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Find the minimum value of
$$
\max (a+b+c, b+c+d, c+d+e, d+e+f, e+f+g)
$$
subject to the constraints
(i) \( a, b, c, d, e, f, g \geq 0 \),
(ii) \( a+b+c+d+e+f+g=1 \).
|
\frac{1}{3}
| 0.625 |
Given the point \( P(-2,5) \) lies on the circle \(\odot C: x^{2}+y^{2}-2x-2y-23=0\), and the line \( l: 3x+4y+8=0 \) intersects \(\odot C\) at points \( A \) and \( B \). Find \(\overrightarrow{AB} \cdot \overrightarrow{BC}\).
|
-32
| 0.875 |
Find the value of the expression \(\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{13 \pi}{24}\).
|
\frac{3}{2}
| 0.75 |
Simplify the expression:
\[ \frac{\sqrt{x-2 \sqrt{2}}}{\sqrt{x^{2}-4 x \sqrt{2}+8}} - \frac{\sqrt{x+2 \sqrt{2}}}{\sqrt{x^{2}+4 x \sqrt{2}+8}} \]
given \( x = 3 \).
|
2
| 0.5 |
A and B are taking turns shooting with a six-shot revolver that has only one bullet. They randomly spin the cylinder before each shot. A starts the game. Find the probability that the gun will fire while A is holding it.
|
\frac{6}{11}
| 0.875 |
Point $M$ is the midpoint of side $BC$ of triangle $ABC$, where $AB=17$, $AC=30$, and $BC=19$. A circle is constructed with diameter $AB$. A point $X$ is chosen arbitrarily on this circle. What is the minimum possible length of the segment $MX$?
|
6.5
| 0.5 |
23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am older than that," Ian's agreeing reply.
Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85.
Ian sighs, "I am even older than that number."
Determine Ian's age.
|
14
| 0.75 |
Point \( P \) is located on side \( AB \) of square \( ABCD \) such that \( AP:PB = 1:2 \). Point \( Q \) lies on side \( BC \) of the square and divides it in the ratio \( BQ:QC = 2 \). Lines \( DP \) and \( AQ \) intersect at point \( E \). Find the ratio of lengths \( PE:ED \).
|
\frac{2}{9}
| 0.25 |
From the same number of squares with sides 1, 2, and 3, form a square of the smallest possible size.
|
14
| 0.5 |
Point \( P(4, -5) \) is outside the circle \( x^{2} + y^{2} = 4 \). Let \( PA \) and \( PB \) be the tangents to the circle from point \( P \). What is the equation of the line containing the points of tangency \( A \) and \( B \)?
|
4x - 5y = 4
| 0.75 |
There are \( n \) lines in general position on a plane. How many triangles are formed by these lines?
|
\binom{n}{3}
| 0.75 |
A thin diverging lens, with an optical power \( D_p = -6 \) diopters, is illuminated by a light beam with a diameter of \( d_1 = 5 \) cm. On a screen parallel to the lens, a bright spot with a diameter of \( d_2 = 20 \) cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power \( D_c \) of the converging lens.
|
10 \text{ diopters}
| 0.25 |
On a line, the points \( 0, \pm 1, \pm 2, \pm 3, \ldots \) are marked.
A particle, upon reaching point \( n \), moves to point \( n+1 \) with probability \( \frac{1}{2} \) and to point \( n-1 \) with probability \( \frac{1}{2} \) in one unit of time. Initially, the particle is at point 0. Find:
a) the probability \( x \) that the particle will ever be at point 1;
b) the probability \( y \) that the particle will ever be at point -1;
c) the probability \( z \) that the particle will ever return to point 0 (i.e., be at point 0 at a moment distinct from the initial one).
|
1
| 0.75 |
There are 52 marbles in five bags in total. No two bags contain the same number of marbles, and some bags may even be empty. It is possible to redistribute the marbles from any (non-empty) bag to the remaining four bags such that they all contain an equal number of marbles.
a) Find a distribution of marbles into the bags that satisfies all the given properties.
b) Show that in any distribution with the given properties, there is exactly one bag containing 12 marbles.
|
12
| 0.75 |
Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces
$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$
|
m = 4\pi
| 0.875 |
Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the service in this online store.
|
0.64
| 0.875 |
Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car?
|
\frac{5}{3}
| 0.875 |
Let \(a, b, c, d\) be positive integers such that the least common multiple (L.C.M.) of any three of them is \(3^{3} \times 7^{5}\). How many different sets of \((a, b, c, d)\) are possible if the order of the numbers is taken into consideration?
|
11457
| 0.375 |
Given a cube $ABCD A_1B_1C_1D_1$ with $O, E, F, G$ being the midpoints of $BD$, $BB_1$, $A_1D_1$, and $D_1C_1$ respectively, and $AB = 1$. Find the volume of the tetrahedron $OEFG$.
|
\frac{5}{48}
| 0.5 |
In a notebook, all irreducible fractions with a numerator of 15 that are greater than $\frac{1}{16}$ and less than $\frac{1}{15}$ are written. How many such fractions are written in the notebook?
|
8
| 0.75 |
Let \( f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3) \), where \( a, b, c \) are real. Given that \( f(x) \) has at least two zeros in the interval \( (0, \pi) \), find all its real zeros.
|
\mathbb{R}
| 0.375 |
Calculate the areas of figures bounded by the lines given in polar coordinates.
\[ r = \frac{1}{2} + \cos \phi \]
|
\frac{3\pi}{4}
| 0.5 |
Let \( f(x) \) be a function defined on \( \mathbf{R} \). Given that \( f(0)=2008 \), and for any \( x \in \mathbf{R} \), it satisfies \( f(x+2)-f(x) \leq 3 \cdot 2^{x} \) and \( f(x+6)-f(x) \geq 63 \cdot 2^{x} \), find \( f(2008) \).
|
2^{2008} + 2007
| 0.75 |
Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is \(1^{2}+2^{2}+4^{2}=21\)).
|
5035485
| 0.125 |
Show that
$$
\frac{\cos 64^{\circ} \cos 4^{\circ}-\cos 86^{\circ} \cos 26^{\circ}}{\cos 71^{\circ} \cos 41^{\circ}-\cos 49^{\circ} \cos 19^{\circ}}=-1
$$
|
-1
| 0.5 |
The number written on the board is allowed to be multiplied by 5 or to rearrange its digits (zero cannot be placed at the first position). Is it possible to obtain a 150-digit number $5222 \ldots 2223$ from the number 1 in this way?
|
\text{No}
| 0.625 |
In how many ways can we arrange 7 white balls and 5 black balls in a line so that there is at least one white ball between any two black balls?
|
56
| 0.125 |
The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\sqrt{85}$, $\sqrt{58}$, and $\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere.
|
\frac{14}{9}
| 0.5 |
Calculate \(\sin (\alpha-\beta)\) if \(\sin \alpha - \cos \beta = \frac{3}{4}\) and \(\cos \alpha + \sin \beta = -\frac{2}{5}\).
|
\frac{511}{800}
| 0.75 |
Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\).
|
\frac{d}{2}
| 0.5 |
Given three points \(A, B, C\) on a plane and three angles \(\angle D, \angle E, \angle F\), each less than \(180^{\circ}\) and summing to \(360^{\circ}\), use a ruler and protractor to construct a point \(O\) on the plane such that \(\angle A O B = \angle D\), \(\angle B O C = \angle E\), and \(\angle C O A = \angle F\). You may use the protractor to measure and set angles.
|
O
| 0.875 |
With eight small cubes each with an edge length of $1 \text{ cm}$ stacked into a three-dimensional shape, and given its top view as shown in the diagram, how many distinct stacking methods are there (considering rotations that result in the same shape as one method)?
|
10
| 0.25 |
The length of a rectangle is 25% greater than its width. A straight cut, parallel to the shorter side, splits this rectangle into a square and a rectangular strip. By what percentage is the perimeter of the square greater than the perimeter of the strip?
|
60\%
| 0.75 |
Petya wrote down a sequence of ten natural numbers as follows: he wrote the first two numbers randomly, and each subsequent number, starting from the third, was equal to the sum of the two preceding numbers. Find the fourth number if the seventh number is 42 and the ninth number is 110.
|
10
| 0.75 |
The non-zero real numbers \(a\) and \(b\) satisfy the equation
$$
\frac{a^{2} b^{2}}{a^{4}-2 b^{4}}=1
$$
Find, with reasoning, all the values taken by the expression
$$
\frac{a^{2}-b^{2}}{a^{2}+b^{2}}
$$
|
\frac{1}{3}
| 0.875 |
Find the non-negative integer-valued functions $f$ defined on the non-negative integers that have the following two properties:
(i) $\quad f(1)>0$;
(ii) $\quad f\left(m^{2}+n^{2}\right)=f^{2}(m)+f^{2}(n)$
for any non-negative integers $m$ and $n$.
|
f(n) = n
| 0.875 |
In the tetrahedron \( P-ABC \), \(\angle APB = \angle BPC = \angle CPA = 90^\circ\). Point \( D \) is inside the base \( ABC \), and \(\angle APD = 45^\circ\), \(\angle BPD = 60^\circ\). Find the cosine of \(\angle CPD\).
|
\frac{1}{2}
| 0.875 |
Mom made homemade currant juice and poured it into bottles. She had two types of bottles: small bottles with a volume of \(500 \ \mathrm{ml}\) and large bottles with a volume of \(750 \ \mathrm{ml}\). In the end, she had 12 small empty bottles, and the remaining bottles were completely filled. Then Mom realized that she could have poured the juice in such a way that only large bottles would be left empty and all others would be completely filled.
How many empty bottles would she have in that case?
|
8
| 0.75 |
Let's call a number "wonderful" if it has exactly 3 different odd natural divisors (and any number of even ones). How many "wonderful" two-digit numbers exist?
|
7
| 0.25 |
Vasya wrote a set of distinct natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers \(a\) and \(b\), the number \(a + b\) is not divisible by the number \(a - b\). What is the maximum number of numbers Vasya might have written?
|
675
| 0.375 |
A divisor of a natural number is called a proper divisor if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is a prime number, and the other is a perfect square, and their sum is also a perfect square (a perfect square is understood as a square of an integer). How many interesting numbers are there not exceeding 1000?
|
70
| 0.5 |
The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on cells of the same color on the board such that no two rooks attack each other? (A rook attacks any cell that is in the same row or column as it.)
|
2880
| 0.375 |
Find all functions \( f \) defined on all real numbers and taking real values such that
\[ f(f(y)) + f(x - y) = f(x f(y) - x) \]
for all real numbers \( x, y \).
|
f(x) = 0
| 0.75 |
Given that \( f(x) \) is an odd function defined on \( \mathbf{R} \), and the function \( y = f(x+1) \) is an even function. When \( -1 \leq x \leq 0 \), \( f(x) = x^3 \). Find \( f\left( \frac{9}{2} \right) \).
|
\frac{1}{8}
| 0.875 |
Given a right trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and right angles at \(A\) and \(D\). Given that the shorter diagonal \(BD\) is perpendicular to the side \(BC\), determine the smallest possible value for the ratio \(CD / AD\).
|
2
| 0.75 |
A ball, sliding on a smooth horizontal surface, catches up with a block that was moving on the same surface. The speed of the ball is perpendicular to the face of the block it hits. The mass of the ball is much smaller than the mass of the block. After an elastic collision, the ball slides on the surface in the opposite direction with a speed that is 2 times less than its initial speed.
Find the ratio of the velocities of the ball and the block before the collision.
|
4
| 0.125 |
Given that in triangle \( \triangle ABC \), the incenter is \( I \), and it satisfies \( 2 \overrightarrow{IA} + 5 \overrightarrow{IB} + 6 \overrightarrow{IC} = \overrightarrow{0} \), determine the value of \( \cos \angle B \).
|
\frac{5}{8}
| 0.875 |
Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n)=1\). Find \(100m+n\).
|
26597
| 0.5 |
Obtain the equation of a line on the plane, given that the line passes through the point $P(2, 3)$ and is perpendicular to the vector $\vec{n} = \{A, B\} = \{4, -1\}$.
|
4x - y - 5 = 0
| 0.875 |
Find the numerical value of the expression
$$
\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{2}{ab+1}
$$
if it is known that \(a\) is not equal to \(b\) and the sum of the first two terms is equal to the third term.
|
2
| 0.75 |
A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse.
|
12
| 0.625 |
Find all pairs \((p, q)\) of prime numbers for which the difference of their fifth powers is also a prime number.
|
(3, 2)
| 0.625 |
Let the sequence \(\{a_n\}\) be defined as:
\[a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{a_n} \quad (n = 1, 2, \ldots)\]
Find the integer part of \(a_{2019}\).
|
63
| 0.625 |
Is it possible to evenly divide 13 identical rectangular cakes among six children so that each cake is either not cut at all, cut into two equal parts, or cut into three equal parts?
|
Yes
| 0.75 |
On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.
Find the expected number of pourings for the first time when the situation is reversed: the first and third glasses are empty, and the second and fourth glasses are full.
|
6
| 0.125 |
From the condition, it follows that the quadrilateral ABCD is inscribed. Then $\mathrm{MD} \cdot \mathrm{MC}=\mathrm{MA} \cdot \mathrm{MB}=(3+2) \cdot 3=15$
|
15
| 0.5 |
Select several numbers from $1, 2, 3, \cdots, 9, 10$ so that every number among $1, 2, 3, \cdots, 19, 20$ can be expressed as either one of the selected numbers or the sum of two selected numbers (which can be the same). What is the minimum number of selections needed?
|
6
| 0.5 |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x, y, z \), it holds that \( f(xy) + f(xz) - 2f(x)f(yz) \geq \frac{1}{2} \). Find the value of \([1 \cdot f(1)] + [2 \cdot f(2)] + \cdots + [2022 \cdot f(2022)]\), where \([x]\) denotes the greatest integer less than or equal to \( x \).
|
1022121
| 0.375 |
Find all injective functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all natural numbers \( n \),
$$
f(f(n)) \leq \frac{f(n) + n}{2}
$$
|
f(n) = n
| 0.875 |
\( p(z) \) and \( q(z) \) are complex polynomials with the same set of roots (but possibly different multiplicities). \( p(z) + 1 \) and \( q(z) + 1 \) also have the same set of roots. Show that \( p(z) \equiv q(z) \).
|
p(z) \equiv q(z)
| 0.75 |
Calculate the value of the square of the expression
$$
1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\frac{5 x^{4}}{128}
$$
to 6 decimal places for $x= \pm 0.1, \pm 0.05, \pm 0.01$.
|
0.990000
| 0.125 |
In a competition, the full score is 100 points. Among the participating students, the highest score is 83 points and the lowest score is 30 points (all scores are integers). There are a total of 8000 students participating. At least how many students have the same score?
|
149
| 0.875 |
How many integers \( n \) are there within the range 1 to 1000, such that \( n^{1993} + 1 \) and \( n^{1994} + 1 \) are coprime?
|
500
| 0.625 |
We consider non-decreasing paths on the integer grid $\mathbb{Z}_{+}^{2}=\{(i, j) \mid i, j=0,1,2, \ldots\}$, which start at the point $(0, 0)$ and end at the point $(n, n)$ while remaining below or touching the "diagonal" (i.e., paths passing through the points in the set $\{(i, j) \in \mathbb{Z}_{+}^{2} \mid 0 \leqslant j \leqslant i \leqslant n\}$).
Show that the number of such paths is equal to $C_{n+1}$, where
$$
C_{n}=\frac{1}{n+1} \binom{2n}{n}, \quad n \geqslant 1
$$
are the Catalan numbers. (Sometimes the Catalan numbers are denoted as $c_{n}=C_{n+1}$, $n \geqslant 0$.)
|
C_{n+1}
| 0.25 |
Two people are playing "Easter egg battle." In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome of each round depends only on which egg has the stronger shell; the winning egg retains its strength). It is known that the first player won the first ten rounds. What is the probability that they will also win the next round?
|
\frac{11}{12}
| 0.625 |
The center of the upper base of a cube with an edge length of \(a\) is connected to the midpoints of the sides of the lower base, which are also connected in sequential order. Calculate the total surface area of the resulting pyramid.
|
2a^2
| 0.625 |
If $A$ is the number of edges, $F$ is the number of faces, and $S$ is the number of vertices of a closed polyhedron, then show that always:
$$
A + 2 = F + S
$$
|
A + 2 = F + S
| 0.875 |
In the quadrilateral \(A B C D\), \(A B = B C = m\), \(\angle A B C = \angle A D C = 120^{\circ}\). Find \(B D\).
|
m
| 0.125 |
Find the value of
$$
\sum_{1 \leq a<b<c} \frac{1}{2^{a} 3^{b} 5^{c}}
$$
(i.e. the sum of \(\frac{1}{2^{a} 3^{b} 5^{c}}\) over all triples of positive integers \((a, b, c)\) satisfying \(a < b < c\)).
|
\frac{1}{1624}
| 0.875 |
Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.
The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total?
|
41
| 0.5 |
In $\triangle ABC$, $\angle BAC = 60^\circ$. The angle bisector of $\angle BAC$, line segment $AD$, intersects $BC$ at $D$ and satisfies $\overrightarrow{AD} = \frac{1}{4} \overrightarrow{AC} + t \overrightarrow{AB}$. Given that $AB = 8$, find the length of $AD$.
|
6\sqrt{3}
| 0.875 |
Along a road, 10 lamp posts were placed at equal intervals, and the distance between the extreme posts was $k$ meters. Along another road, 100 lamp posts were placed at the same intervals, and the distance between the extreme posts was $m$ meters. Find the ratio $m: k$.
|
11
| 0.875 |
In triangle \(ABC\), angle \(C\) is three times the size of angle \(A\). Point \(D\) is chosen on the side \(AB\) such that \(BD = BC\).
Find \(CD\), given that \(AD = 4\).
|
CD = 4
| 0.875 |
As shown in the figure, the small square $EFGH$ is inside the larger square $ABCD$. The total shaded area is 124 square centimeters. Points $E$ and $H$ are on the side $AD$, and $O$ is the midpoint of line segment $CF$. Find the area of the quadrilateral $BOGF$ in square centimeters.
|
31
| 0.25 |
If organisms do not die but only divide, then the population will certainly never die out.
The conditions are satisfied by the function whose graph is highlighted in the image.
$$
x(p)=\left\{\begin{array}{l}
1, \text { if } 0 \leq p \leq \frac{1}{2} \\
\frac{q}{p}, \text { if } \frac{1}{2}<p \leq 1
\end{array}\right.
$$
In our case $p=0.6>\frac{1}{2}$, therefore, $x=\frac{q}{p}=\frac{2}{3}$.
|
\frac{2}{3}
| 0.75 |
In $\triangle ABC$, $\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = 0$ and $\overrightarrow{GA} \cdot \overrightarrow{GB} = 0$. Find $\frac{(\tan A + \tan B) \tan C}{\tan A \tan B} = \quad$.
|
\frac{1}{2}
| 0.25 |
Find the sum of all primes \( p \) for which there exists a prime \( q \) such that \( p^{2} + p q + q^{2} \) is a square.
|
8
| 0.75 |
Split the 2019 natural numbers $1,2,3,4, \ldots ,2019$ into 20 groups such that the average of each group is equal. What is the average of each group?
|
1010
| 0.75 |
If \(\log _{2}\left(\log _{4} P\right)=\log _{4}\left(\log _{2} P\right)\) and \(P \neq 1\), find the value of \(P\).
|
16
| 0.875 |
How many five-digit numbers are exactly divisible by 6, 7, 8, and 9?
|
179
| 0.75 |
Find the number $n$ of sides of a convex $n$-gon if each of its interior angles is at least $143^{\circ}$ and at most $146^{\circ}$.
|
10
| 0.75 |
If \( A = 2011^{2011} \), and \( B = (1 \times 2 \times \cdots \times 2011)^2 \), then \( A \) $\qquad$ B. (Fill in β $>$ β, βοΌβ or βοΌβ)
|
<
| 0.875 |
Let $A$ and $B$ be the vertices on the major axis of the ellipse $\Gamma$, $E$ and $F$ be the foci of $\Gamma$, $|AB|=4$, and $|AF|=2+\sqrt{3}$. Point $P$ is on $\Gamma$ such that $|PE||PF|=2$. Find the area of the triangle $\triangle PEF$.
|
1
| 0.625 |
Given that \(p, q, \frac{2q-1}{p}, \frac{2p-1}{q} \in \mathbf{Z}\), and \(p > 1\), \(q > 1\), find the value of \(p + q\).
|
8
| 0.875 |
Let \( p \) and \( q \) be positive integers such that \(\frac{72}{487}<\frac{p}{q}<\frac{18}{121}\). Find the smallest possible value of \( q \).
|
27
| 0.625 |
Find the number of first-type circular permutations that can be formed using 2 $a$'s, 2 $b$'s, and 2 $c$'s.
|
16
| 0.125 |
Three motorcyclists start simultaneously from the same point on a circular track in the same direction. The first motorcyclist overtakes the second for the first time after 4.5 laps from the start, and 30 minutes before that, he overtakes the third motorcyclist for the first time. The second motorcyclist overtakes the third motorcyclist for the first time three hours after the start. How many laps per hour does the first motorcyclist complete?
|
3
| 0.875 |
The sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1.5\) and \(a_{n}=\frac{1}{n^{2}-1}\) for \(n \in \mathbb{N}, n>1\). Are there any values of \(n\) such that the sum of the first \(n\) terms of this sequence differs from 2.25 by less than 0.01? If yes, find the smallest such value of \(n\).
|
n = 100
| 0.75 |
Let \( x_{1}, x_{2}, \ldots, x_{n} \) be real numbers with absolute values less than 1. What is the minimum value of \( n \) such that
\[
\left| x_{1} \right| + \left| x_{2} \right| + \ldots + \left| x_{n} \right| = 1989 + \left| x_{1} + x_{2} + \ldots + x_{n} \right|
\]
|
1990
| 0.5 |
In trapezoid \(ABCD\), segments \(BC\) and \(AD\) are the bases. Given that \(BC = 9.5\), \(AD = 20\), \(AB = 5\), and \(CD = 8.5\), find its area.
|
59
| 0.875 |
How many positive integers \( n \) less than 2007 can we find such that \(\left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{n}{6} \right\rfloor = n\), where \(\left\lfloor x \right\rfloor\) is the greatest integer less than or equal to \( x \)? (For example, \(\left\lfloor 2.5 \right\rfloor = 2\), \(\left\lfloor 5 \right\rfloor = 5\), \(\left\lfloor -2.5 \right\rfloor = -3\), etc.)
|
334
| 0.75 |
Given seven points in the plane, some of them are connected by segments so that:
(i) among any three of the given points, two are connected by a segment;
(ii) the number of segments is minimal.
How many segments does a figure satisfying (i) and (ii) contain? Give an example of such a figure.
|
9
| 0.875 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \sqrt{n(n+1)(n+2)}\left(\sqrt{n^{3}-3}-\sqrt{n^{3}-2}\right)$$
|
-\frac{1}{2}
| 0.25 |
A circle touches one side of a right angle with vertex $O$ and intersects the other side at points $A$ and $B$. Find the radius of the circle if $O A = a$ and $O B = b$.
|
\frac{a + b}{2}
| 0.5 |
Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee that he will be able to win?
|
2
| 0.875 |
If the real numbers \( x \) and \( y \) satisfy
\[
x - 3 \sqrt{x + 1} = 3 \sqrt{y + 2} - y,
\]
then the maximum value of \( x + y \) is \(\quad\).
|
9 + 3 \sqrt{15}
| 0.875 |
On the extensions of the sides \(AB\), \(BC\), \(CD\), and \(DA\) of the convex quadrilateral \(ABCD\), points \(B_1\), \(C_1\), \(D_1\), and \(A_1\) are taken respectively such that \(BB_1 = AB\), \(CC_1 = BC\), \(DD_1 = CD\), and \(AA_1 = DA\). By how many times is the area of quadrilateral \(A_1B_1C_1D_1\) greater than the area of quadrilateral \(ABCD\)? (10 points)
|
5
| 0.625 |
The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are equal to 65 and 31 respectively, and its lateral sides are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AC} \) and \( \overrightarrow{BD} \).
|
-2015
| 0.875 |
Let \( a_{1}, a_{2}, \ldots \) be an arithmetic sequence and \( b_{1}, b_{2}, \ldots \) be a geometric sequence. Suppose that \( a_{1} b_{1}=20 \), \( a_{2} b_{2}=19 \), and \( a_{3} b_{3}=14 \). Find the greatest possible value of \( a_{4} b_{4} \).
|
\frac{37}{4}
| 0.25 |
How many circles of radius 1 are needed to cover a square with a side length of 2?
|
4
| 0.25 |
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