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As shown in Figure 1, quadrilateral $ACDB$ and parallelogram $AEDF$ satisfy:
$$
\begin{array}{l}
6 \overrightarrow{A E}+3 \overrightarrow{D E}=2 \overrightarrow{A C}, \\
4 \overrightarrow{F B}+5 \overrightarrow{F D}=2 \overrightarrow{A D}.
\end{array}
$$
Find the ratio of the area of quadrilateral $ACDB$ to the area of parallelogram $AEDF$. | \frac{27}{8} | 0.25 |
Compute the number of positive integers that divide at least two of the integers in the set \(\left\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\right\}\). | 22 | 0.25 |
In the cryptarithm
$$
\mathbf{K}<\mathrm{O}<\mathbf{P}>\mathbf{O}>\mathbf{H}>\mathbf{A}>\mathbf{B}>\mathbf{U}>\mathbf{P}>\mathbf{y}>\mathbf{C}
$$
different letters represent different digits. How many solutions does the cryptarithm have? | 0 | 0.25 |
As shown in the figure, three equally sized square sheets of paper in red, yellow, and blue are stacked in a large square box. Given that the exposed areas are 25 for the blue sheet, 19 for the red sheet, and 11 for the yellow sheet, what is the area of the large square box? | 64 | 0.375 |
There is a unique triplet of positive integers \((a, b, c)\) such that \(a \leq b \leq c\) and
$$
\frac{25}{84}=\frac{1}{a}+\frac{1}{a b}+\frac{1}{a b c}.
$$
Determine \(a + b + c\). | 17 | 0.75 |
Find the number of pairs of integers \((x, y)\) that satisfy the equation \(5x^2 - 6xy + y^2 = 6^{100}\). | 19594 | 0.25 |
Suppose \(A, B\) are the foci of a hyperbola and \(C\) is a point on the hyperbola. Given that the three sides of \(\triangle ABC\) form an arithmetic sequence, and \(\angle ACB = 120^\circ\), determine the eccentricity of the hyperbola. | \frac{7}{2} | 0.625 |
Péter took one stamp each of values 1, 2, 3, ..., 37 forints from his stamp collection. He wants to group them so that each group has the same total value of stamps. In how many ways can he do this? | 2 | 0.125 |
Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the river. | 1700 \text{ feet} | 0.25 |
$A B C D E F$ is a regular hexagon, and point \( O \) is its center. How many different isosceles triangles with vertices among these seven points can be formed? Triangles that differ only in the order of the vertices are considered the same (for example, \( AOB \) and \( BOA \)).
The answer is 20. | 20 | 0.875 |
Masha thought of a natural number and found its remainders when divided by 3, 6, and 9. The sum of these remainders turned out to be 15.
Find the remainder when the thought number is divided by 18. | 17 | 0.875 |
Find the number of ordered pairs of positive integers \((x, y)\) with \(x, y \leq 2020\) such that \(3x^2 + 10xy + 3y^2\) is the power of some prime. | 29 | 0.25 |
We call a set of positive integers "tyû-de-jó" if there are no two numbers in the set whose difference is 2. How many "tyû-de-jó" subsets are there of the set $\{1,2,3, \ldots, 10\}$? | 169 | 0.25 |
Find the maximum value of the expression \(a + b + c + d - ab - bc - cd - da\), given that each of the numbers \(a, b, c\), and \(d\) belongs to the interval \([0, 1]\). | 2 | 0.75 |
Find all integers \(a, y \geq 1\) such that \(3^{2a-1} + 3^a + 1 = 7^y\). | (a,y) = (1,1) | 0.875 |
The cube $ABCDEFGH$ standing on a table has its three vertical edges divided by points $K, L$, and $M$ in the ratios $1:2$, $1:3$, and $1:4$, respectively, as shown in the figure. The plane $KLM$ cuts the cube into two parts. What is the ratio of the volumes of the two parts? | \frac{4}{11} | 0.25 |
Five boys and three girls sit around a circular table. If there are no additional requirements, there are $Q(8,8)=7!=5040$ arrangements. How many arrangements are there if the boy $B_{1}$ does not sit next to the girl $G_{1}$? How many arrangements are there if the three girls do not sit next to each other? | 1440 | 0.125 |
Triangle $ABC$ is an isosceles triangle with $BC$ as its base. Let $F$ be the midpoint of $BC$. Point $D$ is an internal point on segment $BF$. The perpendicular to $BC$ at point $D$ intersects side $AB$ at $M$, and a line passing through point $D$ parallel to $AB$ intersects side $AC$ at $P$. Express the ratio of the areas of triangles $AMP$ and $ABC$ as a function of $k = BD : BC$. | k - 2k^2 | 0.375 |
The eleven-digit number ' \(A 123456789 B\) ' is divisible by exactly eight of the numbers \(1, 2, 3, 4, 5, 6, 7, 8, 9\). Find the values of \(A\) and \(B\), explaining why they must have these values. | A = 3, B = 6 | 0.75 |
On an island of knights and liars, knights always tell the truth, and liars always lie. In a school on this island, both knights and liars study in the same class. One day the teacher asked four children: Anu, Banu, Vanu, and Danu, who among them did the homework. They responded:
- Anu: Banu, Vanu, and Danu did the homework.
- Banu: Anu, Vanu, and Danu did not do the homework.
- Vanu: Do not believe them, teacher! Anu and Banu are liars!
- Danu: No, teacher, Anu, Banu, and Vanu are knights!
How many knights are among these children? | 1 | 0.875 |
Given triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) intersect at point \( K \), find the value of \( \frac{|KO|}{|KH|} \). | \frac{3}{2} | 0.75 |
Four squares with integer side lengths are arranged as shown in the diagram, with three vertices of square \( A \) being the centers of squares \( B \), \( C \), and \( D \). If the total area of the red parts is equal to the area of the green part, what is the minimum side length of square \( A \)? | 3 | 0.25 |
How many sequences of ten binary digits are there in which neither two zeroes nor three ones ever appear in a row? | 28 | 0.25 |
Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch? | \frac{1}{2} | 0.25 |
Given the quadratic function \( y = ax^2 + bx + c \) with its graph intersecting the \( x \)-axis at points \( A \) and \( B \), and its vertex at point \( C \):
(1) If \( \triangle ABC \) is a right-angled triangle, find the value of \( b^2 - 4ac \).
(2) Consider the quadratic function
\[ y = x^2 - (2m + 2)x + m^2 + 5m + 3 \]
with its graph intersecting the \( x \)-axis at points \( E \) and \( F \), and it intersects the linear function \( y = 3x - 1 \) at two points, with the point having the smaller \( y \)-coordinate denoted as point \( G \).
(i) Express the coordinates of point \( G \) in terms of \( m \).
(ii) If \( \triangle EFG \) is a right-angled triangle, find the value of \( m \). | -1 | 0.375 |
In which numeral system is the number \( 11111_{d} \) a perfect square? | 3 | 0.375 |
For which values of the parameter \( a \) does the equation
\[
3^{x^{2}-2ax+a^{2}} = ax^{2}-2a^{2}x+a^{3}+a^{2}-4a+4
\]
have exactly one solution? | 1 | 0.625 |
Given the Fibonacci sequence defined as follows: \( F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n} \) (n ≥ 1), find \( \left(F_{2017}, F_{99}F_{101}+1\right) \). | 1 | 0.625 |
In the acute triangle \( \triangle ABC \), \( \sin A = \frac{4 \sqrt{5}}{9} \), and \( a, b, c \) represent the sides opposite to angles \( A, B, \) and \( C \), respectively.
1. Find the value of \( \sin 2(B + C) + \sin^{2} \frac{B+C}{2} \).
2. If \( a = 4 \), find the area \( S_{\triangle ABC} \) of the triangle when \( \overrightarrow{AB} \cdot \overrightarrow{AC} \) reaches its maximum value. | 2\sqrt{5} | 0.75 |
Calculate \(\left\lfloor \sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+\sqrt[2009]{\frac{2009}{2008}} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\). | 2008 | 0.75 |
Using each of the digits 1-9 exactly once, form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them? | 1369 | 0.5 |
Given three quadratic polynomials \( f(x)=a x^{2}+b x+c \), \( g(x)=b x^{2}+c x+a \), and \( h(x)=c x^{2}+a x+b \), where \( a, b, c \) are distinct non-zero real numbers. From these, three equations \( f(x)=g(x) \), \( f(x)=h(x) \), and \( g(x)=h(x) \) are formed. Find the product of all roots of these three equations, assuming each equation has two distinct roots. | 1 | 0.625 |
Let \( a_{n} \) represent the closest positive integer to \( \sqrt{n} \) for \( n \in \mathbf{N}^{*} \). Suppose \( S=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{2000}} \). Determine the value of \( [S] \). | 88 | 0.375 |
Solve:
$$
x^{3}+2 y^{3}=4 z^{3},
$$
where $x$, $y$, and $z$ are integers. | (0,0,0) | 0.625 |
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 value of the length of segment \( MX \)? | 6.5 | 0.375 |
Consider a product \(a_{1} a_{2} \ldots a_{100}\) written on a board, where \(a_{1}, \ldots, a_{100}\) are natural numbers. We consider 99 expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\) that could exist? | 33 | 0.625 |
Find the volume of a parallelepiped where two faces are rhombuses with side length 1 and an acute angle of $60^{\circ}$, and the other faces are squares. | \frac{\sqrt{3}}{2} | 0.375 |
What is the maximum number of colors needed to paint all the cells of a 4 by 4 square so that for each pair of different colors, there exist two cells of these colors located either in the same row or in the same column of the square? | 8 | 0.5 |
Given natural numbers \( m \) and \( n \) where \( n > m > 1 \), the last three digits of the decimal representation of \( 1978^m \) and \( 1978^n \) are the same. Find \( m \) and \( n \) such that \( m+n \) is minimized. | 106 | 0.375 |
A horizontal plane can intersect all 6 lateral edges of a prism. If we tilt this plane in such a way that it intersects the upper base near one of the vertices, it will intersect two edges of the upper base while no longer intersecting one of the lateral edges. This increases the number of intersected edges by 1. Similarly, we can increase this number by 1 at the lower base. Thus, we obtain a plane intersecting 8 edges of the prism.
Why can't we achieve more than 8 intersections? | 8 | 0.75 |
Point \( O \) is the intersection point of the altitudes of an acute triangle \( ABC \). Find \( OC \), given that \( AB = 4 \) and \( \sin \angle C = \frac{5}{13} \). | 9.6 | 0.75 |
The products by four and by five of an integer, when considered together, use each digit from 1 to 9 exactly once. What is this number? | 2469 | 0.375 |
In an isosceles triangle \(ABC\), side \(AC = b\), sides \(BA = BC = a\), \(AM\) and \(CN\) are the angle bisectors of angles \(A\) and \(C\). Find \(MN\). | \frac{ab}{a + b} | 0.625 |
a) Two seventh-grade students participated in a chess tournament along with some eighth-grade students. The two seventh-graders scored a total of 8 points, and each eighth-grader scored the same number of points. How many eighth-graders participated in the tournament? Find all possible solutions.
b) Ninth and tenth-grade students participated in a chess tournament. There were 10 times as many tenth-graders as ninth-graders, and they scored 4.5 times more points in total than all the ninth-graders. How many points did the ninth-graders score? | 10 | 0.25 |
a) Given a piece of wire 120 cm long, is it possible to construct a frame of a cube with an edge length of 10 cm without breaking the wire?
b) What is the minimum number of times the wire needs to be broken in order to construct the required frame? | 3 | 0.25 |
Find the volume of an oblique triangular prism with a base that is an equilateral triangle with side length $a$, given that the lateral edge of the prism is equal to the side of the base and is inclined at an angle of $60^{\circ}$ to the plane of the base. | \frac{3 a^3}{8} | 0.5 |
Let $P$ and $Q$ be two monic polynomials satisfying:
$$
P(P(X)) = Q(Q(X))
$$
Show that $P = Q$. | P = Q | 0.75 |
Given the sequence
$$
a_{n}=\frac{(n+3)^{2}+3}{n(n+1)(n+2)} \cdot \frac{1}{2^{n+1}}
$$
defined by its general term, form the sequence
$$
b_{n}=\sum_{k=1}^{n} a_{k}
$$
Determine the limit of the sequence $b_{n}$ as $n$ approaches $+\infty$. | 1 | 0.75 |
Given a sequence of positive integers $\left\{a_{n}\right\}$ such that:
$$
a_{0}=m, \quad a_{n+1}=a_{n}^{5}+487 \quad (n \in \mathbf{N}),
$$
find the value of the positive integer $m$ such that the sequence $\left\{a_{n}\right\}$ contains the maximum number of perfect squares. | 9 | 0.5 |
For any real number \( x \), the inequality
$$
|x + a| - |x + 1| \leq 2a
$$
holds. What is the minimum value of the real number \( a \)? | \frac{1}{3} | 0.875 |
There are 20 chairs in a room of two colors: blue and red. Seated on each chair is either a knight or a liar. Knights always tell the truth, and liars always lie. Each of the seated individuals initially declared that they were sitting on a blue chair. After that, they somehow changed seats, and now half of the seated individuals claim to be sitting on blue chairs while the other half claim to be sitting on red chairs. How many knights are now sitting on red chairs? | 5 | 0.75 |
Masha's school lessons usually end at 13:00, and her mother picks her up by car to go home. One day, lessons ended at 12:00, and Masha decided to walk home. On her way, she met her mother, who as usual was on her way to pick Masha up at 13:00. They then continued their journey home by car and arrived 12 minutes earlier than usual. At what time did Masha meet her mother on the road? (Assume that both Masha and her mother travel at constant speeds and that no time is spent for Masha to get into the car.) | 12:54 | 0.5 |
The range of the function
$$
f(x)=\frac{\sin x-1}{\sqrt{3-2 \cos x-2 \sin x}} \quad (0 \leqslant x \leqslant 2 \pi)
$$
is . | [-1, 0] | 0.75 |
Consider two regular triangular pyramids \( P-ABC \) and \( Q-ABC \) with the same base inscribed in the same sphere. If the angle between the lateral face and the base of the pyramid \( P-ABC \) is \( 45^\circ \), find the tangent of the angle between the lateral face and the base of the pyramid \( Q-ABC \). | 4 | 0.125 |
A math extracurricular activity group at a certain school designed a tree planting plan on graph paper for a desert as follows: The $k$-th tree is planted at point $P_{k}(x_{k}, y_{k})$, where $x_{1}=1$ and $y_{1}=1$. For $k \geq 2$, the coordinates are determined by:
\[
\begin{cases}
x_{k} = x_{k-1} + 1 - 5 \left[\frac{k-1}{5}\right] + 5 \left[\frac{k-2}{5}\right], \\
y_{k} = y_{k-1} + \left[\frac{k-1}{5}\right] - \left[\frac{k-2}{5}\right],
\end{cases}
\]
where $[a]$ denotes the integer part of the real number $a$, for example, $[2.6] = 2$ and $[0.6] = 0$. According to this plan, the coordinates of the 2008th tree are _____ | (3, 402) | 0.625 |
Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$. | 3\sqrt{2} | 0.25 |
Let \(a, b, c\) be positive integers such that \(ab + bc - ca = 0\) and \(a - c = 101\). Find \(b\). | 2550 | 0.625 |
There are 50 lines on a plane, 20 of which are parallel to each other. The maximum number of regions into which these 50 lines can divide the plane is ________. | 1086 | 0.5 |
In the Karabas Barabas theater, a chess tournament took place among the actors. Each participant played exactly one match against each of the other participants. A win awarded one solido, a draw awarded half a solido, and a loss awarded nothing. It turned out that among every three participants, there was one chess player who earned exactly 1.5 solidos in the games against the other two. What is the maximum number of actors that could have participated in such a tournament? | 5 | 0.125 |
What is the greatest number of natural numbers, not exceeding 2016, that can be selected such that the product of any two selected numbers is a perfect square? | 44 | 0.25 |
Given the numbers \(\log _{\sqrt{x+34}}(2x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2x+23}}(-x-4)\). For which values of \(x\) are two of these numbers equal and the third one greater by 1? | -9 | 0.375 |
In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$. | \frac{\sqrt{3}}{3} | 0.75 |
Petya writes different three-digit natural numbers on the board, each of which is divisible by 3, and the first two digits differ by 2. What is the maximum number of such numbers he can write down if they end in 6 or 7? | 9 | 0.375 |
What is the smallest number of digits that need to be appended to the right of the number 2014 so that the resulting number is divisible by all natural numbers less than 10? | 4 | 0.25 |
Among all the proper fractions where both the numerator and denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{4}{5}$. In your answer, provide its numerator. | 77 | 0.5 |
A line in the plane is called strange if it passes through \((a, 0)\) and \((0, 10-a)\) for some \(a\) in the interval \([0,10]\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points? | \frac{50}{3} | 0.875 |
Let $O$ be the center of the circumscribed circle around the triangle $ABC$. Find the angle $OAC$ if: a) $\angle B=50^{\circ}$; b) $\angle B=126^{\circ}$. | 36^\circ | 0.5 |
Given the circumcenter $O$ of $\triangle ABC$, and $2 \overrightarrow{OA} + 3 \overrightarrow{OB} + 4 \overrightarrow{OC} = \mathbf{0}$, find the value of $\cos \angle BAC$. | \frac{1}{4} | 0.5 |
The natural numbers \(a\) and \(b\) are such that \(5 \, \text{LCM}(a, b) + 2 \, \text{GCD}(a, b) = 120\). Find the largest possible value of \(a\). | 20 | 0.5 |
2000 people registered on a new website. Each person invited 1000 others to be friends. Two people are considered friends if and only if both have invited each other to be friends. What is the minimum number of friend pairs that could have been formed? | 1000 | 0.125 |
Let \( ABC \) be a triangle with orthocenter \( H \), and let \( G \) be the point such that the quadrilateral \( ABGH \) is a parallelogram. Let \( I \) be the point on the line \( GH \) such that the line \( AC \) intersects the segment \( HI \) at its midpoint. The line \( AC \) intersects the circumcircle of triangle \( GCI \) at point \( J \). Show that \( IJ = AH \). | IJ = AH | 0.75 |
The given quadratic polynomial \( f(x) \) has two distinct roots. Can it be the case that the equation \( f(f(x)) = 0 \) has three distinct roots, while the equation \( f(f(f(x))) = 0 \) has seven distinct roots? | \text{No} | 0.5 |
Two spheres touch the plane of triangle \( A B C \) at points \( A \) and \( B \) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \( \sqrt{481} \). The center of a third sphere with radius 9 is located at point \( C \), and this sphere touches each of the first two spheres externally. Find the radius of the circumcircle of triangle \( A B C \). | 3 \sqrt{10} | 0.125 |
Real numbers \(a, b, c\) and positive number \(\lambda\) are such that \(f(x) = x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\), and satisfy:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\). | \frac{3\sqrt{3}}{2} | 0.5 |
Given a cyclic quadrilateral \(ABCD\), it is known that \(\angle ADB = 48^{\circ}\) and \(\angle BDC = 56^{\circ}\). Inside the triangle \(ABC\), a point \(X\) is marked such that \(\angle BCX = 24^{\circ}\), and the ray \(AX\) is the angle bisector of \(\angle BAC\). Find the angle \(CBX\). | 38^\circ | 0.375 |
In triangle \(ABC\), \(AB = 28\), \(BC = 21\), and \(CA = 14\). Points \(D\) and \(E\) are on \(AB\) with \(AD = 7\) and \(\angle ACD = \angle BCE\). Find the length of \(BE\). | 12 | 0.5 |
Solve the equation for integer values: \(\underbrace{\sqrt{n+\sqrt{n+\ldots \sqrt{n}}}}_{\text{1964 times}} = m\) | 0 | 0.125 |
The magician taught Kashtanka to bark the number of times he secretly showed her. When Kashtanka correctly answered how much two times two is, he hid a delicious cake in a suitcase with a combination lock and said:
- The eight-digit code for the suitcase is the solution to the puzzle УЧУЙ $=\text{KE} \times \text{KS}$. You need to replace the same letters with the same digits and different letters with different digits so that the equality is true. Bark the correct number of times for each of the eight characters, and you will get the treat.
But then there was a confusion. Kashtanka, out of excitement, barked 1 more time for each letter than she should have. Of course, the suitcase did not open. Suddenly, a child's voice said: "It's not fair! The dog solved the puzzle correctly!" And indeed, if each digit of the solution the magician had in mind is increased by 1, another solution to the puzzle emerges!
Is it possible to determine: a) what exact solution the magician had in mind; b) what the value of the number УЧУЙ was in that solution? | \text{УЧУЙ} = 2021 | 0.625 |
The quadrilateral \(P Q R S\) is inscribed in a circle. Diagonals \(P R\) and \(Q S\) are perpendicular and intersect at point \(M\). It is known that \(P S = 13\), \(Q M = 10\), and \(Q R = 26\). Find the area of the quadrilateral \(P Q R S\). | 319 | 0.75 |
A palace is shaped like a square and divided into $2003 \times 2003$ rooms, similar to the squares on a large chessboard. There is a door between two rooms if and only if they share a wall. The main entrance allows entry into the palace from the outside into the room located at the northwest corner. A person enters the palace, visits some rooms, and then exits the palace through the entrance, returning to the northwest corner for the first time. It turns out this person visited each of the other rooms exactly 100 times, except for the room located at the southeast corner. How many times has the visitor been in the southeast corner room? | 99 | 0.625 |
From a point \( M \), located inside triangle \( ABC \), perpendiculars are drawn to sides \( BC \), \( AC \), and \( AB \), with lengths \( k \), \( l \), and \( m \) respectively. Find the area of triangle \( ABC \), given that \(\angle CAB = \alpha\) and \(\angle ABC = \beta\). If the answer is not an integer, round it to the nearest whole number.
$$
\alpha = \frac{\pi}{6}, \beta = \frac{\pi}{4}, k = 3, l = 2, m = 4
$$ | 67 | 0.375 |
Given \( |\boldsymbol{a}| = 1 \), \( |\boldsymbol{b}| = |\boldsymbol{c}| = 2 \), \( \boldsymbol{b} \cdot \boldsymbol{c} = 0 \), \( \lambda \in (0, 1) \), find the minimum value of
\[
|a - b + \lambda(b - c)| + \left| \frac{1}{2}c + (1 - \lambda)(b - c) \right|
\] | \sqrt{5} - 1 | 0.25 |
For the quadrilateral $ABCD$, it is known that $AB = BD$, $\angle ABD = \angle DBC$, and $\angle BCD = 90^\circ$. On segment $BC$, point $E$ is marked such that $AD = DE$. Find the length of segment $BD$ if $BE = 7$ and $EC = 5$. | 17 | 0.5 |
A sphere of radius 4 with center at point $Q$ touches three parallel lines at points $F$, $G$, and $H$. It is known that the area of triangle $QGH$ is $4 \sqrt{2}$, and the area of triangle $FGH$ is greater than 16. Find the angle $GFH$. | 67.5^\circ | 0.25 |
Given a triangle $ABC$ with $O$ as the incenter. Find the angle $A$ if the circumradii of triangles $ABC$ and $BOC$ are equal. | 60^\circ | 0.875 |
An archipelago consisting of an infinite number of islands is spread out along the southern shore of an endless sea in a chain. The islands are connected by an infinite chain of bridges, and each island is also connected to the shore by a bridge. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that, after a strong earthquake, it will be possible to travel from the first island to the shore using the remaining bridges? | \frac{2}{3} | 0.625 |
Given \( n \) squares on a plane, where:
1. All the squares are congruent.
2. If two squares share a common intersection point \( P \), then \( P \) must be a vertex of each of the squares.
3. Any square intersects exactly three other squares.
The collection of these \( n \) squares is called "three-connected."
Determine the number of integers \( n \) within the range \( 2018 \leq n \leq 3018 \) for which there exists a three-connected set of \( n \) squares. | 501 | 0.875 |
Select four vertices of a cube such that no two of them are on the same edge. A point inside the cube has distances of \(\sqrt{50}\), \(\sqrt{70}\), \(\sqrt{90}\), and \(\sqrt{110}\) from these four vertices. What is the edge length of the cube? | 10 \text{ units} | 0.875 |
Let \( AH_a \) and \( BH_b \) be the altitudes, and \( AL_a \) and \( BL_b \) be the angle bisectors of triangle \( ABC \). It is known that \( H_aH_b \parallel L_aL_b \). Is it true that \( AC = BC \)? | AC = BC | 0.125 |
A coin is tossed 10 times. Find the probability that two heads never appear consecutively. | \frac{9}{64} | 0.875 |
Let \( z \) be a complex number with a modulus of 1. Then the maximum value of \(\left|\frac{z+\mathrm{i}}{z+2}\right|\) is \(\ \ \ \ \ \ \). | \frac{2\sqrt{5}}{3} | 0.75 |
(1) For which positive integers \( n \) does there exist a finite set of \( n \) distinct positive integers \( S_n \) such that the geometric mean of any subset of \( S_n \) is an integer?
(2) Is there an infinite set of distinct positive integers \( S \) such that the geometric mean of any finite subset of \( S \) is an integer? | \text{No} | 0.125 |
To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. The four digits get reordered (but not flipped or rotated) on his plate, and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed? | 1092 | 0.5 |
In the pyramid \(ABCD\), the dihedral angle at edge \(AC\) is \(90^{\circ}\), \(AB = BC = CD\), and \(BD = AC\). Find the dihedral angle at edge \(AD\). | 60^\circ | 0.5 |
In triangle \(ABC\), the bisector \(BD\) is drawn, and in triangles \(ABD\) and \(CBD\), the bisectors \(DE\) and \(DF\) are drawn, respectively. It turns out that \(EF \parallel AC\). Find the angle \(\angle DEF\). | 45 \text{ degrees} | 0.625 |
In the addition below, identical letters represent the same digit and different letters represent different digits. Find the number $ABCDE$.
\[ \begin{array}{cccccc}
& & & & A & B & C & D & E \\
+ & & & & B & C & D & E \\
+ & & & & & C & D & E \\
+ & & & & & & D & E \\
+ & & & & & & & E \\
\hline
& & A & A & A & A & A \\
\end{array} \] | 52487 | 0.625 |
If the complex number \( z \) satisfies \( |z+\mathrm{i}| + |z-2-\mathrm{i}| = 2 \sqrt{2} \), then the minimum value of \( |z| \) is ______. | \frac{\sqrt{2}}{2} | 0.875 |
Find all pairs of integers \( n, m \) such that \( 7^m = 5^n + 24 \). | (2, 2) | 0.25 |
The diagram shows a square with sides of length \(4 \text{ cm}\). Four identical semicircles are drawn with their centers at the midpoints of the square’s sides. Each semicircle touches two other semicircles. What is the shaded area, in \(\text{cm}^2\)?
A) \(8 - \pi\)
B) \(\pi\)
C) \(\pi - 2\)
D) \(\pi - \sqrt{2}\)
E) \(8 - 2\pi\) | 8 - 2\pi | 0.25 |
Given sets of real numbers \( A, B \), define \( A \otimes B = \{ x \mid x = ab + a + b, \, a \in A, b \in B \} \). Let \( A = \{0, 2, 4, \cdots, 18\} \) and \( B = \{98, 99, 100\} \). Find the sum of all elements in \( A \otimes B \). | 29970 | 0.125 |
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