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Determine all functions from \(\mathbb{R}\) to \(\mathbb{R}\) satisfying:
\[ \forall(x, y) \in \mathbb{R}^{2}, \; f(f(x) + 9y) = f(y) + 9x + 24y \]
|
f(x) = 3x
| 0.75 |
Calculate the length of the arc of the curve given by the polar equation
$$
\varrho = 6 \sin \varphi, \quad 0 \leq \varphi \leq \pi / 3
$$
|
2\pi
| 0.875 |
What is the smallest positive integer divisible by 28 that ends with the digits 28 in decimal representation, and whose sum of digits is 28?
|
18928
| 0.75 |
Calculate the area of the parallelogram formed by the vectors \(a\) and \(b\).
Given:
\[ a = 6p - q \]
\[ b = p + q \]
\[ |p| = 3 \]
\[ |q| = 4 \]
\[ (\widehat{p, q}) = \frac{\pi}{4} \]
|
42\sqrt{2}
| 0.875 |
How many natural numbers $N \leq 1000000$ exist such that $N$ is divisible by $\lfloor \sqrt{N} \rfloor$?
|
2998
| 0.375 |
The numbers \(2^{0}, 2^{1}, \cdots, 2^{15}, 2^{16}=65536\) are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
|
65535
| 0.5 |
Set \( A \) is a subset consisting of 40 elements chosen from \(\{1, 2, 3, \ldots, 50\}\). Let \( S \) be the sum of all elements in set \( A \). Find the number of possible values for \( S \).
|
401
| 0.75 |
Is it possible to cut the figure shown in the picture into four equal parts along the grid lines so that these parts can be rearranged to form a square?
|
Yes
| 0.25 |
A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
|
23
| 0.5 |
The sides of a rectangle were reduced: the length by 10%, the width by 20%. In this case, the perimeter of the rectangle decreased by 12%. By what percentage will the perimeter of the rectangle decrease if its length is reduced by 20% and its width by 10%?
|
18\%
| 0.875 |
In the tetrahedron \(ABCD\), given that \(AB = 1\), \(CD = \sqrt{3}\), the distance between the lines \(AB\) and \(CD\) is 2, and the angle between them is \(\frac{\pi}{3}\), find the volume of the tetrahedron \(ABCD\).
|
\frac{1}{2}
| 0.75 |
There are six children in a family. Five of them are older than the youngest by 2, 6, 8, 12, and 14 years respectively. How old is the youngest if the ages of all children are prime numbers?
|
5
| 0.875 |
Among \( n \) knights, every pair of them is either friends or enemies. Each knight has exactly three enemies, and the enemies of their friends are also their enemies. For which \( n \) is this possible?
|
n=4 \text{ or } n=6
| 0.5 |
Seven fishermen stand in a circle. The fishermen have a professional habit of exaggerating numbers. Each fisherman has an exaggeration factor (unique integer) representing how many times the actual number is exaggerated. For example, if a fisherman with an exaggeration factor of 3 catches two fish, he will say he caught six fish. When asked, "How many fish did your left neighbor catch?", the responses (not necessarily in the order the fishermen are sitting) were $12, 12, 20, 24, 32, 42,$ and $56$. When asked, "How many fish did your right neighbor catch?", six of the fishermen responded $12, 14, 18, 32, 48,$ and $70$. What did the seventh fisherman respond?
|
16
| 0.375 |
Given $m$ points on a plane, where no three points are collinear, and their convex hull is an $n$-gon. Connecting the points appropriately can form a mesh region composed of triangles. Let $f(m, n)$ represent the number of non-overlapping triangles in this region. Find $f(2016, 30)$.
|
4000
| 0.5 |
Find the sum of all real solutions to \( x^{2} + \cos x = 2019 \).
|
0
| 0.875 |
Let \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) be nonnegative real numbers whose sum is 300. Let \( M \) be the maximum of the four numbers \( x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, \) and \( x_{4} + x_{5} \). Find the least possible value of \( M \).
|
100
| 0.25 |
Joey wrote a system of equations on a blackboard, where each of the equations was of the form \( a + b = c \) or \( a \cdot b = c \) for some variables or integers \( a, b, c \). Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:
\[
\begin{array}{ll}
x & z = 15 \\
x & y = 12 \\
x & x = 36
\end{array}
\]
If \( x, y, z \) are integer solutions to the original system, find the sum of all possible values of \( 100x + 10y + z \).
|
2037
| 0.5 |
The sequence $\left\{a_{n}\right\}$ is defined as follows:
$a_{1}=1$, and $a_{n+1}=a_{n}+\frac{1}{a_{n}}$ for $n \geq 1$.
Find the integer part of $a_{100}$.
|
14
| 0.5 |
In a quadrilateral pyramid \( S A B C D \):
- The lateral faces \( S A B, S B C, S C D, S D A \) have areas 9, 9, 27, and 27 respectively.
- The dihedral angles at the edges \( A B, B C, C D, D A \) are equal.
- The quadrilateral \( A B C D \) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \( S A B C D \).
|
54
| 0.25 |
In a convex quadrilateral \(ABCD\), \(\angle ABC = 90^\circ\), \(\angle BAC = \angle CAD\), \(AC = AD\), and \(DH\) is the altitude of triangle \(ACD\). In what ratio does the line \(BH\) divide the segment \(CD\)?
|
1:1
| 0.125 |
A truck and a car are moving in the same direction on adjacent lanes at speeds of 65 km/h and 85 km/h respectively. How far apart will they be 3 minutes after they are even with each other?
|
1 \text{ km}
| 0.75 |
Given the circle \( O: x^{2}+y^{2}=4 \) and the curve \( C: y=3|x-t| \), and points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^*) \) on the curve \( C \), such that the ratio of the distance from any point on the circle \( O \) to point \( A \) and to point \( B \) is a constant \( k (k>1) \), find the value of \( t \).
|
\frac{4}{3}
| 0.5 |
Karlson filled a conical glass with lemonade and drank half of it by height (measured from the surface of the liquid to the top of the cone), and the other half was finished by Kid. By what factor did Karlson drink more lemonade than Kid?
|
7
| 0.75 |
The most common dice is a six-sided die, which is a cube. The 6 faces have 1 to 6 points on them, with the sum of points on opposite faces equal to 7. From a single point in space, you can see multiple faces of the die. The minimum visible sum of points is 1, and the maximum is 15 (15 = 4 + 5 + 6). Determine which sums in the range from 1 to 15 are not possible to see.
|
13
| 0.5 |
Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) > 1\).
Determine the maximum value of \( |S| \).
|
50
| 0.5 |
Players are dividing chips. The first player takes $m$ chips and one-sixth of the remaining chips; the second player takes $2m$ chips and one-sixth of the new remaining chips; the third player takes $3m$ chips and one-sixth of the new remaining chips, and so on. It turns out that the chips were divided equally in this manner. How many players were there?
|
5
| 0.875 |
Let $n$ be a natural number with the following property: If 50 different numbers are randomly chosen from the numbers $1, 2, \ldots, n$, there will necessarily be two numbers among them whose difference is 7. Find the maximum value of such $n$.
|
98
| 0.375 |
Let $x, y, z$ be positive numbers satisfying the following system of equations:
$$
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 9 \\
z^{2} + xz + x^{2} = 21
\end{array}\right.
$$
Find the value of the expression $xy + yz + xz$.
|
12
| 0.5 |
You are in a completely dark room with a drawer containing 10 red, 20 blue, 30 green, and 40 khaki socks. What is the smallest number of socks you must randomly pull out in order to be sure of having at least one of each color?
|
91
| 0.875 |
Find the smallest natural number that cannot be represented in the form \(\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\), where \(a, b, c, d\) are natural numbers.
|
11
| 0.875 |
To the natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, and the result is a power of ten. Find all such \( N \).
|
75
| 0.75 |
Find the largest three-digit number such that the number minus the sum of its digits is a perfect square.
|
919
| 0.5 |
A cuckoo clock produces a number of "cuckoo" sounds equal to the hour it indicates (for example, at 19:00, it sounds "cuckoo" 7 times). One morning, Maxim approaches the clock at 9:05 and starts turning the minute hand forward until the clock shows 7 hours later. How many "cuckoo" sounds are made during this time?
|
43
| 0.75 |
A six-digit number starts with 1. If we move this digit 1 from the first position to the last position on the right, we obtain a new six-digit number, which is three times the original number. What is this number?
|
142857
| 0.75 |
In each of the 16 unit squares of a $4 \times 4$ grid, a + sign is written except in the second square of the first row. You can perform the following three operations:
- Change the sign of each square in a row.
- Change the sign of each square in a column.
- Change the sign of each square in a diagonal (not just the two main diagonals).
Is it possible to obtain a configuration in which every square contains a + sign using a finite number of these operations?
|
\text{No}
| 0.875 |
The non-zero numbers \( a, b, \) and \( c \) are such that the doubled roots of the quadratic polynomial \( x^{2}+a x+b \) are the roots of the polynomial \( x^{2}+b x+c \). What can the ratio \( a / c \) equal?
|
\frac{1}{8}
| 0.75 |
During the draw before the math marathon, the team captains were asked to name the smallest possible sum of the digits in the decimal representation of the number \( n+1 \), given that the sum of the digits of the number \( n \) is 2017. What answer did the captain of the winning team give?
|
2
| 0.5 |
Matvey decided to start eating healthily and each day ate one less bun and one more pear than the previous day. In total, during the period of healthy eating, he ate 264 buns and 187 pears. How many days did Matvey follow his healthy diet?
|
11
| 0.875 |
A carpenter wants to cut a wooden cube with a side length of 3 inches into 27 smaller cubes with a side length of 1 inch each. The question is: "What is the minimum number of cuts required if the pieces can be rearranged arbitrarily during the cutting process, given that the carpenter can make 6 cuts to split the cube into smaller cubes while keeping the pieces together so they don't fall apart?"
Given that without rearranging the pieces, the minimum number of cuts would be 6 because each face needs to be cut to separate the inner cube, what is the minimum number of cuts required if the inner cube is missing and pieces can be rearranged arbitrarily during the process?
|
6
| 0.25 |
Let \( x, y, z \) be positive numbers satisfying the system of equations:
\[ \left\{\begin{array}{l}
x^{2}+xy+y^{2}=75 \\
y^{2}+yz+z^{2}=4 \\
z^{2}+xz+x^{2}=79
\end{array}\right. \]
Find the value of the expression \( xy + yz + xz \).
|
20
| 0.375 |
In the equations of the line \(\frac{x}{2}=\frac{y}{-3}=\frac{z}{n}\), determine the parameter \(n\) so that this line intersects with the line \(\frac{x+1}{3}=\frac{y+5}{2}=\frac{z}{1}\), and find the point of intersection.
|
(2, -3, 1)
| 0.5 |
Given a tetrahedron \(ABCD\), a sphere with a radius of 1 is drawn through vertex \(D\). This sphere is tangent to the circumsphere of the tetrahedron at point \(D\) and also tangent to the face \(ABC\). Given that \(AD=2\sqrt{3}\), \(\angle BAD = \angle CAD = 45^\circ\), and \(\angle BAC = 60^\circ\), find the radius \(r\) of the circumsphere of the tetrahedron.
|
3
| 0.5 |
A magical checkered sheet of paper of size $2000 \times 70$, initially has all cells grey. A painter stands on a certain cell and paints it red. Every second, the painter makes two steps: one cell to the left and one cell down, and paints the cell he lands on red. If the painter is in the leftmost column and needs to step left, he teleports to the rightmost cell of the same row; if the painter is in the bottom row and needs to step down, he teleports to the top cell of the same column. After several moves, the painter returns to the cell where he started. How many cells have been painted red at this point?
|
14000
| 0.125 |
A car starts from point \( O \) and moves along a straight road with a constant speed \( v \). At the same time, a person on a bicycle, located at a distance \( a \) from point \( O \) and \( b \) from the road, begins to ride with the intention of delivering a letter to the car driver. What is the minimum speed the cyclist must maintain to achieve this goal?
|
\frac{v b}{a}
| 0.125 |
Given that \(a\), \(b\), \(c\), and \(d\) are positive integers, and
\[
\log_{a} b=\frac{3}{2}, \quad \log_{c} d=\frac{5}{4}, \quad \text{and} \quad a-c=9,
\]
find the value of \(a + b + c + d\).
|
198
| 0.875 |
In triangle \(ABC\), point \(K\) on side \(AB\) and point \(M\) on side \(AC\) are positioned such that \(AK:KB = 3:2\) and \(AM:MC = 4:5\). Determine the ratio in which the line through point \(K\) parallel to side \(BC\) divides segment \(BM\).
|
\frac{18}{7}
| 0.125 |
One hundred friends, including Petya and Vasya, live in several cities. Petya found out the distance from his city to the city of each of the remaining 99 friends and summed these 99 numbers. Vasya did the same. Petya obtained 1000 km. What is the largest number Vasya might have obtained? (Consider the cities as points on a plane; if two people live in the same city, the distance between their cities is considered to be zero.)
|
99000 \text{ km}
| 0.625 |
A ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Therefore, he decreed that no family should have more than one girl. Thus, every woman in this country who had children would have a girl as the last and only last child because no woman, having given birth to a girl, would dare to have more children. What proportion of boys made up the total mass of children in this country?
|
\frac{1}{2}
| 0.875 |
As shown in the figure, three circles intersect to create seven regions. Fill these regions with the integers from $1$ to $7$, ensuring that the sum of the four numbers in each circle is equal. What is the maximum possible sum for each circle?
|
19
| 0.375 |
The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have?
|
1003
| 0.75 |
Compute the definite integral:
$$
\int_{-14 / 15}^{-7 / 8} \frac{6 \sqrt{x+2}}{(x+2)^{2} \sqrt{x+1}} \, dx
$$
|
1
| 0.625 |
Given the complex number sequence $\left\{z_{n}\right\}$ which satisfies $z_{1}=1$ and $z_{n+1}=\overline{z_{n}}+1+n i$ for $n=1,2, \cdots$, find $z_{2015}$.
|
2015 + 1007i
| 0.875 |
\( z_{1}, z_{2}, z_{3} \) are the three roots of the polynomial
\[ P(z) = z^{3} + a z + b \]
and satisfy the condition
\[ \left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \left|z_{3}\right|^{2} = 250 \]
Moreover, the three points \( z_{1}, z_{2}, z_{3} \) in the complex plane form a right triangle. Find the length of the hypotenuse of this right triangle.
|
5\sqrt{15}
| 0.5 |
Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?
|
145
| 0.5 |
Let the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfy \( f(0) = 1 \) and for any \( x, y \in \mathbf{R} \), \( f(x y + 1) = f(x) f(y) - f(y) - x + 2 \). Find \( f(x) \).
|
f(x) = x + 1
| 0.875 |
Let \( c \) be a positive real number. If \( x^{2} + 2 \sqrt{c} x + b = 0 \) has one real root only, find the value of \( c \).
|
c = b
| 0.875 |
The Fibonacci numbers are defined by \( F_{1} = F_{2} = 1 \), and \( F_{n} = F_{n-1} + F_{n-2} \) for \( n \geq 3 \). If the number
\[ \frac{F_{2003}}{F_{2002}} - \frac{F_{2004}}{F_{2003}} \]
is written as a fraction in lowest terms, what is the numerator?
|
1
| 0.5 |
It is known about the numbers \( x_1 \) and \( x_2 \) that \( x_1 + x_2 = 2 \sqrt{1703} \) and \( \left| x_1 - x_2 \right| = 90 \). Find \( x_1 \cdot x_2 \).
|
-322
| 0.875 |
Given 100 numbers. Each number is increased by 2. The sum of the squares of the numbers remains unchanged. Each resulting number is then increased by 2 again. How has the sum of the squares changed now?
|
800
| 0.75 |
Find the equation of the tangent line to the given curve at the point with the abscissa \( x_{0} = 1 \).
\[ y = \frac{x^{29} + 6}{x^{4} + 1}, \quad x_{0} = 1 \]
|
y = 7.5x - 4
| 0.25 |
The parabola $\Pi_{1}$ with upward-facing branches passes through the points with coordinates $(10,0)$ and $(13,0)$. The parabola $\Pi_{2}$ with upward-facing branches also passes through the point with coordinates $(13,0)$. It is also known that the vertex of $\Pi_{1}$ bisects the segment connecting the origin and the vertex of $\Pi_{2}$. At what abscissa does the parabola $\Pi_{2}$ intersect the $x$-axis again?
|
33
| 0.875 |
Given the sequence \(\left\{a_{n}\right\}\) with the general term
\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \]
find the sum of the first 12 terms \( S_{12} \).
|
104832
| 0.375 |
On a plane, a regular hexagon with side length \( a \) is drawn. For any \( n \in \mathbf{N} \), greater than 1, construct a segment of length \( a / n \) using only a straightedge.
|
\frac{a}{n}
| 0.25 |
In $\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\triangle BPC$?
|
12
| 0.125 |
Exactly half of the population of the island Nevezennya are hares, and all the others are rabbits. If an inhabitant of the island Nevezennya states something, they always sincerely believe in what they are saying. However, hares earnestly make mistakes on average in one out of every four cases, while rabbits earnestly make mistakes on average in one out of every three cases. One day, a creature came to the center of the island and shouted: "I am not a hare!". It thought for a moment and sadly said: "I am not a rabbit." What is the probability that it is indeed a hare?
|
\frac{27}{59}
| 0.875 |
Nine points are drawn on a sheet of paper, as shown in the following figure:
a) In how many ways is it possible to choose three collinear points?
b) In how many ways is it possible to choose four points such that three of them are collinear?
|
48
| 0.75 |
What is the result when we simplify the expression \(\left(1+\frac{1}{x}\right)\left(1-\frac{2}{x+1}\right)\left(1+\frac{2}{x-1}\right)\)?
A. \(1\)
B. \(\frac{1}{x(x+1)}\)
C. \(\frac{1}{(x+1)(x-1)}\)
D. \(\frac{1}{x(x+1)(x-1)}\)
E. \(\frac{x+1}{x}\)
|
\frac{x+1}{x}
| 0.125 |
In a geometric sequence $\left\{a_{n}\right\}$ where all terms are positive, given that $2 a_{4}+a_{3}-2 a_{2}-a_{1}=8$, find the minimum value of $2 a_{8}+a_{7}$.
|
54
| 0.875 |
Pasha and Sasha made three identical toy cars. Sasha did one-fifth of the total work. After that, they sold the cars and divided the proceeds proportionally to the work done. Pasha noticed that if he gave Sasha 400 rubles and Sasha made and sold another similar car, they would have equal amounts of money. How much does one toy car cost?
|
1000 \text{ rubles}
| 0.875 |
Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations
$$
\left\{\begin{array}{l}
\gcd(a, b, c)=15 \\
\text{lcm}(a, b, c)=3^{15} \cdot 5^{18}
\end{array}\right.
$$
|
8568
| 0.375 |
Let \( p \) be a prime number. Find all possible values of the remainder when \( p^{2} - 1 \) is divided by 12.
|
0, 3, 8
| 0.75 |
In the triangle \( ABC \), on the longest side \( BC \) with length \( b \), point \( M \) is chosen. Find the minimum distance between the centers of the circles circumscribed around triangles \( BAM \) and \( ACM \).
|
\frac{b}{2}
| 0.75 |
Compute the limit as \( x \) approaches 0 of the expression \( \frac{e^{x \cos x} - 1 - x}{\sin(x^2)} \).
|
\frac{1}{2}
| 0.625 |
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
| 0.5 |
Given that the sequence \( a_1, a_2, \cdots, a_n, \cdots \) satisfies \( a_1 = a_2 = 1 \) and \( a_3 = 2 \), and for any \( n \in \mathbf{N}^{*} \), it holds that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( \sum_{i=1}^{2023} a_i \).
|
4044
| 0.625 |
What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)?
|
1304
| 0.375 |
In rectangle \(ABCD\), side \(AB\) is 6 and side \(BC\) is 11. From vertices \(B\) and \(C\), angle bisectors are drawn intersecting side \(AD\) at points \(X\) and \(Y\) respectively. Find the length of segment \(XY\).
|
1
| 0.625 |
On Earth, the Autobots discovered a new energy source, "energy crystals," but it was seized by the Decepticons. The Decepticons manufactured cubic containers to transport the energy crystals back to Cybertron. Each energy crystal is a rectangular prism measuring 30 cm in length, 25 cm in width, and 5 cm in height. The container was fully loaded with energy crystals. At least how many energy crystals are there in one container?
|
900
| 0.75 |
Find the value of \((25 + 10\sqrt{5})^{1/3} + (25 - 10\sqrt{5})^{1/3}\).
|
5
| 0.75 |
For a positive integer \( n \), let \( S_{n} \) be the minimum value of \( \sum_{k=1}^{n} \sqrt{(2k-1)^{2} + a_{k}^{2}} \), where \( a_{1}, a_{2}, \cdots, a_{n} \) are positive real numbers whose sum is 17. There exists a unique \( n \) such that \( S_{n} \) is also an integer. Find \( n \).
|
12
| 0.875 |
The café "Buratino" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April?
|
29
| 0.25 |
Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). For example, \([3]=3\), \([2.7]=2\), and \([-2.2]=-3\). Find the last two digits of \(\left[\frac{10^{93}}{10^{31}+3}\right]\).
|
08
| 0.75 |
Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. India was first to finish, followed by Gertrude, Fiona, Janice, and lastly Hannah. What is the lowest total number of points that could have been awarded?
A) 9
B) 8
C) 7
D) 6
E) 5
|
5
| 0.75 |
The number \(123456789(10)(11)(12)(13)(14)\) is written in the base-15 number system, which means the number is equal to:
\[
(14) + (13) \cdot 15 + (12) \cdot 15^{2} + (11) \cdot 15^{3} + \ldots + 2 \cdot 15^{12} + 15^{13}
\]
What remainder does this number give when divided by 7?
|
0
| 0.75 |
In trapezoid \(ABCD\), the side \(AB\) is perpendicular to the base \(BC\). A circle passes through points \(C\) and \(D\) and is tangent to line \(AB\) at point \(E\).
Find the distance from point \(E\) to line \(CD\), if \(AD = 4\) and \(BC = 3\).
|
2 \sqrt{3}
| 0.125 |
2 apples and 3 oranges cost 6 dollars.
4 apples and 7 oranges cost 13 dollars.
16 apples and 23 oranges cost $C$ dollars. Find $C$.
If $K=\frac{6 \cos \theta+5 \sin \theta}{2 \cos \theta+3 \sin \theta}$ and $\tan \theta=2$, find $K$.
|
2
| 0.5 |
It is given that \(\log \frac{x}{2}=0.5\) and \(\log \frac{y}{5}=0.1\). If \(\log xy=c\), find \(c\).
|
1.6
| 0.875 |
The fourth-degree polynomial \(x^{4}-18 x^{3}+k x^{2}+200 x-1984\) has four roots, and the product of two of these roots is \(-32\). Find the real number \(k\).
|
86
| 0.75 |
It is known that the constant term \( a_0 \) of the polynomial \( P(x) \) with integer coefficients is less than 100 in absolute value, and \( P(20) = P(16) = 2016 \). Find \( a_0 \).
|
96
| 0.875 |
Several equally sized teams of guards slept an equal number of nights. Each guard slept more nights than the number of guards in a team but fewer nights than the number of teams. How many guards are there in a team if all the guards together slept a total of 1001 person-nights?
|
7
| 0.875 |
Vanya thought of a two-digit number, then swapped its digits and multiplied the resulting number by itself. The result turned out to be four times the original number. What number did Vanya think of?
|
81
| 0.75 |
How many nonempty subsets of {1, 2, ..., 10} have the property that the sum of their largest element and smallest element is 11?
|
341
| 0.75 |
A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct barcodes are possible?
|
116
| 0.25 |
Find the remainder when \((x-1)^{100} + (x-2)^{200}\) is divided by \(x^{2} - 3x + 2\).
|
1
| 0.75 |
In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.
|
646
| 0.75 |
Solve the equation \( x^2 + y^2 + z^2 = 2xyz \) in integers.
|
(0, 0, 0)
| 0.5 |
If the numbers 826 and 4373 are divided by the same natural number, the remainders are 7 and 8, respectively. Find all possible values of the divisor.
|
9
| 0.75 |
Let \( n \) be a fixed integer, \( n \geq 2 \).
1. Determine the smallest constant \( c \) such that the inequality \[
\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4
\] holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \).
2. For this constant \( c \), determine the necessary and sufficient conditions for equality to hold.
|
\frac{1}{8}
| 0.875 |
The school organized an outing for 1511 people and rented 42-seater and 25-seater buses. If each person must have exactly one seat and each seat is occupied by one person, how many different bus rental arrangements are possible?
|
2
| 0.625 |
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