies of that type. Find the largest possible value of \( n \). | 160 | 1/8 |
Given the function \( f(x) = \frac{1+x}{2-x} \), and for \( n \in \mathbf{N}_{+} \), define \( f_{1}(x) = f(x) \) and \( f_{n+1}(x) = f\left[f_{n}(x)\right] \). If \( f_{13}(x) = f_{31}(x) \), what is the expression for \( f_{16}(x) \)? | \frac{x-1}{x} | 7/8 |
The inverse function of \( f(x) \) is \( y = \frac{x}{1+x} \). Given that \( g_{n}(x) + \frac{1}{f_{n}(x)} = 0 \), let \( f_{1}(x) = f(x) \), and for \( n > 1 \) and \( n \in \mathbf{N}^{*} \), \( f_{n}(x) = f_{n-1}[f_{n-1}(x)] \). Find the analytic expression for \( g_{n}(x) \) for \( n \in \mathbf{N}^{*} \). | g_n(x)=2^{n-1}-\frac{1}{x} | 1/8 |
A right triangle \( ABC \) (with leg \( BC \) longer than leg \( AC \)) is inscribed in a circle. On side \( BC \), a point \( D \) is chosen such that \( BD = AC \). Point \( M \) is the midpoint of the arc \( ACB \). Find the angle \( CDM \). | 45 | 1/8 |
Find the area of triangle \(ABC\) if it is known that \(AB = a\), \(\angle A = \alpha\), \(\angle B = \beta\). | \frac{^2\sin\alpha\sin\beta}{2\sin(\alpha+\beta)} | 4/8 |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, find the maximum value of $(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}) \cdot (\overrightarrow{a} + \overrightarrow{c})$. | 2 + \sqrt{5} | 7/8 |
Show that
$1^{\circ} . \quad \sin \alpha \cos \beta \cos \gamma+\sin \beta \cos \alpha \cos \gamma+\sin \gamma \cos \alpha \cos \beta=\sin \alpha \sin \beta \sin \gamma$.
$$
2^{\circ} . \quad \frac{\operatorname{ctg} \frac{\beta}{2}+\operatorname{ctg} \frac{\gamma}{2}}{\operatorname{ctg} \frac{\alpha}{2}+\operatorname{ctg} \frac{\gamma}{2}}=\frac{\sin \alpha}{\sin \beta}
$$
if
$$
\alpha+\beta+\gamma = 180^{\circ}
$$ | \frac{\sin\alpha}{\sin\beta} | 7/8 |
13 children sat at a round table and agreed that boys would lie to girls and tell the truth to each other, while girls would lie to boys and tell the truth to each other. One of the children said to his right neighbor: "Most of us are boys." That neighbor said to his right neighbor: "Most of us are girls," and he said to his right neighbor: "Most of us are boys," and so on, until the last child said to the first: "Most of us are boys." How many boys were at the table? | 7 | 3/8 |
Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$. | 298 | 7/8 |
Find all functions \( f: [0, +\infty) \rightarrow [0, +\infty) \) such that for all \( x \in [0, +\infty) \):
\[ 4f(x) \geq 3x \]
and
\[ f[4f(x) - 3x] = x. \] | f(x)=x | 1/8 |
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents? | 1009 | 1/8 |
In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table? | 256 | 6/8 |
Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{11 \ldots 11}_{2018} \underbrace{55 \ldots 55}_{2017} 6}$. | 6055 | 7/8 |
There are 20 cards with numbers $1, 2, \cdots, 19, 20$ on them. These cards are placed in a box, and 4 people each draw one card. The two people who draw the smaller numbers will be in one group, and the two who draw the larger numbers will be in another group. If two of the people draw the numbers 5 and 14 respectively, what is the probability that these two people will be in the same group? | $\frac{7}{51}$ | 7/8 |
Given a regular nonagon (9-sided polygon), in how many ways can three of its vertices be selected such that they form an isosceles triangle? | 30 | 6/8 |
Find all triples \((x, y, z)\) of positive integers such that \(x \leq y \leq z\) and
$$
x^{3}(y^{3} + z^{3}) = 2012(xyz + 2).
$$ | (2,251,252) | 1/8 |
Given the sets \(A=\{x \mid x^{2}-4x+3<0, x \in \mathbf{R}\}\) and \(B=\{x \mid 2^{1-x}+a \leq 0, x^{2}-2(a+7)x+5 \leq 0, x \in \mathbf{R}\}\). If \(A \subset B\), find the range of values for the real number \(a\). | [-4,-1] | 5/8 |
Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$ . let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ . Evaluate $a_{n,k}-b_{n,k}$ .
** * ** | (-1)^k\binom{n-1}{k} | 1/8 |
Xiao Wang's home is 750 meters away from the school. One morning, while he was walking to school, he realized he forgot his red scarf when he was about halfway through the journey. So, he went back home to get it and then proceeded to the school again. How many meters did Xiao Wang walk in total that morning? | 1500 | 2/8 |
The Hangzhou Asian Games are underway, and table tennis, known as China's "national sport," is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other side serving two balls, and the service alternates every two balls. The winner of a game is the first side to reach 11 points with a lead of at least 2 points. If the score is tied at 10-10, the service order remains the same, but the service alternates after each point until one side wins by a margin of 2 points. In a singles table tennis match between players A and B, assuming player A serves first, the probability of player A scoring when serving is $\frac{2}{3}$, and the probability of player A scoring when player B serves is $\frac{1}{2}$. The outcomes of each ball are independent.
$(1)$ Find the probability that player A scores 3 points after the first 4 balls in a game.
$(2)$ If the game is tied at 10-10, and the match ends after X additional balls are played, find the probability of the event "X ≤ 4." | \frac{3}{4} | 1/8 |
Given that the base area of a cone is $\pi$, and the lateral area is twice the base area, determine the surface area of the circumscribed sphere of the cone. | \frac{16\pi}{3} | 1/8 |
Find the measure of angle \( B \widehat{A} D \), given that \( D \widehat{A C}=39^{\circ} \), \( A B = A C \), and \( A D = B D \). | 47 | 1/8 |
For which values of the parameter \(a\) does the equation \(x^{3} + 16x^{2} + ax + 64 = 0\) have three distinct real roots that form a geometric progression? | 64 | 7/8 |
Let $x_1, x_2, \ldots, x_7$ be natural numbers, and $x_1 < x_2 < \ldots < x_6 < x_7$, also $x_1 + x_2 + \ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. | 61 | 4/8 |
There are 40 identical looking coins, among which 3 are counterfeit - they weigh the same and are lighter than the genuine coins (the genuine coins also weigh the same). How can you use three weighings on a balance scale without weights to select 16 genuine coins? | 16 | 2/8 |
The circles with centers \(O_{1}\) and \(O_{2}\) have a common chord \(AB\), and \(\angle AO_{1}B = 60^\circ\). The ratio of the length of the first circle to the length of the second circle is \(\sqrt{2}\). Find the angle \(\angle AO_{2}B\). | 90 | 7/8 |
Let \( a \) and \( A \) be natural numbers with \( A \geqslant a \). Prove:
$$
\frac{a}{A}+\frac{A-a}{A} \cdot \frac{a}{A-1}+\frac{A-a}{A} \cdot \frac{A-a-1}{A-1} \cdot \frac{a}{A-2}+\cdots+\frac{A-a}{A} \cdot \frac{A-a-1}{A-1} \cdots \frac{1}{a+1} \cdot \frac{a}{a}=1 .
$$ | 1 | 4/8 |
At a market in Egypt, a tourist is negotiating with a seller for a souvenir worth 10,000 Egyptian pounds. The tourist first reduces the price by \( x \) percent \((0 < x < 100)\), then the seller increases the price by \( x \) percent, and so on. The value of \( x \) does not change during the bargaining process, and the seller increases the price at least once. The negotiation continues until one of the participants gets a non-integer value for the price of the souvenir. Find the largest possible number of price changes during such a negotiation (including the last non-integer price change). | 5 | 1/8 |
Mady now has boxes each capable of holding up to 5 balls instead of 4. Under the same process as described, Mady adds balls and resets boxes. Determine the total number of balls in the boxes after her $2010$th step. | 10 | 3/8 |
Given triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sqrt{3}\sin B + 2\cos^2\frac{B}{2} = 3$ and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A \sin B}{6\sin C}$, find the area of the circumcircle of $\triangle ABC$. | 16\pi | 7/8 |
Given \(a > 0\), \(b > 0\), \(c > 1\), and \(a + b = 1\). Find the minimum value of \(\left(\frac{2a + b}{ab} - 3\right)c + \frac{\sqrt{2}}{c - 1}\). | 4 + 2\sqrt{2} | 6/8 |
Let \( a \) and \( b \) be real numbers such that \( 17\left(a^{2}+b^{2}\right)-30ab-16=0 \). Find the maximum value of \( \sqrt{16a^{2}+4b^{2}-16ab-12a+6b+9} \). | 7 | 4/8 |
Let $[x]$ denote the greatest integer less than or equal to $x$. For any positive integer $n$, compute the sum
$$
\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right].
$$ | n | 4/8 |
Let $\mathrm{ABC}$ be a triangle, and $\mathrm{O}$ a point inside this triangle. The line parallel to $(BC)$ passing through $O$ intersects $[CA]$ at $D$, the line parallel to $(CA)$ passing through $O$ intersects $[AB]$ at $E$, and the line parallel to $(AB)$ passing through $O$ intersects $[BC]$ at $F$. What is the value of the following sum of ratios?
$$
\frac{B F}{B C}+\frac{A E}{A B}+\frac{C D}{A C} ?
$$ | 1 | 7/8 |
A store is having a sale for a change of season, offering discounts on a certain type of clothing. If each item is sold at 40% of the marked price, there is a loss of 30 yuan per item, while selling it at 70% of the marked price yields a profit of 60 yuan per item.
Find:
(1) What is the marked price of each item of clothing?
(2) To ensure no loss is incurred, what is the maximum discount that can be offered on this clothing? | 50\% | 4/8 |
A town has $2k$ people ($k \in \mathbb{Z}^{+}$). Between any two individuals, they are either friends or enemies. For a certain positive integer $t$, each person has at most $t$ enemies, and there exists one person with exactly $t$ enemies. If a group of people are all mutual friends, it is called a "friendly group". It is known that no friendly group has more than $k$ members, and all people can be divided into two disjoint friendly groups, each containing $k$ people.
Prove that the number of friendly groups containing exactly $k$ people does not exceed $2^{k-1} + $ 2^{k-t}$. | 2^{k-1}+2^{k-} | 1/8 |
Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move.
Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board. | 100 | 2/8 |
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