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|---|---|---|
Count the paths composed of \( n \) rises and \( n \) descents of the same amplitude.
|
\binom{2n}{n}
| 0.75 |
Find all three-digit numbers that are five times the product of their digits.
|
175
| 0.625 |
Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?
|
6
| 0.75 |
16 progamers are playing in a single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Each time a progamer wins, he will have a ceremony to celebrate. A player's first ceremony is ten seconds long, and each subsequent ceremony is ten seconds longer than the previous one. What is the total length in seconds of all the ceremonies over the entire tournament?
|
260
| 0.5 |
The plane figure $W$ is the set of all points whose coordinates $(x, y)$ satisfy the inequality: $(|x| + |4 - |y|| - 4)^{2} \leqslant 4$. Draw the figure $W$ and find its area.
|
120
| 0.125 |
Given the set \( A = \{0, 1, 2, \cdots, 9\} \), and a collection of non-empty subsets \( B_1, B_2, \cdots, B_k \) of \( A \) such that for \( i \neq j \), \( B_i \cap B_j \) has at most two elements, determine the maximum value of \( k \).
|
175
| 0.5 |
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have?
|
16
| 0.125 |
Let \( f_{n+1} = \left\{ \begin{array}{ll} f_n + 3 & \text{if } n \text{ is even} \\ f_n - 2 & \text{if } n \text{ is odd} \end{array} \right. \).
If \( f_1 = 60 \), determine the smallest possible value of \( n \) satisfying \( f_m \geq 63 \) for all \( m \geq n \).
|
11
| 0.375 |
A positive integer \( n \) is picante if \( n! \) (n factorial) ends in the same number of zeroes whether written in base 7 or in base 8. How many of the numbers \( 1, 2, \ldots, 2004 \) are picante?
|
4
| 0.25 |
A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?
|
4 - 2\sqrt{3}
| 0.375 |
100 bulbs were hung in a row on the New Year's tree. Then the bulbs began to switch according to the following algorithm: all bulbs were lit initially, then after one second, every second bulb was turned off, and after another second, every third bulb switched: if it was on, it was turned off, and if it was off, it was turned on. Again, after one second, every fourth bulb switched, after another second - every fifth bulb, and so on. This procedure continued for 100 seconds. Determine the probability that a randomly chosen bulb is lit after all this (bulbs do not burn out or break).
|
0.1
| 0.125 |
Determine the maximum value of the sum
$$
\sum_{i<j} x_{i} x_{j}\left(x_{i}+x_{j}\right)
$$
over all \( n \)-tuples \(\left(x_{1}, \ldots, x_{n}\right)\), satisfying \( x_{i} \geq 0 \) and \(\sum_{i=1}^{n} x_{i} = 1 \).
|
\frac{1}{4}
| 0.875 |
As shown in the figure, within the square \(ABCD\), an inscribed circle \(\odot O\) is drawn, tangent to \(AB\) at point \(Q\) and tangent to \(CD\) at point \(P\). Connect \(PA\) and \(PB\) to form an isosceles triangle \(\triangle PAB\). Rotate this figure (the square, the circle, and the isosceles triangle) around the common symmetry axis \(PQ\) for one complete turn to form a cylinder, a sphere, and a cone. What is the volume ratio of these shapes?
|
3:2:1
| 0.5 |
There are a few integers \( n \) such that \( n^{2}+n+1 \) divides \( n^{2013}+61 \). Find the sum of the squares of these integers.
|
62
| 0.625 |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30?
|
21600
| 0.125 |
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?
|
2012
| 0.875 |
Solve the following system of equations:
$$
\frac{80 x+15 y-7}{78 x+12 y}=\frac{2 x^{2}+3 y^{2}-11}{y^{2}-x^{2}+3}=1
$$
|
\text{No solution}
| 0.375 |
Solve the system of equations:
\[ \begin{cases}
3x \geq 2y + 16, \\
x^{4} + 2x^{2}y^{2} + y^{4} + 25 - 26x^{2} - 26y^{2} = 72xy.
\end{cases} \]
|
(6, 1)
| 0.625 |
$V, W, X, Y, Z$ are 5 digits in base-5. The three base-5 three-digit numbers $(V Y Z)_{5}, (V Y X)_{5}, (V V W)_{5}$ form an arithmetic sequence with a common difference of 1. In decimal, what is the value of the three-digit number $(X Y Z)_{5}$?
|
108
| 0.125 |
Let \( F \) be the number of integral solutions of \( x^2 + y^2 + z^2 + w^2 = 3(x + y + z + w) \). Find the value of \( F \).
|
208
| 0.25 |
A sphere of radius 1 is covered in ink and rolling around between concentric spheres of radii 3 and 5. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?
|
\frac{9}{25}
| 0.75 |
All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.
|
111112
| 0.75 |
Determine the highest natural power of 2007 that divides 2007!.
|
9
| 0.875 |
How many solutions does the equation \(\left|\left| |x-1| - 1 \right| - 1 \right| = 1\) have? The modulus function \( |x| \) evaluates the absolute value of a number; for example \( |6| = |-6| = 6 \).
|
4
| 0.75 |
Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.
One day, Uncle Chernomor asked each warrior four questions:
- Is your favorite weapon a sword?
- Is your favorite weapon a spear?
- Is your favorite weapon an axe?
- Is your favorite weapon a bow?
13 warriors answered "yes" to the first question, 15 warriors answered "yes" to the second question, 20 warriors answered "yes" to the third question, and 27 warriors answered "yes" to the fourth question.
How many of the warriors always tell the truth?
|
12
| 0.75 |
Four cars, \( A, B, C, \) and \( D \) start simultaneously from the same point on a circular track. \( A \) and \( B \) drive clockwise, while \( C \) and \( D \) drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, \( A \) meets \( C \) for the first time, and at the same moment, \( B \) meets \( D \) for the first time. After another 46 minutes, \( A \) and \( B \) meet for the first time. After how much time from the start of the race will \( C \) and \( D \) meet for the first time?
|
53
| 0.75 |
The teacher invented a puzzle by replacing the example \(a + b = c\) with the addition of two natural numbers with their digits replaced by letters: identical digits are replaced by identical letters, and different digits by different letters. (For example, if \(a = 23\), \(b = 528\), then \(c = 551\), resulting in the puzzle \(A B + V A \Gamma = VVD\) precisely up to the choice of letters). It turned out that the original example can be uniquely restored from the given puzzle. Find the smallest possible value of the sum \(c\).
|
10
| 0.125 |
There are coins with values of 1, 2, 3, and 5 cruzeiros, each valued at their respective weights in grams (1, 2, 3, and 5 grams). It has been found that one of them is fake and differs in weight from the normal (but it is not known whether it is heavier or lighter than the real one). How can the fake coin be identified with the fewest weighings on a balance scale without weights?
|
2
| 0.625 |
\( f(n) \) is defined on the set of positive integers, and satisfies the following conditions:
1. For any positive integer \( n \), \( f[f(n)] = 4n + 9 \).
2. For any non-negative integer \( k \), \( f(2^k) = 2^{k+1} + 3 \).
Determine \( f(1789) \).
|
3581
| 0.875 |
Four points, \( A, B, C, \) and \( D \), are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle \( ABC \) for which the projection of \( D \) onto the line containing the side lies between the two vertices?
|
\frac{3}{2}
| 0.25 |
Determine, with proof, the value of \(\log _{2} 3 \log _{3} 4 \log _{4} 5 \ldots \log _{255} 256\).
|
8
| 0.875 |
How many necklaces can be made from five white beads and two black beads?
|
3
| 0.375 |
The equation \(x^{2} + a x + 4 = 0\) has two distinct roots \(x_{1}\) and \(x_{2}\); moreover,
\[x_{1}^{2} - \frac{20}{3 x_{2}^{3}} = x_{2}^{2} - \frac{20}{3 x_{1}^{3}}\]
Find all possible values of \(a\).
|
a = -10
| 0.5 |
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders?
|
1009
| 0.25 |
Given a sequence $\left\{a_{n}\right\}$ where all terms are positive and the sum of the first $n$ terms $S_{n}$ satisfies $6 S_{n} = a_{n}^{2} + 3 a_{n} + 2$. If $a_{2}, a_{4}, a_{9}$ form a geometric sequence, find the general formula for the sequence.
|
a_n = 3n - 2
| 0.75 |
How many ordered quadruples \((a, b, c, d)\) of four distinct numbers chosen from the set \(\{1, 2, 3, \ldots, 9\}\) satisfy \(b<a\), \(b<c\), and \(d<c\)?
|
630
| 0.25 |
There are 2005 points inside $\triangle ABC$ that are not collinear, plus the 3 vertices of $\triangle ABC$, making a total of 2008 points. How many non-overlapping small triangles can be formed by connecting these points? $\qquad$
|
4011
| 0.75 |
Given the sequence \(\{a_n\}\) which satisfies \(a_1 = 1, a_2 = 2, a_{n+1} - 3a_n + 2a_{n-1} = 1\) (for \(n \geq 2\), \(n \in \mathbf{N}^*\)), find the general formula for \(\{a_n\}\).
|
2^n - n
| 0.875 |
The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be purchased so that with a probability of 0.99 there will be at least 5 working ones?
|
7
| 0.25 |
The numbers 1, 2, 3, 4, 5, 6, 7 are written in a circle in some order. A number is called "good" if it is equal to the sum of the two numbers written next to it. What is the maximum possible number of "good" numbers among those written?
|
3
| 0.25 |
A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \(a\) and \(b\) such that \(|2^a - 3^b| = n\). Find the smallest value of \(n\).
|
35
| 0.375 |
Is there an integer \( x \) such that
\[ 2010 + 2009x + 2008x^2 + 2007x^3 + \cdots + 2x^{2008} + x^{2009} = 0 ? \
|
\text{No}
| 0.625 |
The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle \(X\) is 9, what is the number in circle \(Y\)?
|
11
| 0.375 |
In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommodate all chairs if selected, and each section is chosen with equal probability. What is the probability that the audience will have to move twice compared to the probability that they will have to move only once?
|
\frac{1}{2}
| 0.125 |
Find the maximum value of the expression
$$
A=\frac{1}{\sin ^{6} \alpha+\cos ^{6} \alpha} \text { for } 0 \leq \alpha \leq \frac{\pi}{2}
$$
|
4
| 0.875 |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(2, 1, 4) \)
\( M_{2}(3, 5, -2) \)
\( M_{3}(-7, -3, 2) \)
\( M_{0}(-3, 1, 8) \)
|
4
| 0.75 |
Does there exist an irrational number \(\alpha > 1\) such that
$$
\left\lfloor \alpha^n \right\rfloor \equiv 0 \pmod{2017}
$$
for all integers \(n \geq 1\)?
|
\text{Yes}
| 0.25 |
Given real numbers \( x \) and \( y \) that satisfy \[ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \], find the maximum value of the function \( U = x + y \).
|
\sqrt{13}
| 0.75 |
$$
\frac{\cos ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)-\sin ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)}{\left(\cos \frac{\alpha}{2}+\sin \frac{\alpha}{2}\right)\left(\cos \left(2 \pi-\frac{\alpha}{2}\right)+\cos \left(\frac{\pi}{2}+\frac{\alpha}{2}\right)\right) \sin \alpha}
$$
|
4 \cos 2 \alpha
| 0.75 |
When a two-digit number is multiplied by a three-digit number, a four-digit number of the form \( A = \overline{abab} \) is obtained. Find the largest \( A \), given that \( A \) is divisible by 14.
|
9898
| 0.375 |
Natural numbers \(a, x\), and \(y\), each greater than 100, are such that \(y^2 - 1 = a^2 (x^2 - 1)\). What is the smallest possible value of the fraction \(a / x\)?
|
2
| 0.625 |
Two swimmers are training in a rectangular quarry. The first swimmer prefers to start at the corner of the quarry, so he swims diagonally to the opposite corner and back. The second swimmer prefers to start at a point that divides one of the sides of the quarry in the ratio \(2018: 2019\). He swims along a quadrilateral, visiting one point on each side, and returns to the starting point. Can the second swimmer choose points on the other three sides in such a way that his path is shorter than the first swimmer's? What is the minimum value that the ratio of the length of the longer path to the shorter one can have?
|
1
| 0.75 |
The perimeter of triangle \( ABC \) is 1. Circle \( \omega \) is tangent to side \( BC \) and the extensions of side \( AB \) at point \( P \) and side \( AC \) at point \( Q \). The line passing through the midpoints of \( AB \) and \( AC \) intersects the circumcircle of triangle \( APQ \) at points \( X \) and \( Y \). Find the length of segment \( XY \).
|
\frac{1}{2}
| 0.125 |
Hooligan Vasily tore out an entire chapter from a book, with the first page numbered 231, and the number of the last page consisted of the same digits. How many sheets did Vasily tear out of the book?
|
41
| 0.5 |
At a ball, there are \(n\) married couples. In each couple, the husband and wife are of the exact same height, but no two couples share the same height. When the waltz begins, all attendees randomly pair up: each man dances with a randomly chosen woman. Find the expected value of the random variable \(X\), which represents the number of men who are shorter than their partners.
|
\frac{n-1}{2}
| 0.625 |
Let \( P \) be any point on the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) other than the endpoints of the major axis, \(F_{1}\) and \(F_{2}\) be the left and right foci respectively, and \(O\) be the center. Then \(\left|P F_{1}\right| \cdot \left|P F_{2}\right| + |O P|^{2} = \, \underline{\hspace{2cm}}\).
|
25
| 0.625 |
Let the line \( y = a \) intersect the curve \( y = \sin x \) (for \( 0 \leqslant x \leqslant \pi \)) at points \( A \) and \( B \). If \( |AB| = \frac{\pi}{5} \), find the value of \( a \) (accurate to 0.0001).
|
0.9511
| 0.875 |
Rózsa, Ibolya, and Viola have decided to solve all the problems in their problem collection. Rózsa solves \(a\), Ibolya \(b\), and Viola \(c\) problems daily. (Only one of them works on a problem.) If Rózsa solved 11 times, Ibolya 7 times, and Viola 9 times more problems daily, they would finish in 5 days; whereas if Rózsa solved 4 times, Ibolya 2 times, and Viola 3 times more problems daily, they would finish in 16 days. How many days will it take for them to complete the solutions?
|
40
| 0.625 |
For how many one-digit positive integers \( k \) is the product \( k \cdot 234 \) divisible by 12?
|
4
| 0.625 |
Sergey wrote down a certain five-digit number and multiplied it by 9. To his surprise, he obtained a number consisting of the same digits, but in reverse order. What number did Sergey write down?
|
10989
| 0.75 |
What is the maximum length of the segment intercepted by the sides of a triangle on the tangent to the inscribed circle, drawn parallel to the base, if the perimeter of the triangle is $2p$?
|
\frac{p}{4}
| 0.25 |
A square with a side length of 36 cm was cut into three rectangles in such a way that the areas of all three rectangles are equal and any two rectangles have a common section of the boundary. What is the total length (in cm) of the made cuts?
|
60 \text{ cm}
| 0.25 |
Show that for any sequence \( a_0, a_1, \ldots, a_n, \ldots \) consisting of ones and minus ones, there exist such \( n \) and \( k \) such that
\[ |a_0 a_1 \cdots a_k + a_1 a_2 \cdots a_{k+1} + \cdots + a_n a_{n+1} \cdots a_{n+k}| = 2017. \]
|
S = 2017
| 0.5 |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of segment \( BM \). Given \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).
|
\sqrt{3} + 1
| 0.75 |
Two spheres of one radius and two spheres of another radius are arranged so that each sphere touches three other spheres and a given plane. Find the ratio of the radii of the spheres.
|
2 + \sqrt{3}
| 0.625 |
Find the point \( M' \) symmetric to the point \( M \) with respect to the plane.
\( M(-1, 0, 1) \)
\(2x + 4y - 3 = 0\)
|
(0, 2, 1)
| 0.75 |
The brother left home 6 minutes later than his sister, following her, and caught up with her in 12 minutes. How many minutes would it take him to catch up with her if he walked twice as fast? Both the brother and sister walk at a constant speed.
|
3 \text{ minutes}
| 0.625 |
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \( C: x^{2} - \frac{y^{2}}{24} = 1 \) respectively, and let \( P \) be a point on the hyperbola \( C \) in the first quadrant. If \( \frac{\left|P F_{1}\right|}{\left|P F_{2}\right|} = \frac{4}{3} \), find the radius of the inscribed circle of the triangle \( P F_{1} F_{2} \).
|
2
| 0.875 |
Let \( S \) be the set of points whose coordinates \( x \), \( y \), and \( z \) are integers that satisfy \( 0 \leq x \leq 2 \), \( 0 \leq y \leq 3 \), and \( 0 \leq z \leq 4 \). Two distinct points are randomly chosen from \( S \). Find the probability that the midpoint of the two chosen points also belongs to \( S \).
|
\frac{23}{177}
| 0.875 |
On side \(AB\) of an equilateral triangle \(ABC\), a point \(K\) is chosen, and on side \(BC\), points \(L\) and \(M\) are chosen such that \(KL = KM\) and point \(L\) is closer to \(B\) than \(M\).
a) Find the angle \(MKA\) if it is known that \(\angle BKL = 10^\circ\).
b) Find \(MC\) if \(BL = 2\) and \(KA = 3\).
Provide justification for each part of the answers.
|
5
| 0.375 |
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\operatorname{tg}\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$
|
1
| 0.75 |
A train passes a post in 5 seconds. In how many seconds will the train and an electric train pass by each other, if the speed of the train is 2 times the speed of the electric train, and the length of the train is 3 times the length of the electric train?
|
\frac{40}{9}
| 0.875 |
Find all integer values of the parameter \(a\) for which the system has at least one solution:
$$
\left\{\begin{array}{l}
y - 2 = x(x + 2) \\
x^{2} + a^{2} + 2x = y(2a - y)
\end{array}\right.
$$
In the answer, indicate the sum of the found values of parameter \(a\).
|
3
| 0.5 |
In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. What is the maximum number of marked points that lie on one circle?
|
1976
| 0.5 |
Given the function $f(x)=\frac{1}{3} x^{3}-(a-1) x^{2}+b^{2} x$, where $a \in\{1,2,3,4\}$ and $b \in\{1,2,3\}$, find the probability that the function $f(x)$ is an increasing function on $\mathbf{R}$.
|
\frac{3}{4}
| 0.125 |
Calculate the area enclosed by the cardioid $\rho = 1 + \cos \varphi$ for $0 \leqslant \varphi \leqslant 2\pi$.
|
\frac{3\pi}{2}
| 0.625 |
Let $\{a_n\}$ be a sequence of positive numbers with $b_n$ representing the sum of the first $n$ terms of $\{a_n\}$. The product of the first $n$ terms of the sequence $\{b_n\}$ is denoted by $c_n$, and $b_n + c_n = 1$. What is the number in the sequence $\left\{\frac{1}{a_n}\right\}$ that is closest to 2009?
|
1980
| 0.25 |
Two circles of radius 12 have their centers on each other. \( A \) is the center of the left circle, and \( AB \) is a diameter of the right circle. A smaller circle is constructed tangent to \( AB \) and the two given circles, internally to the right circle and externally to the left circle. Find the radius of the smaller circle.
|
3\sqrt{3}
| 0.5 |
Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be a permutation of the numbers $1, 2, 3, 4, 5$. If there does not exist $1 \leq i < j < k \leq 5$ such that $a_{i} < a_{j} < a_{k}$, then the number of such permutations is ______.
|
42
| 0.625 |
A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \( 4 \sqrt{6} \).
Find the area of the quadrilateral.
|
18 \sqrt{161}
| 0.125 |
Find the greatest common divisor (GCD) of all the numbers of the form \( n^{13} - n \).
|
2730
| 0.5 |
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer.
|
1530
| 0.625 |
A weight supported by a helical spring is lifted by a distance $b$ and then released. It begins to fall, with its acceleration described by the equation $g = -p^2 s$, where $p$ is a constant and $s$ is the distance from the equilibrium position. Find the equation of motion.
|
s(t) = b \cos(pt)
| 0.875 |
Suppose \( f \) is a function that satisfies \( f(2) = 20 \) and \( f(2n) + n f(2) = f(2n+2) \) for all positive integers \( n \). What is the value of \( f(10) \)?
|
220
| 0.375 |
Given \( S = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \cdots + \frac{1}{\text{1000 nines}} \), what is the 2016th digit after the decimal point in the value of \( S \)?
|
4
| 0.25 |
On the median \(A A_{1}\) of triangle \(A B C\), a point \(M\) is taken such that \(A M : M A_{1} = 1 : 3\). In what ratio does the line \(B M\) divide the side \(A C\)?
|
1:6
| 0.875 |
In rectangle \( ABCD \), a circle \(\omega\) is constructed using side \( AB \) as its diameter. Let \( P \) be the second intersection point of segment \( AC \) with circle \(\omega\). The tangent to \(\omega\) at point \( P \) intersects segment \( BC \) at point \( K \) and passes through point \( D \). Find \( AD \), given that \( KD = 36 \).
|
24
| 0.75 |
Given some triangles with side lengths \(a \,\text{cm}, 2 \,\text{cm}\) and \(b \,\text{cm}\), where \(a\) and \(b\) are integers and \(a \leq 2 \leq b\). If there are \(q\) non-congruent classes of triangles satisfying the above conditions, find the value of \(q\).
|
3
| 0.75 |
Given a regular tetrahedron \(P-ABC\) with edge length 1, let \(E\), \(F\), \(G\), and \(H\) be points on edges \(PA\), \(PB\), \(CA\), and \(CB\) respectively. If \(\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}\) and \(\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}\), then find \(\overrightarrow{EG} \cdot \overrightarrow{FH}\).
|
\frac{5}{18}
| 0.375 |
$\bigcirc \bigcirc \div \square=14 \cdots 2$, how many ways are there to fill the square?
|
4
| 0.5 |
Grandmother has apples of three varieties in her garden: Antonovka, Grushovka, and Bely Naliv. If the Antonovka apples were tripled, the total number of apples would increase by 70%. If the Grushovka apples were tripled, the total number of apples would increase by 50%. By what percentage would the total number of apples change if the Bely Naliv apples were tripled?
|
80\%
| 0.875 |
The sequence \(a_{0}, a_{1}, a_{2}, \cdots, a_{n}\) satisfies \(a_{0}=\sqrt{3}\), \(a_{n+1}=\left\lfloor a_{n}\right\rfloor + \frac{1}{\left\{a_{n}\right\}}\), where \(\left\lfloor a_{n}\right\rfloor\) and \(\left\{a_{n}\right\}\) represent the integer part and fractional part of \(a_{n}\), respectively. Find \(a_{2016}\).
|
3024+\sqrt{3}
| 0.625 |
Compute the number of quadruples \((a, b, c, d)\) of positive integers satisfying
\[ 12a + 21b + 28c + 84d = 2024. \]
|
2024
| 0.375 |
Positive integers \( a, b \), and \( c \) have the property that \( a^b \), \( b^c \), and \( c^a \) end in 4, 2, and 9, respectively. Compute the minimum possible value of \( a + b + c \).
|
17
| 0.25 |
Find the sum of the first twelve terms of an arithmetic sequence if its fifth term $a_{5} = 1$ and its seventeenth term $a_{17} = 18$.
|
37.5
| 0.375 |
There are 700 cards in a box, in six colors: red, orange, yellow, green, blue, and white. The ratio of the number of red, orange, and yellow cards is $1: 3: 4$, and the ratio of the number of green, blue, and white cards is $3:1:6$. Given that there are 50 more yellow cards than blue cards, determine the minimum number of cards that must be drawn to ensure that there are at least 60 cards of the same color among the drawn cards.
|
312
| 0.5 |
It is known that \( A \) is the largest number that is the product of several natural numbers whose sum is 2011.
What is the largest power of three that divides \( A \)?
|
3^{669}
| 0.125 |
A \(10 \times 1\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at least one green and one yellow tile should be used, and adjacent tiles should have different colors?
|
1022
| 0.75 |
Let \( a, b, c \) be pairwise distinct positive integers such that \( a+b, b+c \) and \( c+a \) are all square numbers. Find the smallest possible value of \( a+b+c \).
|
55
| 0.875 |
In the tetrahedron $ABCD$, the angles $ABD$ and $ACD$ are obtuse. Show that $AD > BC$.
|
AD > BC
| 0.875 |
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