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Given that \( g(x) = \sqrt{x(1-x)} \) is a function defined on the interval \([0,1]\), find the area enclosed by the graph of the function \( x g(x) \) and the x-axis.
\frac{\pi}{16}
0.875
Find the number of natural numbers \( k \) not exceeding 353500 such that \( k^{2} + k \) is divisible by 505.
2800
0.25
The sequence \(a_{n}\) is defined as follows: \(a_{1}=1\), \(a_{2n}=a_{n}\), \(a_{2n+1}+a_{n}=1\). What is \(a_{2006}\)?
0
0.25
Forty-two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average?
\frac{43}{14}
0.25
Let \( n = 1990 \), then what is \( \frac{1}{2^{n}}\left(1-3 \mathrm{C}_{n}^{2}+3^{2} \mathrm{C}_{n}^{4}-3^{3} \mathrm{C}_{n}^{6}+\cdots+3^{994} \mathrm{C}_{n}^{1988}-3^{9995} \mathrm{C}_{n}^{1990}\right) \)?
-\frac{1}{2}
0.625
Is it possible in a calendar year that no Sunday falls on the seventh day of any month?
No
0.375
Let \( k \in \mathbb{N}^* \). Suppose that all positive integers are colored using \( k \) different colors, and there exists a function \( f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) satisfying: 1. For positive integers \( m \) and \( n \) of the same color (they can be the same), \( f(m+n) = f(m) + f(n) \); 2. There exist positive integers \( m \) and \( n \) (they can be the same) such that \( f(m+n) \neq f(m) + f(n) \). Find the minimum value of \( k \).
3
0.375
The height of a right triangle, dropped to the hypotenuse, is $h$. What is the minimum length that the median, drawn from the vertex of the larger acute angle, can have?
\frac{3h}{2}
0.625
From the numbers $1,2,3, \cdots, 10$, 3 numbers are drawn at random. What is the probability that at least two of these numbers are consecutive integers?
\frac{8}{15}
0.375
Let \(\{a_{n}\}\) be a geometric sequence with each term greater than 1, then the value of \(\lg a_{1} \lg a_{2012} \sum_{i=1}^{20111} \frac{1}{\lg a_{i} \lg a_{i+1}}\) is ________ .
2011
0.75
Find the number of points in the plane \(xOy\) that have natural coordinates \((x, y)\) and lie on the parabola \(y = -\frac{x^2}{4} + 5x + 39\).
12
0.875
Find the minimum value of the function \( y = \sqrt{x^2 + 2x + 2} + \sqrt{x^2 - 2x + 2} \).
2\sqrt{2}
0.75
Around a circle with a radius of 1, a rhombus and a triangle are described. Two sides of the triangle are parallel to the diagonals of the rhombus, and the third side is parallel to one of the sides of the rhombus and is equal to 5. Find the side of the rhombus.
\frac{25}{12}
0.25
Ali Baba went to a cave where there are gold and diamonds. He had with him one large bag. It is known that a full bag of gold weighs 200 kg, and if the bag is filled only with diamonds, it weighs 40 kg (the empty bag weighs nothing). A kilogram of gold costs 20 dinars, and a kilogram of diamonds costs 60 dinars. What is the maximum amount of money Ali Baba can earn if he can carry no more than 100 kg in this bag? Provide justification for your answer.
3000 \text{ dinars}
0.875
Around the circle, a pentagon is described, the lengths of the sides of which are integers, and the first and third sides are equal to 1. Into which segments does the point of tangency divide the second side?
\frac{1}{2}
0.125
Replace the asterisk $(*)$ in the expression $\left(x^{3}-2\right)^{2}+\left(x^{2}+*\right)^{2}$ with a monomial such that, after squaring and combining like terms, the resulting expression has four terms.
2x
0.75
A point \( P \) lies inside a regular hexagon \( A B C D E F \). The distances from \( P \) to the sides \( A B, B C, C D, D E, E F \), and \( F A \) are respectively \( 1, 2, 5, 7, 6 \), and \( x \). Find \( x \).
3
0.875
A quadrilateral \(ABCD\), whose diagonals are perpendicular to each other, is inscribed in a circle with center \(O\). Find the distance from point \(O\) to side \(AB\), given that \(CD = 8\).
4
0.75
Five identical balls are moving in one direction along a straight line, and five other identical balls are moving in the opposite direction. The speeds of all the balls are the same. When any two balls collide, they rebound in opposite directions with the same speed they had before the collision. How many collisions will occur in total between the balls?
25
0.75
Grass in a given area grows equally fast and dense. It is known that 70 cows would eat all the grass in 24 days, and 30 cows would eat it in 60 days. How many cows would eat all the grass in 96 days? (Assume that the cows eat the grass uniformly).
20
0.875
In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table. Each dragon said: - 1st head: "The dragon to my left is green." - 2nd head: "The dragon to my right is blue." - 3rd head: "There is no red dragon next to me." What is the maximum number of red dragons that could have been at the table?
176
0.75
As shown in the figure, identical equilateral triangles are joined downward progressively to form larger equilateral triangles. The number of vertices (counting overlapping vertices just once) of the smallest triangles in sequence are $3, 6, 10, 15, 21, \cdots$. What is the 9th term in this sequence?
55
0.875
There are 20 cards numbered $1, 2, \cdots, 19, 20$. They are placed in a box, and 4 people each draw one card. The two people who draw the smaller numbers are placed in one group, and the two who draw the larger numbers are placed in another group. If two of the people draw the numbers 5 and 14, what is the probability that these two people are in the same group?
\frac{7}{51}
0.625
A three-digit number, when a decimal point is added in an appropriate place, becomes a decimal. This decimal is 201.6 less than the original three-digit number. What is the original three-digit number?
224
0.875
Let $\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\circ}$ around these lines maps $\mathcal{O}$ onto itself?
13
0.25
Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked?
64
0.5
Let \( x, y, z \) be distinct positive integers such that \( xyz \mid (xy-1)(yz-1)(zx-1) \). Find all possible values of \( x, y, z \).
(2,3,5)
0.875
Find all values of \( x \) for which the greater of the numbers \( \sqrt{\frac{x}{2}} \) and \( \operatorname{tg} x \) is not greater than 1. Provide the total length of the intervals on the number line that satisfy this condition, rounding the result to the nearest hundredth if necessary.
1.21
0.5
If the real and imaginary parts of both $\frac{z}{20}$ and $\frac{20}{z}$ are between 0 and 1 (exclusive), find the area of the figure formed by the corresponding points of such complex numbers $z$.
300-50\pi
0.5
What is the maximum number of months in a year that can have five Sundays?
5
0.375
The circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively. Find the angle $ABC$, given that $AE = 1$ and $BD = 3$.
30^\circ
0.75
What are the last two digits of the number $$ 8 + 88 + 888 + \cdots + \overbrace{88 \cdots 88}^{2008} ? $$
24
0.5
What is the maximum area that a quadrilateral with side lengths of 1, 4, 7, and 8 can have?
18
0.75
From cities $A$ and $B$, which are 240 km apart, two cars simultaneously start driving towards each other. One car travels at 60 km/h and the other at 80 km/h. How far from the point $C$, located at the midpoint between $A$ and $B$, will the cars meet? Give the answer in kilometers, rounding to the nearest hundredth if necessary.
17.14
0.875
Given the set \( A = \{ a \mid -1 \leqslant a \leqslant 2 \} \), find the area of the plane region \( B = \{ (x, y) \mid x, y \in A, x + y \geqslant 0 \} \).
7
0.75
After walking \( \frac{2}{5} \) of the length of a narrow bridge, a pedestrian noticed a car approaching the bridge from behind. He then walked back and met the car at the beginning of the bridge. If the pedestrian had continued walking forward, the car would have caught up with him at the end of the bridge. Find the ratio of the speed of the car to the speed of the pedestrian.
5
0.875
Find the smallest positive period of the function \( f(x)=\sin x \sin \frac{1}{2} x \sin \frac{1}{3} x \).
12\pi
0.75
If \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\) satisfy the following system of equations: \[ \begin{cases} 2x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 6, \\ x_{1} + 2x_{2} + x_{3} + x_{4} + x_{5} = 12, \\ x_{1} + x_{2} + 2x_{3} + x_{4} + x_{5} = 24, \\ x_{1} + x_{2} + x_{3} + 2x_{4} + x_{5} = 48, \\ x_{1} + x_{2} + x_{3} + x_{4} + 2x_{5} = 96, \end{cases} \] find the value of \(3x_{4} + 2x_{5}\).
181
0.875
There were seven boxes. Some of these boxes were filled with seven additional boxes each (not nested within one another) and so on. As a result, there were 10 non-empty boxes. How many boxes are there in total?
77
0.125
In a $3 \times 4$ table, 12 numbers are arranged such that all seven sums of these numbers in the rows and columns of the table are distinct. What is the maximum number of numbers in this table that can be equal to zero?
8
0.25
At the end of a day of activities, an amusement park collected 100 reais from the tickets of 100 people. We know that each adult had to pay 3 reais to enter, each youth 2 reais, and each child 30 cents. What is the smallest number of adults that entered the park that day?
2
0.875
If integers \( a \) and \( b \) are neither relatively prime nor does one divide the other, then \( a \) and \( b \) are called a "union" pair. Let \( A \) be an \( n \)-element subset of the set \( M = \{1, 2, \ldots, 2017\} \), such that any two numbers in \( A \) are a "union" pair. Determine the maximum value of \( n \).
504
0.125
Given the real numbers \( a, b, c \) satisfy \( a + b + c = 6 \), \( ab + bc + ca = 5 \), and \( abc = 1 \), determine the value of \( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \).
38
0.75
The parabola \( y = ax^2 + bx + 1 \) has parameters \( a \) and \( b \) satisfying \( 8a^2 + 4ab = b^3 \). Find the equation of the locus of the vertex \((s, t)\) of the parabola as \( a \) and \( b \) vary.
s t = 1
0.5
Ksyusha runs twice as fast as she walks (both speeds are constant). On Tuesday, when she left home for school, she first walked, and then, when she realized she was late, she started running. The distance Ksyusha walked was twice the distance she ran. As a result, she reached the school from home in exactly 30 minutes. On Wednesday, Ksyusha left home even later, so she had to run twice the distance she walked. How many minutes did it take her to get from home to school on Wednesday?
24
0.875
In an acute-angled triangle, two altitudes are 3 and \(2 \sqrt{2}\), and their intersection point divides the third altitude in the ratio 5:1, counting from the vertex of the triangle. Find the area of the triangle.
6
0.75
A regular \( n \)-gon \( P_{1} P_{2} \ldots P_{n} \) satisfies \( \angle P_{1} P_{7} P_{8} = 178^{\circ} \). Compute \( n \).
630
0.25
A pedestrian and a cyclist started simultaneously in the same direction from a point on a circular track. The speed of the cyclist is 55% greater than the speed of the pedestrian, which results in the cyclist overtaking the pedestrian from time to time. How many different points on the track will these overtakes occur?
11
0.75
Masha braided her dolls' hair: half of the dolls got one braid each, a quarter of the dolls got two braids each, and the remaining quarter of the dolls got four braids each. She used one ribbon for each braid. How many dolls does Masha have if she needed 24 ribbons in total?
12
0.875
Given a regular triangular pyramid $S-ABC$ with a height $SO = 3$ and a base side length of 6. From point $A$, a perpendicular is drawn to the opposite face $SBC$, with the foot of the perpendicular being $O'$. On $AO'$, find a point $P$ such that $\frac{AP}{PO'} = 8$. Find the area of the cross-section passing through point $P$ and parallel to the base.
\sqrt{3}
0.5
Find all real numbers $p$ such that the cubic equation $5x^3 - 5(p+1)x^2 + (71p - 1)x + 1 = 66p$ has three roots that are all positive integers.
76
0.125
Three cones are placed on a table, standing on their bases and touching each other. The radii of their bases are \(2r\), \(3r\), and \(10r\). A truncated cone (frustum) is placed on the table with its smaller base downward, sharing a slant height with each of the other cones. Find \(r\), if the radius of the smaller base of the truncated cone is 15.
29
0.25
Determine all natural numbers \( n \geq 2 \) for which there exist two permutations \((a_{1}, a_{2}, \ldots, a_{n})\) and \((b_{1}, b_{2}, \ldots, b_{n})\) of the numbers \(1, 2, \ldots, n\) such that \((a_{1}+b_{1}, a_{2}+b_{2}, \ldots, a_{n}+b_{n})\) are consecutive natural numbers.
n \text{ is odd}
0.25
Starting from her house to get to school, Julia needs to walk eight blocks to the right and five blocks up, as indicated in the given diagram. She knows that there are many different ways to take the house-to-school route, always following the shortest path. Since she is very curious, she would like to always take different routes. How many such routes exist from Julia's house to the school?
1287
0.875
The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers.
10
0.625
Given that $a$ and $b$ are positive integers satisfying $\frac{1}{a} - \frac{1}{b} = \frac{1}{2018}$, find the number of all positive integer pairs $(a, b)$.
4
0.625
The function \( f \) has the following properties: 1. Its domain is all real numbers. 2. It is an odd function, i.e., \( f(-x) = -f(x) \) for every real number \( x \). 3. \( f(2x-3) - 2f(3x-10) + f(x-3) = 28 - 6x \) for every real number \( x \). Determine the value of \( f(4) \).
8
0.875
Let \( x \) and \( y \) be real numbers. Determine the maximum value of \( \frac{2x + \sqrt{2}y}{2x^4 + 4y^4 + 9} \).
\frac{1}{4}
0.5
Add together all natural numbers less than 1980 for which the sum of their digits is even!
979605
0.125
In a trapezoid, the smaller base is equal to 2, and the adjacent angles are each $135^{\circ}$. The angle between the diagonals, facing the base, is $150^{\circ}$. Find the area of the trapezoid. ![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0942.jpg?height=243&width=679&top_left_y=918&top_left_x=595)
2
0.5
Find all real numbers \( x, y, z \) such that \[ \begin{aligned} x^{2} y + y^{2} z + z^{2} &= 0, \\ z^{3} + z^{2} y + z y^{3} + x^{2} y &= \frac{1}{4}\left(x^{4} + y^{4}\right). \end{aligned} \]
x = y = z = 0
0.125
Convex quadrilateral \(ABCD\) has sides \(AB = BC = 7\), \(CD = 5\), and \(AD = 3\). Given additionally that \( \angle ABC = 60^{\circ} \), find \(BD\).
8
0.875
Mice built an underground house consisting of chambers and tunnels: - Each tunnel leads from one chamber to another (i.e., none are dead ends). - From each chamber, exactly three tunnels lead to three different chambers. - From each chamber, it is possible to reach any other chamber through tunnels. - There is exactly one tunnel such that, if it is filled in, the house will be divided into two separate parts. What is the minimum number of chambers the mice's house could have? Draw a possible configuration of how the chambers could be connected.
10
0.125
The coefficients of the quadratic polynomials \(f(x) = x^2 + mx + n\) and \(p(x) = x^2 + kx + l\) satisfy the condition \(k > m > n > l > 0\). Is it possible for \(f(x)\) and \(p(x)\) to have a common root?
\text{No}
0.875
Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of the caught fish himself. How many fish did Vasya catch?
10
0.75
Two signals, $A$ and $B$, are transmitted over a communication line with probabilities of 0.72 and 0.28, respectively. Due to noise, $\frac{1}{6}$ of $A$ signals are distorted and received as $B$ signals, and $\frac{1}{7}$ of transmitted $B$ signals are received as $A$ signals. a) Determine the probability that an $A$ signal will be received at the receiving end. b) Given that an $A$ signal is received, what is the probability that it was originally transmitted as an $A$ signal?
0.9375
0.625
One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters in the first minute and in each subsequent minute covered 0.5 meters more than in the previous minute. How many minutes will it take for the second particle to catch up with the first?
17
0.875
You have a set of n biased coins. The mth coin has a probability of \( \frac{1}{2m+1} \) of landing heads (for \( m = 1, 2, ... , n \)) and the results for each coin are independent. What is the probability that if each coin is tossed once, you get an odd number of heads?
\frac{n}{2n+1}
0.625
Lisa considers the number $$ x=\frac{1}{1^{1}}+\frac{1}{2^{2}}+\cdots+\frac{1}{100^{100}} . $$ Lisa wants to know what $x$ is when rounded to the nearest integer. Help her determine its value.
1
0.875
Given that \( x \geq 1 \), \( y \geq 1 \), and \(\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}\), find the maximum value of \( u=\lg x y \).
2 + 2 \sqrt{2}
0.625
Given \( x, y \geq 1, x + y = 8 \), find the maximum value of \( \left|\sqrt{x - \frac{1}{y}} + \sqrt{y - \frac{1}{x}} \right| \).
\sqrt{15}
0.75
In the decimal system, a $(n+1)$-digit number $A=\overline{a_{n} a_{n-1} \ldots a_{1} a_{0}}$ is called the reverse of $A*=\overline{a_{0} a_{1} \ldots a_{n}}$. (Thus, the reverse of 759 is 957, and the reverse of 980 is 89.) Find the four-digit numbers that "reverse" when multiplied by 9, meaning $9A = A^*$.
1089
0.125
Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - f(x) = 0 \) has exactly three solutions. Find the ordinate of the vertex of the polynomial \( f(x) \).
0
0.75
Let the set \( S = \{1, 2, \cdots, 3n\} \), where \( n \) is a positive integer. Let \( T \) be a subset of \( S \) such that for any \( x, y, z \in T \) (where \( x, y, z \) can be the same), the condition \( x + y + z \notin T \) is satisfied. Find the maximum number of elements in such a subset \( T \).
2n
0.875
Points \( A, B, C, D \) are marked on a sheet of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between two given points in centimeters; b) compare two given numbers. What is the minimum number of operations this device needs to perform to definitively determine whether quadrilateral \( A B C D \) is a square?
10
0.125
Find the maximum possible value of the GCD \((x + 2015y, y + 2015x)\), given that \(x\) and \(y\) are coprime numbers.
4060224
0.5
Let \( P(x) \) be a \( 2n \)-degree polynomial such that \( P(0) = P(2) = \cdots = P(2n) = 0 \) and \( P(1) = P(3) = \cdots = P(2n-1) = 2 \), with \( P(2n+1) = -30 \). Find \( n \).
n = 2
0.5
Vasya wrote all natural numbers from 104 to 203 in the cells of a $10 \times 10$ table. He calculated the product of numbers in each row of the table and obtained a set of ten numbers. Then he calculated the product of numbers in each column of the table and also obtained a set of ten numbers. Could the resulting sets be identical?
\text{No}
0.75
Given the sequence \(\left\{x_{n}\right\}\), and \[ x_{n+1} = \frac{x_{n}+(2-\sqrt{3})}{1-(2-\sqrt{3}) x_{n}}, \] find the value of \(x_{1001} - x_{401}\).
0
0.875
The three midlines of a triangle divide it into four parts. The area of one of them is \( S \). Find the area of the given triangle.
4S
0.75
Let \( A_{n} \) be the area outside a regular \( n \)-gon of side length 1 but inside its circumscribed circle, and let \( B_{n} \) be the area inside the \( n \)-gon but outside its inscribed circle. Find the limit as \( n \) tends to infinity of \(\frac{A_{n}}{B_{n}}\).
2
0.5
Given a point $P$ on the inscribed circle of a square $ABCD$, considering the angles $\angle APC = \alpha$ and $\angle BPD = \beta$, find the value of $\tan^2 \alpha + $\tan^2 \beta$.
8
0.75
A set of real numbers is called "simple" if it contains elements such that \( x + y = z \). Given the set \( \{1, 2, \cdots, 2n+1\} \), find the maximum number of elements in one of its simple subsets.
n+1
0.125
All natural numbers whose digit sum is equal to 5 are arranged in ascending order. Which number is in the 125th position?
41000
0.25
If the price of a product increased from 5.00 reais to 5.55 reais, what was the percentage increase?
11\%
0.375
An urn initially contains one white ball. Another ball, either white or black (with equal probabilities), is added to the urn. After that, a ball is randomly drawn from the urn, and it turns out to be white. What is the conditional probability that the remaining ball in the urn is also white?
\frac{2}{3}
0.625
Given an ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$ respectively, and upper and lower vertices $B_{1}$ and $B_{2}$ respectively, the right vertex is $A$. The line $A B_{1}$ intersects $B_{2} F_{1}$ at point D. If $2\left|A B_{1}\right|=3\left|B_{1} D\right|$, determine the eccentricity of ellipse $C$.
\frac{1}{4}
0.75
Does there exist a natural number \( n \) such that \[ n + S(n) = 125 \] where \( S(n) \) is the sum of the digits of \( n \)?
121
0.625
In the rhombus \(ABCD\), the measure of angle \(B\) is \(40^\circ\). \(E\) is the midpoint of \(BC\), and \(F\) is the foot of the perpendicular dropped from \(A\) to \(DE\). Find the measure of angle \(DFC\).
110^\circ
0.125
Determine the volume of a regular quadrangular prism if its diagonal forms an angle of $30^{\circ}$ with the plane of one of its lateral faces, and the side of its base is equal to $a$.
a^3 \sqrt{2}
0.875
In the sequence \(\left\{a_{n}\right\}\), \(a_{1} = -1\), \(a_{2} = 1\), \(a_{3} = -2\). Given that for all \(n \in \mathbf{N}_{+}\), \(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\), and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\), find the sum of the first 4321 terms of the sequence \(S_{4321}\).
-4321
0.375
On a rectangular sheet of paper, a picture in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$ is drawn, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
52.5
0.625
Find all positive integers \( n \) with the following property: the \( k \) positive divisors of \( n \) have a permutation \(\left( d_{1}, d_{2}, \ldots, d_{k} \right)\) such that for every \( i = 1, 2, \ldots, k \), the number \( d_{1} + \cdots + d_{i} \) is a perfect square.
1 \text{ and } 3
0.875
In the number \(2 * 0 * 1 * 6 * 0 * 2 *\), replace each of the 6 asterisks with any digit from the set \(\{0,1,2,3,4,5,6,7,8\}\) (digits may repeat) so that the resulting 12-digit number is divisible by 45. How many ways can this be done?
13122
0.5
Find all real numbers \( x \) that satisfy the equation \(\lg (x+1)=\frac{1}{2} \log _{3} x\).
9
0.5
Let \( \triangle ABC \) be a triangle with \( AB = 16 \) and \( AC = 5 \). Suppose the bisectors of angles \( \angle ABC \) and \( \angle BCA \) meet at point \( P \) in the triangle's interior. Given that \( AP = 4 \), compute \( BC \).
14
0.75
In triangle \( \triangle ABC \), given \( AB = 4 \), \( AC = 3 \), and \( P \) is a point on the perpendicular bisector of \( BC \), find \( \overrightarrow{BC} \cdot \overrightarrow{AP} \).
-\frac{7}{2}
0.875
In a trapezoid, the midline is equal to 7, the height is $\frac{15 \sqrt{3}}{7}$, and the angle between the diagonals opposite the base is $120^{\circ}$. Find the diagonals of the trapezoid.
10
0.375
In the number $2 * 0 * 1 * 6 * 02 *$, each of the 5 asterisks should be replaced with any of the digits $0, 2, 4, 7, 8, 9$ (the digits may repeat) such that the resulting 11-digit number is divisible by 12. How many ways can this be done?
1296
0.5
Find all natural numbers \( N \) such that the remainder when 2017 is divided by \( N \) is 17. Indicate the number of such \( N \).
13
0.875