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|---|---|---|
Find the minimum value of the function \( u(x, y) = x^{2} + \frac{81}{x^{2}} - 2xy + \frac{18}{x} \sqrt{2 - y^{2}} \) with real variables \( x \) and \( y \).
|
6
| 0.5 |
Given real numbers \( x \) and \( y \) satisfying
\[ 2^x + 3^y = 4^x + 9^y, \]
determine the range of values of \( U = 8^x + 27^y \).
|
(1, 2]
| 0.25 |
10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?
|
50
| 0.625 |
In triangle \(ABC\), a median \(AM\) is drawn. Find the angle \(AMC\) if the angles \(BAC\) and \(BCA\) are \(45^\circ\) and \(30^\circ\) respectively.
|
135^\circ
| 0.25 |
Seryozha wrote a five-digit natural number on the board. It turned out that among any two neighboring digits, the one on the right is greater. Valera multiplied this number by 9 and calculated the sum of the digits of the resulting product. Find all possible values that this sum can take. Justify your answer.
|
9
| 0.375 |
Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:
$$
\left|x^3 + y^3 + z^3 + 1\right| \leqslant C \left|x^5 + y^5 + z^5 + 1\right|.
$$
|
\frac{9}{10}
| 0.625 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{x}\right)^{\frac{2}{x+2}}
$$
|
4
| 0.875 |
The hypotenuse \(AB\) of a right triangle \(ABC\) is equal to 2 and is a chord of a certain circle. The leg \(AC\) is equal to 1 and lies within the circle, and its extension intersects the circle at point \(D\) with \(CD = 3\). Find the radius of the circle.
|
2
| 0.5 |
Let \( D \) be an internal point of the acute triangle \( \triangle ABC \), such that \( \angle ADB = \angle ACB + 90^{\circ} \) and \( AC \cdot BD = AD \cdot BC \). Find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \).
|
\sqrt{2}
| 0.5 |
On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has two equal digits. What is the smallest of the 14 house numbers?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.)
|
506
| 0.375 |
There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amount of grass)?
|
20
| 0.75 |
Given \(a + b + c = 1\), where \(a, b, c \in \mathbf{R}^{+}\), find the maximum value of \(m\) such that \(\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1} \geq m\).
|
2 + \sqrt{5}
| 0.5 |
Divide a circle into \( n \) equal parts and color each point either red or blue. Starting from any point, record the colors of \( k \) points in a counterclockwise direction, which is called a " \( k \)-order color sequence." Two \( k \)-order color sequences are considered different if there is at least one position where the colors differ. If any two 3-order color sequences are different, what is the maximum value of \( n \)?
|
8
| 0.375 |
Given a function \( f(x) \) defined on the set of real numbers \(\mathbf{R}\) that satisfies:
1. \( f(-x) = f(x) \)
2. \( f(4-x) = f(x) \)
For \( x \in [0, 2] \), \( f(x) = -x^2 + 1 \). Determine \( f(x) \) for \( x \in [-6, -4] \).
|
-x^2 - 8x - 15
| 0.25 |
All natural numbers whose digits sum up to 5 are ordered in increasing order. What number is in the 125th place?
|
41000
| 0.25 |
Given the sequence \(\{x_n\}\) defined by \(x_0 = 0\) and \(x_{n+1} = x_n + a + \sqrt{b^2 + 4ax_n}\) for \(n = 0, 1, 2, \ldots\), where \(a\) and \(b\) are given positive real numbers, find the general term of this sequence.
|
x_n = an^2 + bn
| 0.125 |
Let the circle \( O: x^{2} + y^{2} = 5 \) intersect the parabola \( C: y^{2} = 2px \) (where \( p > 0 \)) at the point \( A(x_{0}, 2) \). Let \( AB \) be a diameter of the circle \( O \), and a line passing through \( B \) intersects the parabola \( C \) at two distinct points \( D \) and \( E \). Find the product of the slopes of the lines \( AD \) and \( AE \).
|
2
| 0.875 |
Find all seven-digit numbers that contain each of the digits from 0 to 6 exactly once and satisfy the following conditions: both the first and last two digits are divisible by 2, both the first and last three digits are divisible by 3, both the first and last four digits are divisible by 4, both the first and last five digits are divisible by 5, and both the first and last six digits are divisible by 6.
|
3216540
| 0.25 |
Given \(a, b, c \in \mathbb{R}\), with \(a \neq 0\), consider the graph of the function \(P(x) = a x^{2} + b x + c\). Show that by using a dilation and a translation, one can obtain the graph of the function \(Q(x) = x^{2}\).
|
Q(x) = x^2
| 0.25 |
Let \( S \) be the set of rational numbers \( r \) where \( 0 < r < 1 \), and \( r \) can be expressed as a repeating decimal \( \overline{0.abcabcabc\cdots} = \overline{0.a\dot{b}\dot{c}} \), where \( a, b, \) and \( c \) are not necessarily distinct. How many distinct numerators can elements of \( S \) have when expressed in simplest fractional form?
(10th Annual American Mathematics Invitational, 1992)
|
660
| 0.25 |
In a tournament, there are 16 chess players. Determine the number of different possible schedules for the first round (schedules are considered different if they differ by the participants of at least one match; the color of the pieces and the board number are not considered).
|
2027025
| 0.25 |
Using matchsticks of the same length to form a $3 \times 1996$ grid (with each edge of the small squares being one matchstick, as shown in the figure). In total, $\qquad$ matchsticks are needed.
|
13975
| 0.625 |
The numbers \( a \) and \( b \) are such that the polynomial \( x^4 + 3x^3 + x^2 + ax + b \) is the square of some other polynomial. Find \( b \).
|
\frac{25}{64}
| 0.5 |
In triangle \(ABC\), side \(AB\) is 21, the bisector \(BD\) is \(8 \sqrt{7}\), and \(DC\) is 8. Find the perimeter of the triangle \(ABC\).
|
60
| 0.625 |
If $f$ is a circulation, then for any set $X \subseteq V$, we have $f(X, \bar{X})=0$.
|
f(X, \bar{X}) = 0
| 0.25 |
A lattice point in the plane is a point of the form \((n, m)\), where \(n\) and \(m\) are integers. Consider a set \(S\) of lattice points. We construct the transform of \(S\), denoted by \(S^{\prime}\), by the following rule: the pair \((n, m)\) is in \(S^{\prime}\) if and only if any of \((n, m-1)\), \((n, m+1)\), \((n-1, m)\), \((n+1, m)\), or \((n, m)\) is in \(S\). How many elements are in the set obtained by successively transforming \(\{(0,0)\}\) 14 times?
|
421
| 0.625 |
Write a three-digit number after 1220 to form a seven-digit number. If this seven-digit number is a multiple of 2014, then what is this three-digit number?
|
484
| 0.875 |
Point \( P \) and line \( \ell \) are such that the distance from \( P \) to \( \ell \) is 12. Given that \( T \) is a point on \( \ell \) such that \( PT = 13 \), find the radius of the circle passing through \( P \) and tangent to \( \ell \) at \( T \).
|
\frac{169}{24}
| 0.875 |
Compute the smallest positive integer \( n \) for which
\[
\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}
\]
is an integer.
|
6156
| 0.25 |
Find all 4-digit numbers that are 7182 less than the number written with the same digits in reverse order.
|
1909
| 0.25 |
Determine the number of ordered 9-tuples of positive integers \((a_{1}, a_{2}, \cdots, a_{9})\) (elements in the array can be the same) that satisfy the following condition: for any \(1 \leq i < j < k \leq 9\), there exists a different \(l (1 \leq l \leq 9)\) such that
$$
a_{i} + a_{j} + a_{k} + a_{l} = 100.
$$
|
1
| 0.625 |
Let \(a\), \(b\), and \(c\) be real constants such that \(x^{2}+x+2\) is a factor of \(ax^{3}+bx^{2}+cx+5\), and \(2x-1\) is a factor of \(ax^{3}+bx^{2}+cx-\frac{25}{16}\). Find \(a+b+c\).
|
\frac{45}{11}
| 0.75 |
On a semicircle with diameter $[AD]$, and we place two points $B$ and $C$ such that $AB = BC = 1$. We assume that $AD = 3$. Calculate the length $CD$.
|
\frac{7}{3}
| 0.75 |
In the figure, in cube \(ABCD-A_{1}B_{1}C_{1}D_{1}\), what is the measure of the dihedral angle \(A-BD_{1}-A_{1}\)?
|
60^\circ
| 0.75 |
The year of the Tiger, 2022, has the following property: it is a multiple of 6 and the sum of its digits is 6. A positive integer with such properties is called a "White Tiger number". How many "White Tiger numbers" are there among the first 2022 positive integers?
|
30
| 0.125 |
Given a line \( l \) and points \( A \) and \( B \) on opposite sides of it, use a compass and straightedge to construct a point \( M \) such that the angle between \( AM \) and \( l \) is half the angle between \( BM \) and \( l \), provided these angles do not share common sides.
|
M
| 0.875 |
The wages of a worker for October and November were in the ratio $3 / 2: 4 / 3$, and for November and December, they were in the ratio $2: 8 / 3$. In December, he received 450 rubles more than in October, and for exceeding the quarterly plan, he was awarded a bonus equal to $20 \%$ of his three-month earnings. Find the size of the bonus.
|
1494 \text{ rubles}
| 0.75 |
A family of 4 people, consisting of a mom, dad, and two children, arrived in city $N$ for 5 days. They plan to make 10 trips on the subway each day. What is the minimum amount they will have to spend on tickets if the following rates are available in city $N$?
| Adult ticket for one trip | 40 rubles |
| --- | --- |
| Child ticket for one trip | 20 rubles |
| Unlimited day pass for one person | 350 rubles |
| Unlimited day pass for a group of up to 5 people | 1500 rubles |
| Unlimited three-day pass for one person | 900 rubles |
| Unlimited three-day pass for a group of up to 5 people | 3500 rubles |
|
5200 \text{ rubles}
| 0.25 |
A trapezoid is circumscribed around a circle, and its perimeter is 12. Find the median of the trapezoid.
|
3
| 0.875 |
From a single point on a circular track, a pedestrian and a cyclist start simultaneously in the same direction. The speed of the cyclist is 55% greater than the speed of the pedestrian, and therefore the cyclist periodically overtakes the pedestrian. At how many different points on the track will the overtakes occur?
|
11
| 0.625 |
A school's emergency telephone tree operates as follows: The principal calls two students. In the first level, there is only the principal. In the second level, two students are contacted. For each subsequent level, each student contacts two new students who have not yet been contacted. This process continues with each student from the previous level contacting two new students. After the 8th level, how many students in total have been contacted?
|
254
| 0.75 |
Let \( x_{1} = a \), \( x_{2} = b \), and \( x_{n+1} = \frac{1}{2}\left(x_{n} + x_{n-1}\right) \) for \( n \geq 1 \). Calculate \(\lim_{n \rightarrow \infty} x_{n}\).
|
\frac{a + 2b}{3}
| 0.875 |
What is \( n \) if
a) the total number of different permutations of \( n \) elements is 6,227,020,800?
b) the total number of fifth-order variations without repetition of \( n \) elements is 742,560?
c) the total number of third-order variations with at least one repetition of \( n \) elements is 20,501?
|
n=83
| 0.125 |
The parabola \( P \) has its focus a distance \( m \) from the directrix. The chord \( AB \) is normal to \( P \) at \( A \). What is the minimum length for \( AB \)?
|
3\sqrt{3} m
| 0.5 |
Given the real numbers \(x_1, x_2, x_3\) satisfy the equations \(x_1 + \frac{1}{2} x_2 + \frac{1}{3} x_3 = 1\) and \(x_1^2 + \frac{1}{2} x_2^2 + \frac{1}{3} x_3^2 = 3\), what is the minimum value of \(x_3\)?
|
-\frac{21}{11}
| 0.875 |
\(ABC\) is a triangle such that \(BC = 10\), \(CA = 12\). Let \(M\) be the midpoint of side \(AC\). Given that \(BM\) is parallel to the external bisector of \(\angle A\), find the area of triangle \(ABC\). (Lines \(AB\) and \(AC\) form two angles, one of which is \(\angle BAC\). The external bisector of \(\angle A\) is the line that bisects the other angle.)
|
8 \sqrt{14}
| 0.375 |
The following figure shows a square \(ABCD\), and two points \(P\) and \(Q\) on the sides \(\overline{BC}\) and \(\overline{DA}\), respectively.
We now fold the square along the segment \(\overline{PQ}\), bringing vertex \(B\) to the midpoint of segment \(\overline{CD}\).
It is known that the side of the square measures 24.
a) Calculate the length of segment \(\overline{PC}\).
b) Calculate the length of segment \(\overline{AQ}\).
c) Calculate the length of segment \(\overline{PQ}\).
|
12 \sqrt{5}
| 0.25 |
Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer.
|
1680
| 0.125 |
Given \( n = 1990 \), find the value of
\[
\frac{1}{2^{n}}\left(1 - 3 \binom{n}{2} + 3^{2} \binom{n}{4} - 3^{3} \binom{n}{6} + \cdots + 3^{994} \binom{n}{1988} - 3^{995} \binom{n}{1990}\right).
\]
|
-\frac{1}{2}
| 0.875 |
Calculate the value of the expression \(\frac{2a - b}{3a - b} + \frac{5b - a}{3a + b}\) given that \(10a^2 - 3b^2 + 5ab = 0\) and \(9a^2 - b^2 \neq 0\).
|
-3
| 0.75 |
If the product of three consecutive integers is divided by their sum, the quotient is 5. What are these numbers?
In general: when will the quotient obtained in the aforementioned manner be an integer? For instance, which group of numbers will directly follow the group identified in the previous part in magnitude?
|
4, 5, 6
| 0.25 |
Find the greatest integer \( N \) such that
\[
N \leq \sqrt{2007^{2}-20070+31}.
\]
|
2002
| 0.5 |
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 positive integers in \( S \) less than 2008.
|
127
| 0.375 |
A regular hexagon is divided into 6 congruent equilateral triangular regions denoted as \(A, B, C, D, E, F\). Each region is to be planted with ornamental plants, with the condition that the same plant is used in any one region and different plants are used in adjacent regions. Given there are 4 different types of plants available, how many different planting schemes are possible?
|
732
| 0.375 |
In a square piece of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if, after cutting out the hole, 209 cells remained?
|
225
| 0.75 |
What is the maximum area that a rectangle can have if the coordinates of its vertices satisfy the equation \( |y-x| = (y+x+1)(5-x-y) \), and its sides are parallel to the lines \( y = x \) and \( y = -x \)? Give the square of the value of the maximum area found as the answer. (12 points)
|
432
| 0.375 |
Expanding the expression \((1+\sqrt{7})^{205}\) using the Binomial theorem, we obtain terms of the form \(C_{205}^k (\sqrt{7})^k\). Find the value of \(k\) for which such a term takes the maximum value.
|
149
| 0.875 |
In Sunny Valley, there are 10 villages. Statisticians conducted a study on the population of these villages and found the following:
1. The number of residents in any two villages differs by no more than 100 people.
2. The village of Znoinie has exactly 1000 residents, which is 90 more than the average population of the villages in the valley.
How many residents are in the village of Raduzhny, which is also located in Sunny Valley?
|
900
| 0.5 |
Find the radius of the circumcircle of an isosceles triangle with a base of 6 and a side length of 5.
|
\frac{25}{8}
| 0.5 |
The sides of a triangle are \(a, b, c\) and its area is given by \(\frac{(a+b+c)(a+b-c)}{4}\). What is the measure of the largest angle of the triangle?
|
90^\circ
| 0.625 |
Inside a sphere with radius \( R \), there are four smaller spheres each with radius \( r \). Determine the maximum possible value of \( r \). ___
|
r=(\sqrt{6}-2)R
| 0.125 |
Grisha: Masha, Sasha, and Natasha told the truth.
How many children actually told the truth?
|
1
| 0.125 |
Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest.
|
\sqrt{2} + 1
| 0.5 |
Find the minimum value of \(x^{2} + 4y^{2} - 2x\), where \(x\) and \(y\) are real numbers that satisfy \(2x + 8y = 3\).
|
-\frac{19}{20}
| 0.875 |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the relations
$$
\left\{
\begin{array}{c}
a_{1} b_{1} + a_{2} b_{3} = 1 \\
a_{1} b_{2} + a_{2} b_{4} = 0 \\
a_{3} b_{1} + a_{4} b_{3} = 0 \\
a_{3} b_{2} + a_{4} b_{4} = 1
\end{array}
\right.
$$
It is known that \( a_{2} b_{3} = 7 \). Find \( a_{4} b_{4} \).
|
a_{4} b_{4} = -6
| 0.875 |
Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations
\[
\begin{cases}
\gcd(a, b, c) = 22 \\
\mathrm{lcm}(a, b, c) = 2^{16} \cdot 11^{19}
\end{cases}
\]
|
9720
| 0.375 |
Compute the sum of all positive integers \( n < 1000 \) for which \( a_{n} \) is a rational number, given the sequence \( \{a_{n}\} \) defined by \( a_{1} = 1 \) and for all integers \( n \geq 1 \):
$$
a_{n+1} = \frac{a_{n} \sqrt{n^{2} + n}}{\sqrt{n^{2} + n + 2 a_{n}^{2}}}.
$$
|
131
| 0.25 |
Allison has a coin which comes up heads \(\frac{2}{3}\) of the time. She flips it 5 times. What is the probability that she sees more heads than tails?
|
\frac{64}{81}
| 0.875 |
In the figure, the graph of the function \( y = x^2 + ax + b \) is shown. It is known that the line \( AB \) is perpendicular to the line \( y = x \). Find the length of the segment \( OC \).
|
1
| 0.25 |
Point \( P \) moves on the circle \( x^2 + (y-3)^2 = \frac{1}{4} \), and point \( Q \) moves on the ellipse \( x^2 + 4y^2 = 4 \). Find the maximum value of \( |PQ| \) and the coordinates of the corresponding point \( Q \).
|
\left(0, -1\right)
| 0.375 |
Let \( a_{n} = 6^{n} + 8^{n} \). Find the remainder when \( a_{2018} \) is divided by 49.
|
2
| 0.5 |
An inspector, checking the quality of 400 items, determined that 20 of them belong to the second grade, and the rest to the first grade. Find the frequency of first-grade items and the frequency of second-grade items.
|
0.95, 0.05
| 0.75 |
Find the smallest positive integer \( n \) such that the polynomial \( (x+1)^{n} - 1 \) is divisible by \( x^2 + 1 \) modulo 3.
This can be interpreted through either of the following equivalent conditions:
- There exist polynomials \( P(x) \) and \( Q(x) \) with integer coefficients such that \( (x+1)^{n} - 1 = (x^2 + 1) P(x) + 3 Q(x) \).
- The remainder when \( (x+1)^{n} - 1 \) is divided by \( x^2 + 1 \) is a polynomial with integer coefficients all divisible by 3.
|
8
| 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 |
There are 64 small wooden cubes with an edge length of 1. Each cube has two faces painted red, and the other faces are white. Among these cubes, 20 have two adjacent faces painted red, and 44 have two opposite faces painted red. If these cubes are assembled to form one large cube, what is the maximum possible area of the red surface on the large cube?
|
76
| 0.625 |
\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there?
|
24
| 0.375 |
Find all values of the parameter \(a\), for each of which the set of solutions to the inequality \(\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0\) is the union of three non-overlapping intervals. In your answer, specify the sum of the three smallest integer values of \(a\) from the obtained interval.
|
9
| 0.75 |
Two circles touch internally at point \( A \). From the center \( O \) of the larger circle, a radius \( O B \) is drawn, which touches the smaller circle at point \( C \). Find \(\angle BAC\).
|
45^\circ
| 0.25 |
The minute hand of a clock moves along the circumference of the dial. A spider is watching its tip from a certain point on the same circumference. By what angle does the tip of the minute hand rotate per minute from the spider's perspective?
|
3^\circ
| 0.5 |
Let \( x_{1}, \ldots, x_{100} \) be positive real numbers such that \( x_{i} + x_{i+1} + x_{i+2} \leq 1 \) for all \( i \leq 100 \) (with the convention that \( x_{101} = x_{1} \) and \( x_{102} = x_{2} \)). Determine the maximum value of
\[
S = \sum_{i=1}^{100} x_{i} x_{i+2}
\]
|
\frac{25}{2}
| 0.125 |
As illustrated in Figure 3.1.7, D is the incenter of triangle ABC, E is the incenter of triangle ABD, and F is the incenter of triangle BDE. If the measure of angle BFE is an integer, find the smallest possible measure of angle BFE.
|
113
| 0.125 |
Given the function \( f(x) = -\frac{1}{2} x^{2} + x \) with the domain \([m, n] (m < n)\) and the range \([k m, k n] (k > 1)\), determine the value of \( n \).
|
0
| 0.75 |
Two circles \(\omega_{1}\) and \(\omega_{2}\) with radii 10 and 13, respectively, are externally tangent at point \(P\). Another circle \(\omega_{3}\) with radius \(2 \sqrt{2}\) passes through \(P\) and is orthogonal to both \(\omega_{1}\) and \(\omega_{2}\). A fourth circle \(\omega_{4}\), orthogonal to \(\omega_{3}\), is externally tangent to \(\omega_{1}\) and \(\omega_{2}\). Compute the radius of \(\omega_{4}\).
|
\frac{92}{61}
| 0.625 |
Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\).
|
3
| 0.625 |
Find the limit
$$
\lim _{x \rightarrow 0}\left(\int_{0}^{\operatorname{arctg} x} e^{\sin x} \, dx \Big/ \int_{0}^{x} \cos \left(x^{2}\right) \, dx\right)
$$
|
1
| 0.875 |
Given that \(a\), \(b\), and \(c\) are three distinct real numbers, and in the quadratic equations
\[
x^{2} + ax + b = 0
\]
\[
x^{2} + bx + c = 0
\]
\[
x^{2} + cx + a = 0
\]
any two of the equations have exactly one common root, find the value of \(a^{2} + b^{2} + c^{2}\).
|
6
| 0.75 |
Two schools, A and B, each send out 5 students to participate in a long-distance race. The rules are: The $K$th student to reach the finish line scores $K$ points (no students tie for any position). The school with the lower total score wins. How many possible total scores can the winning team have?
|
13
| 0.5 |
Determine all functions \( f: \mathbb{Z} \longrightarrow \mathbb{Z} \), where \( \mathbb{Z} \) is the set of integers, such that
\[
f(m+f(f(n)))=-f(f(m+1))-n
\]
for all integers \( m \) and \( n \).
|
f(n) = -n - 1
| 0.875 |
Consider the sequence \(1, 3, 4, 9, 10, 12, 13, \ldots\) consisting of integers greater than or equal to 1, in increasing order, that are either powers of 3 or sums of distinct powers of 3 (for example: \(4=3^{1}+3^{0}, 10=3^{2}+3^{0}, 13=3^{2}+3^{1}+3^{0}\)).
Which integer is in the hundredth position?
|
981
| 0.5 |
What is the largest number of integers that can be selected from the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5?
|
1000
| 0.625 |
Four friends, Adam, Mojmír, and twins Petr and Pavel, received a total of 52 smileys in their math class, with each receiving at least 1. The twins together have 33 smileys, but Mojmír was the most successful. How many smileys did Adam receive?
|
1
| 0.75 |
Ottó decided to assign a number to each pair \((x, y)\) and denote it as \((x \circ y)\). He wants the following relationships to hold:
a) \(x \circ y = y \circ x\)
b) \((x \circ y) \circ z = (x \circ z) \circ (y \circ z)\)
c) \((x \circ y) + z = (x + z) \circ (y + z)\).
What number should Ottó assign to the pair \((1975, 1976)\)?
|
1975.5
| 0.75 |
The real numbers \(a, b, c\) satisfy \(5^{a}=2^{b}=\sqrt{10^{c}}\), and \(ab \neq 0\). Find the value of \(\frac{c}{a}+\frac{c}{b}\).
|
2
| 0.75 |
A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called "good" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs?
|
22
| 0.375 |
Three circles of different radii are pairwise tangent to each other. The segments connecting their centers form a right triangle. Find the radius of the smallest circle if the radii of the largest and the medium circles are 6 cm and 4 cm.
|
2 \text{ cm}
| 0.875 |
Solve the equation \(3 \cdot 4^{\log_{x} 2} - 46 \cdot 2^{\log_{x} 2 - 1} = 8\).
|
\sqrt[3]{2}
| 0.875 |
The opposite sides of a quadrilateral inscribed in a circle intersect at points \(P\) and \(Q\). Find \(PQ\), if the tangents to the circle from points \(P\) and \(Q\) are equal to \(a\) and \(b\).
|
\sqrt{a^2 + b^2}
| 0.75 |
In a $29 \times 29$ table, the numbers $1, 2, 3, \ldots, 29$ are written, each number repeated 29 times. It turns out that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number written in the central cell of the table.
|
15
| 0.25 |
Let \( n \) be a positive integer. By removing the last three digits of \( n \), one gets the cube root of \( n \). Find a possible value of \( n \).
|
32768
| 0.875 |
For any \( x, y \in \mathbf{R} \), the function \( f(x, y) \) satisfies:
1. \( f(0, y) = y + 1 \);
2. \( f(x + 1, 0) = f(x, 1) \);
3. \( f(x + 1, y + 1) = f(x, f(x + 1, y)) \).
Then find \( f(3, 2016) \).
|
2^{2019}-3
| 0.5 |
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