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Given a quadrilateral \( ABCD \) such that \( AB = BD \), \( \angle ABD = \angle DBC \), and \( \angle BCD = 90^\circ \). Point \( E \) is marked on segment \( BC \) such that \( AD = DE \). What is the length of segment \( BD \) if it is known that \( BE = 7 \) and \( EC = 5 \)? | 17 | 0.625 |
In the equation \(x^{2} + p x + 3 = 0\), determine \(p\) such that one root of the equation is three times the other root. | \pm 4 | 0.875 |
Let $n$ be a given integer greater than 2. There are $n$ indistinguishable bags, and the $k$-th bag $(k= 1,2, \cdots, n)$ contains $k$ red balls and $n-k$ white balls. After mixing these bags, one bag is chosen at random, and three balls are drawn consecutively without replacement. Find the probability that the third ball drawn is white. | \frac{n-1}{2n} | 0.875 |
In trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(AD = 3 \cdot BC\). A line intersects the non-parallel sides of the trapezoid at points \(M\) and \(N\) such that \(AM:MB = 3:5\) and \(CN:ND = 2:7\). Find the ratio of the areas of quadrilaterals \(MBCN\) and \(AMND\). | \frac{9}{23} | 0.375 |
Given that $\lg 2 = a$ and $\log_{2} 7 = b$, find $\lg 56$. | a(b+3) | 0.5 |
Juliana wants to assign each of the 26 letters $A, B, C, D, \ldots, W, X, Y, Z$ of the alphabet a nonzero numerical value, such that $A \times C = B, B \times D = C, C \times E = D$, and so on, up to $X \times Z = Y$.
a) If Juliana assigns the values 5 and 7 to $A$ and $B$ respectively, what will be the values of $C, D$, and $E$?
b) Show that $G = A$, regardless of the values Juliana assigns to $A$ and $B$.
c) If Juliana assigns values to $A$ and $B$ such that $A \times B = 2010$, what will be the value of the product $A \times B \times C \times D \times \cdots \times W \times X \times Y \times Z$? | 2010 | 0.5 |
On the sides \(AB\) and \(AC\) of an isosceles triangle \(ABC\) (\(AB = AC\)), points \(M\) and \(N\) are marked respectively such that \(AN > AM\). The lines \(MN\) and \(BC\) intersect at point \(K\). Compare the lengths of segments \(MK\) and \(MB\). | MK > MB | 0.625 |
Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords? | \sqrt{2-\sqrt{2}} | 0.75 |
Construct a square \(A B C D\) with side length \(6 \text{ cm}\). Construct a line \(p\) parallel to the diagonal \(A C\) passing through point \(D\). Construct a rectangle \(A C E F\) such that vertices \(E\) and \(F\) lie on the line \(p\).
Using the given information, calculate the area of rectangle \(A C E F\). | 36 \text{ cm}^2 | 0.75 |
In the trapezoid $ABCD$, the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the diagonal $AC$, $N$ the midpoint of the diagonal $BD$, and $P$ the midpoint of the side $AB$. Given that $AB = 15 \text{ cm}$, $CD = 24 \text{ cm}$, and the height of the trapezoid is $h = 14 \text{ cm}$:
a) Calculate the length of the segment $MN$.
b) Calculate the area of the triangle $MNP$. | 15.75 \text{ cm}^2 | 0.625 |
Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked? | 64 | 0.5 |
Tanya sequentially wrote down numbers of the form \( n^{7} - 1 \) for natural numbers \( n = 2, 3, \ldots \) and noticed that for \( n = 8 \), the resulting number is divisible by 337. For what smallest \( n > 1 \) will she get a number divisible by 2022? | 79 | 0.25 |
What is the minimum value that the product of two positive numbers \(a\) and \(b\) can take if it is known that \(ab = a + b\)? | 4 | 0.875 |
If \( a > b > c > d \), and \( \frac{1}{a-b} + \frac{1}{h-c} + \frac{1}{c-d} \geq \frac{n}{a-d} \), then the maximum value of the integer \( n \) is ___ | 9 | 0.75 |
Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]
Find the sum of all possible values of \( m \). | 93 | 0.625 |
Find the number of natural numbers \( k \) not exceeding 267000 such that \( k^{2} -1 \) is divisible by 267. | 4000 | 0.125 |
Vasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya rotates the triangle 6 times clockwise around its vertex numbered 2 by an angle equal to the angle at this vertex, it will return to its original position. Vasya claims that if the triangle is rotated $n$ times around its vertex numbered 3 by an angle equal to the angle at this vertex, it will return to its original position. What is the minimum $n$ that Vasya might name so that his claim is true for at least some cardboard triangle? | 4 | 0.625 |
Find the cosine of the angle between the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
$A(-1, -2, 1), B(-4, -2, 5), C(-8, -2, 2)$ | \frac{1}{\sqrt{2}} | 0.125 |
Let \( u, v, w \) be positive real numbers, all different from 1. If
\[
\log_{u}(vw) + \log_{v}(w) = 5 \quad \text{and} \quad \log_{v}(u) + \log_{w}(v) = 3,
\]
find the value of \( \log_{w}(u) \). | \frac{4}{5} | 0.875 |
Plot the set of points on the $(x, y)$ plane that satisfy the equation $|4x| + |3y| + |24 - 4x - 3y| = 24$ and find the area of the resulting figure. | 24 | 0.625 |
How many ways are there to insert plus signs (+) between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30? | 2002 | 0.375 |
How many four-digit numbers contain the digit 9 followed immediately by the digit 5? | 279 | 0.25 |
There are four piles of stones: one with 6 stones, two with 8 stones, and one with 9 stones. Five players numbered 1, 2, 3, 4, and 5 take turns, in the order of their numbers, choosing one of the piles and dividing it into two smaller piles. The loser is the player who cannot do this. State the number of the player who loses. | 3 | 0.875 |
There are 36 students in a club. If any 33 of them attend a session, girls will always be in the majority. However, if 31 students attend, it might happen that boys are in the majority. How many girls are in the club? | 20 | 0.75 |
Extract the square root of the polynomial:
$$
16ac + 4a^2 - 12ab + 9b^2 - 24bc + 16c^2
$$ | 2a - 3b + 4c | 0.875 |
Through the origin, lines (including the coordinate axes) are drawn that divide the coordinate plane into angles of $1^{\circ}$. Find the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100 - x$. | 8950 | 0.25 |
A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM? | \frac{5}{16} | 0.5 |
Eight singers sing songs at a festival. Each song is sung once by a group of four singers. Every pair of singers sings the same number of songs together. Find the smallest possible number of songs. | 14 | 0.875 |
Find the largest real number \( x \) such that \( \sqrt[3]{x} + \sqrt[3]{4 - x} = 1 \). | 2 + \sqrt{5} | 0.875 |
The length of the road is 300 km. Car $A$ starts at one end of the road at noon and moves at a constant speed of 50 km/h. At the same time, car $B$ starts at the other end of the road with a constant speed of 100 km/h, and a fly starts with a speed of 150 km/h. Upon meeting car $A$, the fly turns around and heads towards car $B$.
1) When does the fly meet car $B$?
2) If upon meeting car $B$, the fly turns around, heads towards car $A$, meets it, turns around again, and continues flying back and forth between $A$ and $B$ until they collide, when do the cars crush the fly? | 2 \ \text{hours} | 0.625 |
On a $3 \times 3$ grid, there are knights who always tell the truth and liars who always lie. Each one stated: "Among my neighbors, exactly three are liars." How many liars are on the grid?
Neighbors are considered to be people located on cells that share a common side. | 5 | 0.625 |
Given a $5 \times 5$ grid where the number in the $i$-th row and $j$-th column is denoted by \( a_{ij} \) (where \( a_{ij} \in \{0, 1\} \)), with the condition that \( a_{ij} = a_{ji} \) for \( 1 \leq i, j \leq 5 \). Calculate the total number of ways to fill the grid such that there are exactly five 1's in the grid. | 326 | 0.375 |
The cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ has an edge length of 1. What is the distance between the lines $A_{1} C_{1}$ and $B D_{1}$? | \frac{\sqrt{6}}{6} | 0.875 |
If \( a + b + c = 1 \), what is the maximum value of \( \sqrt{3a+1} + \sqrt{3b+1} + \sqrt{3c+1} \)? | 3\sqrt{2} | 0.75 |
A circle is circumscribed around an isosceles triangle with a $45^\circ$ angle at the vertex. Another circle is inscribed within this triangle, touching the first circle internally and also touching the two equal sides of the triangle. The distance from the center of the second circle to this vertex of the triangle is 4 cm. Find the distance from the center of this second circle to the center of the circle inscribed in the triangle. | 4 \text{ cm} | 0.125 |
Let \( D \) be a point inside an acute triangle \( \triangle ABC \) such that \( \angle ADB = \angle ACB + 90^\circ \), and \( AC \cdot BD = AD \cdot BC \). Calculate the ratio: \(\frac{AB \cdot CD}{AC \cdot BD} \). | \sqrt{2} | 0.5 |
Using small cubes with edge length \( m \) to form a larger cube with an edge length of 12, the surface of the larger cube (6 faces) is painted red. The number of small cubes with only one face painted red is equal to the number of small cubes with two faces painted red. Find \( m \). | 3 | 0.875 |
Professor Fernando's tree grows according to the following rule:
- In the first week, the tree starts to grow with only one branch;
- After growing for two weeks, this branch produces a new branch every week;
- Each new branch continues to grow and, after growing for two weeks, produces a new branch every week.
The figure below illustrates Professor Fernando's tree after five weeks:
(Note: The figure is omitted here)
Observe that after three weeks, there were two branches; after four weeks, there were three branches; and after five weeks, there were five branches.
a) How many branches will there be after six weeks?
b) How many branches will there be after seven weeks?
c) How many branches will there be after thirteen weeks? | 233 | 0.75 |
Alice and Bob play on a $20 \times 20$ grid. Initially, all the cells are empty. Alice starts and the two players take turns placing stones on unoccupied cells. On her turn, Alice places a red stone on an empty cell that is not at a distance of $\sqrt{5}$ from any other cell containing a red stone. On his turn, Bob places a blue stone on an unoccupied cell. The game ends when a player can no longer place a stone. Determine the largest $K$ such that Alice can ensure to place at least $K$ red stones regardless of how Bob places his stones. | 100 | 0.625 |
At a chamber music festival, six musicians have gathered. At each concert, some of the musicians perform while the others listen from the audience. What is the minimum number of concerts needed for each of the six musicians to listen (from the audience) to all the others? | 4 | 0.25 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{\sqrt{\cos x}-1}{\sin ^{2} 2 x}$$ | -\frac{1}{16} | 0.875 |
As shown in the figure, a rectangle is divided into four smaller rectangles. The perimeters of rectangles $A$, $B$, and $C$ are 10 cm, 12 cm, and 14 cm, respectively. What is the maximum possible area of rectangle $D$ in square centimeters? | 16 \ \text{cm}^2 | 0.625 |
Dragomir has 6 pairs of socks in a drawer. Each pair of socks is different from every other pair in the drawer. He randomly removes 4 individual socks from the drawer, one at a time. What is the probability that there is exactly 1 matching pair of socks among these 4 socks? | \frac{16}{33} | 0.875 |
Tetrahedron \(ABCD\) has base \( \triangle ABC \). Point \( E \) is the midpoint of \( AB \). Point \( F \) is on \( AD \) so that \( FD = 2AF \), point \( G \) is on \( BD \) so that \( GD = 2BG \), and point \( H \) is on \( CD \) so that \( HD = 2CH \). Point \( M \) is the midpoint of \( FG \) and point \( P \) is the point of intersection of the line segments \( EH \) and \( CM \). What is the ratio of the volume of tetrahedron \( EBCP \) to the volume of tetrahedron \( ABCD \)? | \frac{1}{10} | 0.625 |
Given an isosceles triangle \( ABC \) with \( AB = BC \) and the vertex angle equal to \( 102^\circ \). A point \( O \) is located inside the triangle \( ABC \) such that \( \angle OCA = 30^\circ \) and \( \angle OAC = 21^\circ \). Find the measure of angle \( \angle BOA \). | 81^{\circ} | 0.125 |
In a shooting competition, there are 8 clay targets arranged in 3 columns as shown in the diagram. A sharpshooter follows these rules to shoot down all the targets:
1. First, select a column from which one target will be shot.
2. Then, target the lowest remaining target in the selected column.
How many different sequences are there to shoot down all 8 targets? | 560 | 0.375 |
The sequence \(\{a_n\}\) has consecutive terms \(a_n\) and \(a_{n+1}\) as the roots of the equation \(x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0\), with initial term \(a_1 = 2\). Find the sum of the infinite series \(c_1, c_2, \cdots, c_n, \cdots\). | \frac{9}{2} | 0.75 |
Grandpa prepared a pile of hazelnuts for his six grandchildren and told them to take some. First, Adam came, counted out half of the hazelnuts, took one extra hazelnut, and left. The second, Bob did the same, as did the third, Cyril, fourth, Dan, and fifth, Eda. Franta was left looking sadly at the empty table; there were no hazelnuts left for him.
How many hazelnuts were originally in the pile?
Hint: How many hazelnuts did Eda take? | 62 | 0.875 |
A square has a tens digit of 7. What is the units digit? | 6 | 0.5 |
The diagram below shows \( \triangle ABC \), which is isosceles with \( AB = AC \) and \( \angle A = 20^\circ \). The point \( D \) lies on \( AC \) such that \( AD = BC \). The segment \( BD \) is constructed as shown. Determine \( \angle ABD \) in degrees. | 10 | 0.75 |
Find a point that is at distances $m$, $n$, and $p$ from three planes. How many solutions does this problem have? | 8 | 0.875 |
In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances from \(P\) to vertices \(A, B,\) and \(C\). Find \(d_{\min} + d_{\max}\). | 160 | 0.375 |
In triangle \( \triangle ABC \), it is known that
\[ \cos C = \frac{2 \sqrt{5}}{5} \]
\[ \overrightarrow{A H} \cdot \overrightarrow{BC} = 0 \]
\[ \overrightarrow{AB} \cdot (\overrightarrow{CA} + \overrightarrow{CB}) = 0 \]
Determine the eccentricity of the hyperbola passing through point \( C \) with foci at points \( A \) and \( H \). | \sqrt{5} + 2 | 0.125 |
Inside the tetrahedron \(ABCD\), there is a point \(O\) such that the lines \(AO, BO, CO,\) and \(DO\) intersect the faces \(BCD, ACD, ABD,\) and \(ABC\) at the points \(A_{1}, B_{1}, C_{1},\) and \(D_{1}\) respectively. Given that \(\frac{AO}{OA_{1}}=\frac{BO}{OB_{1}}=\frac{CO}{OC_{1}}=\frac{DO}{OD_{1}}=k\), find the value of \(k\). | 3 | 0.75 |
Find the number of pairs of integers $(a, b)$ such that $1 \leq a \leq 70, 1 \leq b \leq 50$, and the area $S$ of the figure defined by the system of inequalities
$$
\left\{\begin{array}{l}
\frac{x}{a}+\frac{y}{b} \geq 1 \\
x \leq a \\
y \leq b
\end{array}\right.
$$
is such that the number $2S$ is a multiple of 5. | 1260 | 0.5 |
Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move.
Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board. | 100 | 0.5 |
Find the sum of the first 10 elements that appear both in the arithmetic progression $\{5, 8, 11, 14, \ldots\}$ and in the geometric progression $\{20, 40, 80, 160, \ldots\}$. | 6990500 | 0.125 |
Given that \( a b c = 8 \) with \( a, b, c \) being positive, what is the minimum value of \( a + b + c \)? | 6 | 0.75 |
Without solving the equation \(3x^{2} - 5x - 2 = 0\), find the sum of the cubes of its roots. | \frac{215}{27} | 0.875 |
Mr. Li goes to a toy store to buy balls. The money he has can buy exactly 60 plastic balls, or exactly 36 glass balls, or 45 wooden balls. Mr. Li decides to buy 10 plastic balls and 10 glass balls, and spends the remaining money on wooden balls. How many balls does Mr. Li buy in total? | 45 | 0.875 |
Compute
$$
\int_{L} \frac{60 e^{z}}{z(z+3)(z+4)(z+5)} d z
$$
where \( L \) is a unit circle centered at the origin. | 2 \pi i | 0.875 |
On a line, 5 points \( P, Q, R, S, T \) are marked in that order. It is known that the sum of the distances from \( P \) to the other 4 points is 67, and the sum of the distances from \( Q \) to the other 4 points is 34. Find the length of segment \( PQ \). | 11 | 0.5 |
Find the length of the arc of a cycloid defined by \( x = a(t - \sin t) \) and \( y = a(1 - \cos t) \). | 8a | 0.625 |
$$
n < p_{Z_{k}^{k}}^{k} \text{,} $$
where $p_{2 k}$ denotes the $2k$-th prime number. | n < p_{2k}^{k} | 0.5 |
Recall that the sum of the angles of a triangle is 180 degrees. In triangle $ABC$, angle $A$ is a right angle. Let $BM$ be the median of the triangle and $D$ be the midpoint of $BM$. It turns out that $\angle ABD = \angle ACD$. What are the measures of these angles? | 30^\circ | 0.75 |
Let \( n \) be a given positive integer. Find the smallest positive integer \( u_n \) such that, for every positive integer \( d \), the number of numbers divisible by \( d \) in any \( u_n \) consecutive positive odd numbers is no less than the number of numbers divisible by \( d \) in the set of odd numbers \( 1, 3, 5, \ldots, 2n-1 \). | 2n - 1 | 0.875 |
Suppose \( x = \frac{13}{\sqrt{19 + 8 \sqrt{3}}} \). Find the exact value of
\[
\frac{x^{4} - 6 x^{3} - 2 x^{2} + 18 x + 23}{x^{2} - 8 x + 15}.
\] | 5 | 0.75 |
Given that \( m \) and \( n \) are integers such that \( m, n \in \{1, 2, \dots, 1981\} \), and \( (n^2 - mn - m^2)^2 = 1 \), determine the maximum value of \( m^2 + n^2 \). | 3524578 | 0.5 |
Given a grid of size $m \times n$ (with $m, n > 1$). The centers of all the cells are marked. What is the maximum number of marked centers that can be chosen such that no three of them form the vertices of a right triangle? | m + n - 2 | 0.625 |
Find the value of \( n \) for which the following equality holds:
$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2014
$$ | 4060224 | 0.875 |
Calculate the volumes of bodies bounded by the surfaces.
$$
\frac{x^{2}}{27}+y^{2}=1, z=\frac{y}{\sqrt{3}}, z=0(y \geq 0)
$$ | 2 | 0.625 |
If we can always select 4 numbers from any set of $\mathrm{n}$ numbers such that their sum is a multiple of 4, what is the smallest value of $n$? | 7 | 0.25 |
A car license plate contains three letters and three digits, for example, A123BE. The allowable letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Katya considers a plate number lucky if the second letter is a consonant, the first digit is odd, and the third digit is even (there are no restrictions on the other characters). How many license plates does Katya consider lucky? | 288000 | 0.5 |
In the rhombus \(ABCD\), point \(M\) is chosen on side \(BC\). Perpendiculars are drawn from \(M\) to the diagonals \(BD\) and \(AC\), intersecting line \(AD\) at points \(P\) and \(Q\) respectively. It is known that lines \(PB\), \(QC\), and \(AM\) intersect at a single point. What is the ratio \(\frac{BM}{MC}\)? | \frac{1}{2} | 0.5 |
Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously? | 35 \text{ minutes} | 0.625 |
The secret object is a rectangle measuring $200 \times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along the outer perimeter (the intruder remained stationary). Three guards collectively ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder? | 150 \text{ meters} | 0.625 |
On the clock tower at the train station, there is an electronic clock. Along the boundary of the circular clock face, there are small colorful lights at each minute mark. At 9:35:20 PM, there are how many small colorful lights in the acute angle formed by the minute hand and the hour hand? | 12 | 0.375 |
Calculate the integral $\int_{1}^{4} x^{2} \, dx$ using Simpson's Rule. | 21 | 0.625 |
There are 10 consecutive natural numbers written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square? | 4 | 0.5 |
Vanya received three sets of candies for New Year. Each set contains three types of candies: hard candies, chocolates, and gummy candies. The total number of hard candies in all three sets is equal to the total number of chocolates in all three sets, and also to the total number of gummy candies in all three sets. In the first set, there are equal numbers of chocolates and gummy candies, and 7 more hard candies than chocolates. In the second set, there are equal numbers of hard candies and chocolates, and 15 fewer gummy candies than hard candies. How many candies are in the third set if it is known that there are no hard candies in it? | 29 | 0.875 |
Given a circle, inscribe an isosceles triangle with an acute angle. Above one of the legs of this triangle, construct another isosceles triangle whose apex is also on the circumference of the circle. Continue this process and determine the limit of the angles at the vertices of the triangles. | 60^\circ | 0.5 |
From head to tail of the zebra Hippotigris, there are 360 stripes of equal width. Fleas Masha and Dasha started crawling from the head to the tail of the zebra. At the same time, flea Sasha started crawling from the tail to the head. Flea Dasha crawls twice as fast as flea Masha. Before meeting flea Sasha, Masha covered exactly 180 stripes. How many stripes will Dasha cover before meeting flea Sasha? | 240 \text{ stripes} | 0.875 |
The microcalculator MK-97 can perform only three operations on numbers stored in memory:
1) Check if two chosen numbers are equal,
2) Add the chosen numbers,
3) For chosen numbers \( a \) and \( b \), find the roots of the equation \( x^2 + ax + b = 0 \), and if there are no roots, display a message indicating so.
All results of actions are stored in memory. Initially, one number \( x \) is stored in the memory. Using the MK-97, how can you determine if this number is equal to one? | x = 1 | 0.875 |
The distance between the non-intersecting diagonals of two adjacent lateral faces of a cube is \( d \). Determine the total surface area of the cube. | 18d^2 | 0.625 |
Compute the definite integral:
$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{d x}{(6 - \tan x) \sin 2 x}
$$ | \frac{\ln 5}{6} | 0.25 |
Let \( a_{1}, a_{2}, \cdots, a_{21} \) be a permutation of \( 1, 2, \cdots, 21 \) that satisfies
\[ \left|a_{20} - a_{21}\right| \geq \left|a_{19} - a_{21}\right| \geq \left|a_{18} - a_{21}\right| \geq \cdots \geq \left|a_{1} - a_{21}\right|. \]
Determine the number of such permutations. | 3070 | 0.5 |
Inside the cube \(A B C D A_{1} B_{1} C_{1} D_{1}\) there is a center \(O\) of a sphere with radius 10. The sphere intersects the face \(A A_{1} D_{1} D\) creating a circle with radius 1, the face \(A_{1} B_{1} C_{1} D_{1}\) creating a circle with radius 1, and the face \(C D D_{1} C_{1}\) creating a circle with radius 3. Find the length of the segment \(O D_{1}\). | 17 | 0.5 |
Circle \(\Omega\) has radius 13. Circle \(\omega\) has radius 14 and its center \(P\) lies on the boundary of circle \(\Omega\). Points \(A\) and \(B\) lie on \(\Omega\) such that chord \(AB\) has length 24 and is tangent to \(\omega\) at point \(T\). Find \(AT \cdot BT\). | 56 | 0.25 |
If you flip a fair coin 1000 times, what is the expected value of the product of the number of heads and the number of tails? | 249750 | 0.875 |
Solve the following system of equations in the set of real numbers:
$$
\begin{gathered}
\log _{3} a+\log _{3} b+\log _{3} c=0 \\
3^{3^{a}}+3^{3^{b}}+3^{3^{c}}=81
\end{gathered}
$$ | a = 1, b = 1, c = 1 | 0.125 |
Analyzing natural numbers with 4 digits:
a) How many of them have all different digits?
b) How many have the digit 1 exactly once and all other digits are different?
c) How many have the digit 1? | 3,168 | 0.5 |
The rational numbers \( x \) and \( y \), when written in lowest terms, have denominators 60 and 70, respectively. What is the smallest possible denominator of \( x + y \)? | 84 | 0.75 |
Given two distinct numbers \(a\) and \(b\) such that \(\frac{a}{b} + a = \frac{b}{a} + b\), find \(\frac{1}{a} + \frac{1}{b}\). | -1 | 0.625 |
The altitude drawn to the base of an isosceles triangle is $h$ and is twice its projection onto one of the legs. Find the area of the triangle. | h^2 \sqrt{3} | 0.25 |
How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them. | 50 | 0.625 |
Write the equation of a plane passing through point $A$ and perpendicular to the vector $\overrightarrow{B C}$.
$A(2, 1, 7)$
$B(9, 0, 2)$
$C(9, 2, 3)$ | 2y + z - 9 = 0 | 0.875 |
On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either "You are a knight!" or "You are a liar!". The phrase "You are a liar!" was said exactly 230 times. How many times was the phrase "You are a knight!" said? | 526 | 0.75 |
Let \( x \) and \( y \) be two non-zero numbers such that \( x^2 + xy + y^2 = 0 \) (where \( x \) and \( y \) are complex numbers, but that is not too important). Find the value of
\[ \left( \frac{x}{x+y} \right)^{2013} + \left( \frac{y}{x+y} \right)^{2013} \] | -2 | 0.75 |
Each of the 12 knights sitting around a round table has chosen a number, and all the numbers are different. Each knight claims that the number they have chosen is greater than the numbers chosen by their neighbors on the right and left. What is the maximum number of these claims that can be true? | 6 | 0.75 |
Determine the area of a triangle given its three sides:
1) \( a=13, b=14, c=15 \)
2) \( a=5, b=12, c=13 \) | 30 | 0.875 |
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