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Given \( n \) new students such that among any 3 students, there are at least 2 students who know each other, and among any 4 students, there are at least 2 students who do not know each other. Determine the maximum value of \( n \). | 8 | 0.25 |
There are 5 integers written on a board. By summing these numbers in pairs, the following set of 10 numbers is obtained: $-1, 2, 6, 7, 8, 11, 13, 14, 16, 20$. Determine which numbers are written on the board. Write their product as the answer. | -2970 | 0.5 |
Let $S$ be the set of seven-element ordered arrays $\left(a_{1}, a_{2}, \ldots, a_{7}\right)$, where $a_i = 1$ or 0. For any two elements $a = \left(a_{1}, a_{2}, \ldots, a_{7}\right)$ and $b = \left(b_{1}, b_{2}, \ldots, b_{7}\right)$ in $S$, define the distance between $a$ and $b$ as $\sum_{i=1}^{7}\left|a_{i}-b_{i}\right|$. Let $T$ be a subset of $S$ such that the distance between any two elements is at least 3. Find the maximum value of $|T|$. | 16 | 0.875 |
Tanya had a set of identical sticks. She arranged them into a large triangle, each side of which consists of 11 sticks, and created a pattern inside the triangle such that the triangle was divided into small triangles with a side of 1 stick. How many sticks did Tanya use in total? | 198 | 0.5 |
How many numbers from 1 to 100 are divisible by 3, but do not contain the digit 3? | 26 | 0.5 |
Compute the circumradius of cyclic hexagon \( ABCDEF \), which has side lengths \( AB = BC = 2 \), \( CD = DE = 9 \), and \( EF = FA = 12 \). | 8 | 0.25 |
Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{2014}}{3}\right]$. | 15 | 0.125 |
What is the smallest three-digit positive integer which can be written in the form \( p q^{2} r \), where \( p, q \), and \( r \) are distinct primes? | 126 | 0.625 |
Suppose that \( a = \cos^4 \theta - \sin^4 \theta - 2 \cos^2 \theta \), find the value of \( a \).
If \( x^y = 3 \) and \( b = x^{3y} + 10a \), find the value of \( b \).
If there is (are) \( c \) positive integer(s) \( n \) such that \( \frac{n+b}{n-7} \) is also a positive integer, find the value of \( c \).
Suppose that \( d = \log_{4} 2 + \log_{4} 4 + \log_{4} 8 + \ldots + \log_{4} 2^{c} \), find the value of \( d \). | 18 | 0.75 |
Determine the value of
$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$
if \( n \) is infinitely large. | a^2 | 0.5 |
Given the function \( f(x) = A \sin (\omega x + \varphi) \) where \( A \neq 0 \), \( \omega > 0 \), \( 0 < \varphi < \frac{\pi}{2} \), if \( f\left(\frac{5\pi}{6}\right) + f(0) = 0 \), find the minimum value of \( \omega \). | \frac{6}{5} | 0.625 |
Given the system of equations involving \( x \) and \( y \):
$$
\left\{\begin{array}{l}
\sin x = m \sin^3 y, \\
\cos x = m \cos^3 y
\end{array}\right.
$$
If real solutions exist, what is the range of values for the positive real number \( m \)? | [1,2] | 0.625 |
Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers \( n \) satisfy \( f(n) = f(2005) \) ? | 225 | 0.375 |
Given \( f(x) = x^{5} - 10x^{3} + ax^{2} + bx + c \), and knowing that the equation \( f(x) = 0 \) has all real roots, let \( m \) be the largest of these 5 real roots. Find the maximum value of \( m \). | 4 | 0.875 |
A rubber band is 4 inches long. An ant begins at the left end. Every minute, the ant walks one inch rightwards along the rubber band, but then the band is stretched uniformly by one inch. For what value of \( n \) will the ant reach the right end during the \( n \)-th minute? | 7 | 0.75 |
Find all functions \( f: \mathbb{Z} \longrightarrow \mathbb{Z} \) such that \( f(p) > 0 \) for every prime \( p \) and for every prime \( p \) and every \( x \in \mathbb{Z} \):
$$
p \mid (f(x)+f(p))^{f(p)} - x
$$ | f(x) = x | 0.75 |
Find the measure of the angle
$$
\delta = \arccos \left( \left( \sin 2195^\circ + \sin 2196^\circ + \cdots + \sin 5795^\circ \right)^{\cos 2160^\circ} + \cos 2161^\circ + \cdots + \cos 5760^\circ \right)
$$ | 55^\circ | 0.25 |
There are 1990 piles of stones, with the number of stones in each pile being $1, 2, \cdots, 1990$. You can perform the following operation: in each step, you can choose any number of piles and remove the same number of stones from each of the chosen piles. What is the minimum number of steps required to remove all the stones from all the piles? | 11 | 0.625 |
For natural numbers \(a > b > 1\), define the sequence \(x_1, x_2, \ldots\) by the formula \(x_n = \frac{a^n - 1}{b^n - 1}\). Find the smallest \(d\) such that for any \(a\) and \(b\), this sequence does not contain \(d\) consecutive terms that are prime numbers. | 3 | 0.75 |
Suppose a die is rolled. There are two opportunities to roll: you can choose to roll just once, or twice, but the result is determined by the last roll. What is the expected value of the outcome in this scenario? If there are three opportunities, still taking the value of the last roll, what is the expected value of the outcome? | \frac{14}{3} | 0.625 |
The numbers \( a, b, c, d \) belong to the interval \([-5, 5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 110 | 0.5 |
Little Rabbit and Little Turtle start from point $A$ to the Forest Amusement Park simultaneously. Little Rabbit jumps forward 36 meters in 1 minute and rests after every 3 minutes of jumping. The first rest period is 0.5 minutes, the second rest period is 1 minute, the third rest period is 1.5 minutes, and so on, with the $k$th rest period being $0.5k$ minutes. Little Turtle does not rest or play on the way. It is known that Little Turtle reaches the Forest Amusement Park 3 minutes and 20 seconds earlier than Little Rabbit. The distance from point $A$ to the Forest Amusement Park is 2640 meters. How many meters does Little Turtle crawl in 1 minute? | 12 | 0.5 |
Given that the equation \( a x(x+1) + b x(x+2) + c (x+1)(x+2) = 0 \) has roots 1 and 2, and that \( a + b + c = 2 \), find the value of \( a \). | 12 | 0.75 |
The sequence $\mathrm{Az}\left(a_{n}\right)$ is defined as follows:
$$
a_{0}=a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{a_{n-1}}
$$
Show that $a_{n} \geq \sqrt{n}$. | a_n \geq \sqrt{n} | 0.5 |
The number \( n \) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \( n \)? | 2891 | 0.625 |
The graph of the function \( f(x)=\sin \left(\frac{\pi}{4} x-\frac{\pi}{6}\right)-2 \cos ^{2} \frac{\pi}{8} x+1 \) is symmetrical with the graph of the function \( y=g(x) \) about the line \( x=1 \). When \( x \in\left[0, \frac{4}{3}\right] \), the maximum value of \( g(x) \) is \(\qquad\) . | \frac{\sqrt{3}}{2} | 0.75 |
A ring has beads placed at positions $0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}$ in a clockwise direction. The beads can be either red or blue. How many distinct equivalence classes of arrangements are there? In other words, how many different arrangements exist, considering rotations of the ring as equivalent?
This problem can also be viewed as the case of four slots on a square (as shown in Figure 4-8), where the square is a transparent glass panel and is colored with either red or blue. How many coloring schemes are there? | 6 | 0.875 |
Let $(x, y, z, t) \in\left(\mathbb{N}^{*}\right)^{4}$ be a quadruple satisfying $x+y=z+t$ and $2xy=zt$, with $x \geq y$. Find the largest value of $m$ such that $m \leq x / y$ for sure. | 3 + 2\sqrt{2} | 0.125 |
Let
$$
2^{x}=\left(1+\tan 0.01^{\circ}\right)\left(1+\tan 0.02^{\circ}\right)\left(1+\tan 0.03^{\circ}\right) \ldots\left(1+\tan 44.99^{\circ}\right)
$$
Find \( x \). If necessary, round the answer to the nearest 0.01. | 2249.5 | 0.625 |
Points $A, B, C, D, X$ are located on a plane. Some segment lengths are known: $AC = 2$, $AX = 5$, $AD = 11$, $CD = 9$, $CB = 10$, $DB = 1$, $XB = 7$. Find the length of segment $CX$. | 3 | 0.625 |
Given 95 numbers \(a_{1}, a_{2}, \cdots, a_{95}\), where each can only take the values +1 or -1, what is the minimum positive value of the sum of their pairwise products \(a_{1} a_{2}+a_{1} a_{3}+\cdots+a_{94} a_{95}\)? | 13 | 0.875 |
A three-stage launch vehicle consists of cylindrical stages. All these cylinders are similar to each other. The length of the middle stage is half the sum of the lengths of the first and third stages. When fueled, the mass of the middle stage is $13 / 6$ times less than the combined mass of the fueled first and third stages. Find the ratio of the lengths of the first and third stages. The masses of the engines, instruments, and the shells of the stages can be neglected. | \frac{7}{5} | 0.75 |
A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left. | \frac{2}{3} | 0.875 |
There are 12 shapes made from matches - 3 triangles, 4 squares, and 5 pentagons. The shapes do not share sides. Petya and Vasya take turns removing one match at a time. Vasya wants as few untouched shapes as possible to remain, while Petya wants as many untouched shapes as possible to remain. How many shapes will remain after 10 turns? Each player makes 5 moves, and Petya starts first. | 6 | 0.25 |
In a winter camp, Vanya and Grisha share a room. Each night they draw lots to determine who will turn off the light before bed. The switch is near the door, so the loser has to walk back to bed in complete darkness, bumping into chairs.
Usually, Vanya and Grisha draw lots without any particular method, but this time Grisha has come up with a special way:
- "Let's flip a coin. If heads come up on an even-numbered throw, we stop flipping the coin: I win. If tails come up on an odd-numbered throw, you win."
a) What is the probability that Grisha wins?
b) Find the expected number of coin flips until the outcome is decided. | 2 | 0.25 |
Find the probability of a simultaneous occurrence of a specific digit when tossing two coins once. | \frac{1}{4} | 0.25 |
Let \( F_{n} \) be the Fibonacci sequence, that is, \( F_{0}=0 \), \( F_{1}=1 \), and \( F_{n+2}=F_{n+1}+F_{n} \). Compute \( \sum_{n=0}^{\infty} \frac{F_{n}}{10^{n}} \). | \frac{10}{89} | 0.625 |
Construct polynomials \( f(x) \) of degree no higher than 2 that satisfy the conditions:
a) \( f(0)=1, f(1)=3, f(2)=3 \)
b) \( f(-1)=-1, f(0)=2, f(1)=5 \)
c) \( f(-1)=1, f(0)=0, f(2)=4 \). | x^2 | 0.875 |
Given \( a_{i}, b_{i} \in \mathbf{R} \) \((i=1,2, \cdots, n)\), \( \sum_{i=1}^{n} a_{i}^{2}=4 \), and \( \sum_{i=1}^{n} b_{i}^{2}=9 \), find the maximum value of \( \sum_{i=1}^{n} a_{i} b_{i} \). | 6 | 0.625 |
Determine which is greater without using a calculator or logarithm table: \(200!\) or \(100^{200}\). | 100^{200} | 0.625 |
For a positive integer $n$, define $n! = n(n-1)(n-2) \cdots 2 \cdot 1$. Let $S_{n}=n!\left[\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}-1\right]$. Find the value of $S_{2017} = \quad$. | -\frac{1}{2018} | 0.75 |
From eleven school students and three teachers, a school committee needs to be formed consisting of eight people. At least one teacher must be included in the committee. In how many ways can the committee be formed? | 2838 | 0.875 |
Let integers \( x \) and \( y \) satisfy \( x^2 + y^2 < 16 \), and \( xy > 4 \). Determine the maximum value of \( x^2 - 2xy - 3y \). | 3 | 0.75 |
Find the smallest prime \( p > 100 \) for which there exists an integer \( a > 1 \) such that \( p \) divides \( \frac{a^{89} - 1}{a - 1} \). | 179 | 0.625 |
A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number. | 29 | 0.625 |
Previously, on an old truck, I traveled from village $A$ through $B$ to village $C$. After five minutes, I asked the driver how far we were from $A$. "Half as far as from $B," was the answer. Expressing my concerns about the slow speed of the truck, the driver assured me that while the truck cannot go faster, it maintains its current speed throughout the entire journey.
$13$ km after $B$, I inquired again how far we were from $C$. I received exactly the same response as my initial inquiry. A quarter of an hour later, we arrived at our destination. How many kilometers is the journey from $A$ to $C$? | 26 \text{ km} | 0.75 |
In the rectangular parallelepiped $ABCDA_{1}B_{1}C_{1}D_{1}$, the lengths of the edges are given as $AB=18$, $AD=30$, and $AA_{1}=20$. Point $E$ is marked on the midpoint of edge $A_{1}B_{1}$, and point $F$ is marked on the midpoint of edge $B_{1}C_{1}$. Find the distance between the lines $AE$ and $BF$. | 14.4 | 0.75 |
One of two parallel lines is tangent to a circle of radius $R$ at point $A$, while the other intersects this circle at points $B$ and $C$. Express the area of triangle $ABC$ as a function of the distance $x$ between the parallel lines. | x \sqrt{2R x - x^2} | 0.5 |
Compute the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx
$$ | \frac{3\pi}{8} | 0.875 |
$M$ is a moving point on the circle $x^{2}+y^{2}-6 x-8 y=0$, $O$ is the origin, and $N$ is a point on the ray $O M$. If $|O M| \cdot |O N| = 150$, find the equation of the locus of point $N$. | 3 x + 4 y = 75 | 0.125 |
Given that point \( F \) is the common focus of the parabola \( C: x^{2}=4y \) and the ellipse \( \frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 \) (\(a > b > 0\)), and the maximum distance from point \( M \) on the ellipse to point \( F \) is 3:
1. Find the equation of the ellipse.
2. Draw two tangent lines from point \( M \) to the parabola \( C \), with points of tangency labeled \( A \) and \( B \). Find the maximum area of \( \triangle MAB \). | 8 \sqrt{2} | 0.375 |
In triangle \( \triangle ABC \), \( BD \) is a median, point \( P \) lies on \( BD \) with the ratio \( |BP| : |PD| = 3:1 \), and \( AP \) intersects \( BC \) at \( K \). Find the ratio of the area of \( \triangle ABK \) to the area of \( \triangle ACK \). | \frac{3}{2} | 0.875 |
Let \( a_{1}, a_{2}, \cdots, a_{2006} \) be 2006 positive integers (they can be the same) such that \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all different from each other. What is the minimum number of distinct numbers in \( a_{1}, a_{2}, \cdots, a_{2006} \)? | 46 | 0.5 |
Triangle \(ABC\) is isosceles \((AB = BC)\). Segment \(AM\) divides it into two isosceles triangles with bases \(AB\) and \(MC\). Find angle \(B\). | 36^\circ | 0.625 |
Solve the cryptarithm:
$$
C, B A + A, A A = B, A
$$
(Different letters represent different digits, and the same letters represent the same digits.) | A=5, B=9, C=3 | 0.125 |
On the lateral sides \( A B \) and \( C D \) of trapezoid \( A B C D \), points \( P \) and \( Q \) are marked such that line \( P Q \) is parallel to \( A D \), and segment \( P Q \) is divided by the diagonals of the trapezoid into three equal parts. Find the length of the base \( B C \), given that \( A D = a \), \( P Q = m \), and the intersection point of the diagonals of the trapezoid lies inside the quadrilateral \( B P C Q \). | \frac{am}{3a - 2m} | 0.125 |
Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$. | 89 | 0.375 |
\(AB\) is the diameter of a circle, \(BC\) is a tangent, and \(CDA\) is a secant. Find the ratio \(CD : DA\) if \(BC\) is equal to the radius of the circle. | 1:4 | 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 |
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. | 12 | 0.75 |
Consider the cards $A, 2, \cdots, J, Q, K$ as the numbers $1, 2, \cdots, 11, 12, 13$. If we take the 13 cards of spades and 13 cards of hearts together and randomly draw 2 cards, what is the probability that the two cards are of the same suit and the product of the two numbers is a perfect square? | \frac{2}{65} | 0.125 |
The side \(A D\) of rectangle \(A B C D\) is three times longer than side \(A B\). Points \(M\) and \(N\) divide \(A D\) into three equal parts. Find \(\angle A M B + \angle A N B + \angle A D B\). | 90^\circ | 0.625 |
How can you find the lightest and heaviest stones among $2N$ stones, where any two stones have different weights, with $3N-2$ weighings? All weighings are done using a two-pan balance scale without any weights. | 3N-2 | 0.625 |
Let \( N \) be a natural number whose base-2016 representation is \( ABC \). Working now in base-10, what is the remainder when \( N - (A + B + C + k) \) is divided by 2015, if \( k \in \{ 1, 2, \ldots, 2015 \} \)? | 2015 - k | 0.875 |
A regular octahedron has a side length of 1. What is the distance between two opposite faces? | \frac{\sqrt{6}}{3} | 0.75 |
A particular coin has a $\frac{1}{3}$ chance of landing on heads (H), $\frac{1}{3}$ chance of landing on tails (T), and $\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT? | \frac{1}{4} | 0.375 |
In the city of Perpendicularka, new multi-story houses are planned to be built, with some potentially being single-story houses, so that the total number of floors is equal to 30. The city’s architect Parallelnikov proposed a project according to which, if you climb to the roof of each new house, count the number of new houses that are shorter, and add all such numbers, the resulting sum will be maximized. What is this sum? How many houses are proposed to be built, and what are their floor counts? | 112 | 0.125 |
\(\left(16 \cdot 5^{2x-1} - 2 \cdot 5^{x-1} - 0.048\right) \log \left(x^{3} + 2x + 1\right) = 0\). | 0 | 0.75 |
In triangle ABC, angle CAB is 30 degrees, and angle ABC is 80 degrees. The point M lies inside the triangle such that angle MAC is 10 degrees and angle MCA is 30 degrees. Find angle BMC in degrees. | 110^\circ | 0.25 |
Show that if two integers \( m \) and \( n \) are such that \( p(m) = p(n) \), then \( m = n \). | m = n | 0.75 |
Let \( x_{1} \) and \( x_{2} \) be the roots of the equation \( x^{2} - 6x + 1 = 0 \). Define \( a_{n} = x_{1}^{n} + x_{2}^{n} \pmod{5} \), and \( 0 \leq a_{n} < 5 \). Find \( a_{20 \times 2} \). | 4 | 0.75 |
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."
What is the minimum number of troublemakers that can be in the class? | 10 | 0.75 |
In rectangle \( J K L M \), the bisector of angle \( K J M \) cuts the diagonal \( K M \) at point \( N \). The distances between \( N \) and sides \( L M \) and \( K L \) are \( 8 \) cm and \( 1 \) cm respectively. The length of \( K L \) is \( (a+\sqrt{b}) \) cm. What is the value of \( a+b \)? | 16 | 0.875 |
There are 52 cards numbered from 1 to 52. It is considered that 1 is older than 52, and in all other pairs, the card with the higher number is older. The cards are lying face down on the table in a random order. In one question, Petya can find out which of any two cards is older. Can Petya guarantee to find the card numbered 52 in no more than 64 questions? | Yes | 0.625 |
The rank of a rational number \( q \) is the unique \( k \) for which \( q=\frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}} \), where each \( a_{i} \) is the smallest positive integer such that \( q \geq \frac{1}{a_{1}}+\cdots+\frac{1}{a_{i}} \). Let \( q \) be the largest rational number less than \( \frac{1}{4} \) with rank 3, and suppose the expression for \( q \) is \( \frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}} \). Find the ordered triple \( \left( a_{1}, a_{2}, a_{3} \right) \). | (5, 21, 421) | 0.875 |
In the following figure, the square \(ABCD\) is divided into three rectangles of equal area.
If the length of the segment \(BM\) is equal to 4, calculate the area of the square \(ABCD\). | 144 | 0.875 |
The smallest four two-digit primes are written in different squares of a \(2 \times 2\) table. The sums of the numbers in each row and column are calculated. Two of these sums are 24 and 28. The other two sums are \(c\) and \(d\), where \(c<d\). What is the value of \(5c + 7d\)? | 412 | 0.625 |
Positive integers \( m \) and \( n \) satisfy \( mn = 5000 \). If \( m \) is not divisible by 10 and \( n \) is not divisible by 10, what is the value of \( m + n \)? | 633 | 0.75 |
The accrued salary of a citizen from January to June inclusive was 23,000 rubles per month, and from July to December, it was 25,000 rubles. In August, the citizen, participating in a poetry contest, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs to be paid to the budget? (Provide the answer as a whole number, without spaces or units of measurement.)
Answer: 39540. | 39540 | 0.125 |
a) Find the last digit of the numbers \(9^{(9^8)}\) and \(2^{(3^4)}\).
b) Find the last two digits of the numbers \(2^{999}\) and \(3^{999}\).
c) * Find the last two digits of the number \(14^{(14^4)}\). | 36 | 0.625 |
Given \( A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty) \), where \( a < b \), if \( A \cup B=\mathbf{R} \), what is the minimum value of \( a - b \) ? | -1 | 0.625 |
Quadrilateral \(ABCD\) is such that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). How many degrees is \(\angle ABC\)? | 83^\circ | 0.625 |
\( a \) and \( b \) are positive integers. After rounding to three decimal places, the expression \(\frac{a}{5} + \frac{b}{7} = 1.51\). Find \( a + b \). | 9 | 0.625 |
Calculate the limit of the function:
\[ \lim_{x \to 0} \frac{e^{3x} - e^{2x}}{\sin 3x - \tan 2x} \] | 1 | 0.75 |
A metallic weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times the amount of the second metal. The mass of the second metal relates to the mass of the third metal as $3:4$, and the mass of the third metal to the mass of the fourth metal as $5:6$. Determine the mass of the fourth metal. Give your answer in kilograms, rounding to the nearest hundredth if necessary. | 5.89 \text{ kg} | 0.875 |
There were cards with the digits from 1 to 9 (a total of 9 cards) on the table. Katya selected four cards such that the product of the digits on two of them was equal to the product of the digits on the other two. Then Anton took one more card from the table. As a result, the cards left on the table were 1, 4, 5, and 8. Which digit was on the card that Anton took? | 7 | 0.75 |
A nine-digit number is allowed to undergo the following operation: any digit of the number can be replaced with the last digit of the sum of the digits of the number. Is it possible to obtain the number 123456789 from the number 133355555 using such operations? | \text{No} | 0.375 |
A dandelion blossoms in the morning, blooms yellow for three days, turns white on the morning of the fourth day, and by the evening of the fifth day, its seeds disperse. On Monday afternoon, there were 20 yellow and 14 white dandelions in the meadow, and on Wednesday, there were 15 yellow and 11 white. How many white dandelions will be in the meadow on Saturday? | 6 | 0.125 |
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)? | 15^\circ | 0.25 |
Let \( M = \{1, 2, \cdots, 1995\} \). \( A \) is a subset of \( M \) that satisfies the condition: if \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)? | 1870 | 0.375 |
A random variable \(\xi\) is uniformly distributed on the interval \([0 ; 6]\). Find the probability that the inequality \(x^{2}+ (2 \xi + 1)x + 3 - \xi \leq 0\) holds for all \(x \in [-2 ;-1]\). | \frac{5}{6} | 0.625 |
The number \( a \) is a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). Specify all natural values of \( n \) for which the equality \( a^{4} + a^{3} = a^{n} + 1 \) holds. | 15 | 0.25 |
A student plotted the graph of the function \( y = ax^2 + bx + c \) (\(a \neq 0\)) by choosing 7 different values for the variable \( x \): \( x_1 < x_2 < \cdots < x_7 \). They set \( x_2 - x_1 = x_3 - x_2 = \cdots = x_7 - x_6 \) and calculated the corresponding \( y \) values, creating the following table:
\[
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ \\
\hline
$y$ & 51 & 107 & 185 & 285 & 407 & 549 & 717 \\
\hline
\end{tabular}
\]
However, due to a calculation error, one of the \( y \) values is incorrect. Which \( y \) value is incorrect, what should the correct value be, and why? | 551 | 0.5 |
Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000? | 125 | 0.625 |
Two smaller equal circles are tangent to a given circle, one from the inside and one from the outside, such that the arc between the points of tangency is $60^{\circ}$. The radii of the smaller circles are $r$, and the radius of the larger circle is $R$. Find the distance between the centers of the smaller circles. | \sqrt{R^2 + 3r^2} | 0.875 |
Compute the limit of the function:
\[
\lim _{x \rightarrow 3} \frac{\sqrt{x+13}-2 \sqrt{x+1}}{\sqrt[3]{x^{2}-9}}
\] | 0 | 0.875 |
Compute
$$
\sum_{\substack{a+b+c=12 \\ a \geq 6, b, c \geq 0}} \frac{a!}{b!c!(a-b-c)!},
$$
where the sum runs over all triples of nonnegative integers \((a, b, c)\) such that \(a+b+c=12\) and \(a \geq 6\). | 2731 | 0.625 |
Six consecutive numbers were written on a board. When one of them was crossed out and the remaining were summed, the result was 10085. What number could have been crossed out? Specify all possible options. | 2020 | 0.25 |
A smooth ball with a radius of 1 cm was dipped in blue paint and placed between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball ended up outside the smaller sphere but inside the larger one). When touching both spheres, the ball leaves a blue mark. During its motion, the ball traveled along a closed route, resulting in a blue-bordered region on the smaller sphere with an area of 27 square cm. Find the area of the region bordered in blue on the larger sphere. Provide your answer in square centimeters, rounded to two decimal places if necessary. | 60.75 | 0.875 |
It is known that the numbers \( x, y, z \) form an arithmetic progression with a common difference of \( \alpha = \arccos \left(-\frac{2}{5}\right) \), and the numbers \( 3 + \sin x, 3 + \sin y, 3 + \sin z \) form a non-constant geometric progression. Find \( \sin y \). | -\frac{1}{10} | 0.875 |
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