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57,391 | 5. A square with a side of 12 needs to be completely cut into four squares with an integer side $a$, three squares with an integer side $b$, and ten rectangles with sides $a$ and $b$. Find all values of $a$ and $b$ for which this is possible. | =2,b=4 |
60,360 | 8. On the sides $B C, C A, A B$ of an equilateral triangle $A B C$ with side length 7, points $A_{1}, B_{1}, C_{1}$ are taken respectively. It is known that $A C_{1}=B A_{1}=C B_{1}=3$. Find the ratio of the area of triangle $A B C$ to the area of the triangle formed by the lines $A A_{1}, B B_{1}, C C_{1}$. | 37 |
61,780 | Example 2. Solve the equation $x e^{y} d x+\left(y+y x^{2}\right) d y=0$. | \ln\sqrt{1+x^{2}}-(y+1)e^{-y}=C |
57,268 | 1. (13 points) What is the minimum number of participants that could have been in the school drama club if the number of fifth graders was more than $25 \%$ but less than $35 \%$, the number of sixth graders was more than $30 \%$ but less than $40 \%$, and the number of seventh graders was more than $35 \%$ but less than $45 \%$ (there were no participants from other grades). | 11 |
57,407 | 7-5. In a row, there are 1000 toy bears. The bears can be of three colors: white, brown, and black. Among any three consecutive bears, there is a toy of each color. Iskander is trying to guess the colors of the bears. He made five guesses:
- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
- The 400th bear from the left is black;
- The 600th bear from the left is brown;
- The 800th bear from the left is white.
It turned out that exactly one of his guesses is incorrect. What could be the number of the bear whose color Iskander did NOT guess correctly? Select all possible answers. | 20 |
68,649 | 21 Find the integer part of
$$
\frac{1}{\frac{1}{2003}+\frac{1}{2004}+\frac{1}{2005}+\frac{1}{2006}+\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}} .
$$ | 286 |
24,965 | In how many ways can you rearrange the letters of "HMMTHMMT" such that the consecutive substring "HMMT" does not appear? | 361 |
27,025 | On the calculator keyboard, there are digits from 0 to 9 and signs of two operations (see the figure). Initially, the display shows the number 0. You can press any keys. The calculator performs operations in the sequence of key presses. If the operation sign is pressed several times in a row, the calculator will only remember the last press. The Absent-Minded Scientist pressed a lot of buttons in a random sequence. Find the approximate probability that the result of the resulting chain of operations is an odd number?
# | \frac{1}{3} |
58,628 | We erect a straight cone over the base of a hemisphere with radius $r$, the height of which is 2r. Calculate the volume of the common solid formed by the two bodies. | \frac{14}{25}r^{3}\pi |
68,300 | B1. The digit sum of a number is obtained by adding the digits of the number. The digit sum of 1303, for example, is $1+3+0+3=7$.
Find the smallest positive integer $n$ for which the digit sum of $n$ and the digit sum of $n+1$ are both divisible by 5. | 49999 |
22,659 | Let \( x, y, z, w \) be four consecutive vertices of a regular \( A \)-gon. If the length of the line segment \( xy \) is 2 and the area of the quadrilateral \( xyzw \) is \( a + \sqrt{b} \), find the value of \( B = 2^a \cdot 3^b \). | 108 |
17,911 | 1. Calculate the value of the expression
$$
S=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}
$$
where $a_{1}>0, a_{2}>0, \ldots, a_{n}>0$ and $a_{1}, a_{2}, \ldots, a_{n}$ form an arithmetic progression. | \dfrac{n-1}{\sqrt{a_1} + \sqrt{a_n}} |
67,652 | 3. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=B C=4, A A_{1}=2, P$ is a point on the plane $A_{1} B C$, and $\overrightarrow{D P} \cdot \overrightarrow{P B}=0$. Then, the area of the plane region enclosed by all points $P$ that satisfy the above conditions is $\qquad$. | \frac{36\pi}{5} |
23,921 | A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that its length is 75 meters. | 50 |
51,706 | The base of the pyramid SABCD is a parallelogram ABCD. What figure is obtained by the section of this pyramid by the plane $ABM$, where $M$ is a point on the edge SC? | Trapezoid |
66,204 | 10. 78 students stand in a row. Starting from the left, give a badminton shuttlecock to the first person, and then give a badminton shuttlecock every 2 people. Starting from the right, give a table tennis ball to the first person, and then give a table tennis ball every 3 people. The number of students who have both a badminton shuttlecock and a table tennis ball is $\qquad$ people. | 6 |
16,877 | 69. Faulty Locomotive. We set off by railway from Angchester to Clinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\frac{3}{5}$ of the original. As a result, we arrived in Clinkerton 2 hours late, and the driver said that if the malfunction had occurred 50 miles further, the train would have arrived 40 minutes earlier.
What is the distance from Angchester to Clinkerton? | 200 |
57,608 | 2. For an integer $x$, the following holds
$$
|\ldots||| x-1|-10|-10^{2}\left|-\ldots-10^{2006}\right|=10^{2007}
$$
Find the hundredth digit of the number $|x|$. | 1 |
12,703 |
The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minute hand at moment $Y$ will be where the hour hand is at moment $X$. How many interesting moments will there be from 00:01 to 12:01? | 143 |
62,517 | 8. Given $a, b, c \in \mathbf{R}_{+}, a^{2}+b^{2}+c^{2}=1$. Let
$$
M=\max \left\{a+\frac{1}{b}, b+\frac{1}{c}, c+\frac{1}{a}\right\},
$$
then the minimum value of $M$ is . $\qquad$ | \frac{4\sqrt{3}}{3} |
7,829 | The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\triangle A P F$. | 3\sqrt{2} + \sqrt{10} |
14,891 | In the parallelogram \(ABCD\), \(AC = AB \sqrt{2}\). Prove that the angles between the diagonals are equal to the angles between the sides. | \text{The angles between the diagonals are equal to the angles between the sides.} |
55,972 | Example 5. Find the flux of the vector field $\mathbf{a} = \boldsymbol{y}^{2} \mathbf{j} + z \mathbf{k}$ through the part of the surface $z = x^{2} + y^{2}$, cut off by the plane $z = 2$. The normal is taken outward with respect to the region bounded by the paraboloid.
 | -2\pi |
59,386 | 9.9. On a plane, $N$ points are marked. Any three of them form a triangle, the angles of which in degrees are expressed by natural numbers. For what largest $N$ is this possible
$$
\text { (E. Bakayev) }
$$ | 180 |
57,345 | 8.1. Find the area of the figure defined on the coordinate plane by the inequality $2(2-x) \geq\left|y-x^{2}\right|+\left|y+x^{2}\right|$. | 15 |
59,823 | 1. In the equation $\overline{x 5} \cdot \overline{3 y} \bar{z}=7850$, restore the digits $x, y, z$ | x=2, y=1, z=4 |
23,861 | A positive integer \( n \) is said to be good if \( 3n \) is a re-ordering of the digits of \( n \) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11. | 2475 |
58,342 | 2. Choose any three vertices from a regular nonagon to form a triangle, then the probability that the center of the nonagon is inside the triangle is | \frac{5}{14} |
5,010 | Let $f$ be a function defined on and taking values in the set of non-negative real numbers. Find all functions $f$ that satisfy the following conditions:
(i) $f[x \cdot f(y)] \cdot f(y)=f(x+y)$;
(ii) $f(2)=0$;
(iii) $f(x) \neq 0$, when $0 \leq x < 2$. | f(x) = \begin{cases} \dfrac{2}{2 - x} & \text{if } 0 \leq x < 2, \\ 0 & \text{if } x \geq 2. \end{cases} |
64,191 | 2. The chord $AB$ divides a circle with radius $r$ into two arcs in the ratio $1:2$. In the larger segment, a square is inscribed such that one of its sides lies on this chord. Express the length of the side of the square in terms of the radius $r$. | \frac{2+\sqrt{19}}{5}r |
9,730 | Given triangle \( ABC \) with \( AB = 12 \), \( BC = 10 \), and \( \angle ABC = 120^\circ \), find \( R^2 \), where \( R \) is the radius of the smallest circle that can contain this triangle. | 91 |
9,740 | Two circles meet at A and B and touch a common tangent at C and D. Show that triangles ABC and ABD have the same area. | \text{Triangles } ABC \text{ and } ABD \text{ have the same area.} |
60,117 | 3. Let $A B C D$ be a quadrilateral for which $\angle B A C=\angle A C B=20^{\circ}, \angle D C A=30^{\circ}$ and $\angle C A D=40^{\circ}$. Determine the measure of the angle $\angle C B D$. | 80 |
6,558 | A necklace consists of 100 red beads and some number of blue beads. It is known that on any segment of the necklace containing 10 red beads, there are at least 7 blue beads. What is the minimum number of blue beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last bead is adjacent to the first one.) | 78 |
68,663 | 4. The given figure is an isosceles triangle $ABC$ with base $\overline{AB}$ of length $65 \, \text{cm}$ and legs $\overline{AC}$ and $\overline{BC}$ of length $80 \, \text{cm}$. On the base $\overline{AB}$, a point $D$ is chosen such that the perimeter of triangle $ADC$ is $173 \, \text{cm}$, and the perimeter of triangle $DBC$ is $220 \, \text{cm}$. What are the lengths of segments $\overline{CD}, \overline{AD}$, and $\overline{DB}$? | 84 |
30,273 | Through the midpoint of the hypotenuse of a right triangle, a perpendicular is drawn to it. The segment of this perpendicular, enclosed within the triangle, is equal to c, and the segment enclosed between one leg and the extension of the other is equal to 3c. Find the hypotenuse.
# | 4c |
60,853 | Example 3 If real numbers $x, y$ satisfy
$$
\begin{array}{l}
\frac{x}{3^{3}+4^{3}}+\frac{y}{3^{3}+6^{3}}=1, \\
\frac{x}{5^{3}+4^{3}}+\frac{y}{5^{3}+6^{3}}=1,
\end{array}
$$
then $x+y=$ $\qquad$
(2005, National Junior High School Mathematics Competition) | 432 |
25,465 | Problem 2. The numbers $d$ and $e$ are the roots of the quadratic trinomial $a x^{2}+b x+c$. Could it be that $a, b, c, d, e$ are consecutive integers in some order? | Yes |
12,164 | Given that \( S \) is a set consisting of \( n \) (\( n \geq 3 \)) positive numbers. If three different elements in \( S \) can form the sides of a triangle, then \( S \) is called a "triangular set." Consider the set of consecutive positive integers \(\{4,5, \cdots, m\}\). All of its 10-element subsets are triangular sets. What is the maximum possible value of \( m \)? | 253 |
24,488 | For any set \( S \), let \( |S| \) represent the number of elements in set \( S \) and let \( n(S) \) represent the number of subsets of set \( S \). If \( A \), \( B \), and \( C \) are three finite sets such that:
(1) \( |A|=|B|=2016 \);
(2) \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \),
then the maximum value of \( |A \cap B \cap C| \) is ________. | 2015 |
63,520 | 8-1. Find all common points of the graphs
$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text { and } \quad y=\cos 9 \pi x
$$
with abscissas belonging to the segment $x \in[0 ; 1]$. In your answer, specify the sum of the abscissas of the found points. | 3.5 |
66,176 | 3.57. A line is drawn through the vertex of angle $\alpha$ at the base of an isosceles triangle, intersecting the opposite lateral side and forming an angle $\beta$ with the base. In what ratio does this line divide the area of the triangle? | \frac{\sin(\alpha-\beta)}{2\cos\alpha\sin\beta} |
65,056 | 5. Three Thieves, Bingo, Bunko, and Balko, robbed a bank and carried away 22 bags of banknotes. They placed them in a row so that the first bag contained the least amount of money, and each subsequent bag contained one more stack of banknotes than the previous one. Chief Bingo divided the stolen bags of money according to the principle: the first one for me (Bingo), the second one for you (Bunko), the third one for me (Bingo), the fourth one for you (Balko), the fifth one for me (Bingo), the sixth one for you (Bunko)... Then they counted the money. Bunko and Balko found that together they had a whopping total of 6710000 SIT. They were also delighted to find that together they had 110000 SIT more than their chief. Your task is to determine how much money was in the first bag. Write down the answer.[^3]
## Solutions to the Problems and Scoring
A contestant who arrives at the solution by any correct method (even if the scoring does not provide for it) receives all possible points.
For a correct method, any procedure that
- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.
A contestant who has only partially solved the problem, from otherwise correct solving procedures but does not show the way to the final solution of the problem, cannot receive more than half of the possible points.
## First Year
| 500000 |
31,798 | (EGMO 2017) Find the smallest positive integer $k$ for which there exists a $k$-coloring of the positive integers and a function $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that
1. $\forall n, m \in \mathbb{N}^{*}$ of the same color, $f(n+m)=f(n)+f(m)$
2. $\exists n, m \in \mathbb{N}^{*}$ such that $f(n+m) \neq f(n)+f(m)$ | 3 |
28,371 | [ $\quad$ Coloring
What is the minimum number of colors needed to color all vertices, sides, and diagonals of a convex $n$-gon, if the following two conditions must be satisfied:
1) any two segments emanating from the same vertex must be of different colors;
2) the color of any vertex must be different from the color of any segment emanating from it?
# | n |
31,451 | 4. A trapezoid $ABCD (AD \| BC)$ and a rectangle $A_{1}B_{1}C_{1}D_{1}$ are inscribed in a circle $\Omega$ with radius 13, such that $AC \perp B_{1}D_{1}, BD \perp A_{1}C_{1}$. Find the ratio of the area of $ABCD$ to the area of $A_{1}B_{1}C_{1}D_{1}$, given that $AD=10, BC=24$. | \frac{289}{338} |
3,283 | Let three circles \(\Gamma_{1}, \Gamma_{2}, \Gamma_{3}\) with centers \(A_{1}, A_{2}, A_{3}\) and radii \(r_{1}, r_{2}, r_{3}\) respectively be mutually tangent to each other externally. Suppose that the tangent to the circumcircle of the triangle \(A_{1} A_{2} A_{3}\) at \(A_{3}\) and the two external common tangents of \(\Gamma_{1}\) and \(\Gamma_{2}\) meet at a common point \(P\). Given that \(r_{1}=18 \mathrm{~cm}, r_{2}=8 \mathrm{~cm}\) and \(r_{3}=k \mathrm{~cm}\), find the value of \(k\). | 12 |
66,208 | 12. Determine the total number of pairs of integers $x$ and $y$ that satisfy the equation
$$
\frac{1}{y}-\frac{1}{y+2}=\frac{1}{3 \cdot 2^{x}}
$$ | 6 |
18,219 | Given that the axis of symmetry of a parabola is \(2x + y - 1 = 0\), the directrix of the parabola is \(x - 2y - 5 = 0\), and the parabola is tangent to the line \(2y + 3 = 0\), find the equation of the parabola. | 4x^2 + 4xy + y^2 - 10y - 15 = 0 |
17,994 | Let \( N_{0} \) be the set of non-negative integers, and \( f: N_{0} \rightarrow N_{0} \) be a function such that \( f(0)=0 \) and for any \( n \in N_{0} \), \( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \) and \( f(2n) > f(n) \). Determine how many elements in \( f(N_{0}) \) are less than 2004.
| 128 |
8,684 | The sum of the three largest natural divisors of a natural number \( N \) is 10 times the sum of its three smallest natural divisors. Find all possible values of \( N \). | 40 |
54,926 | C2. In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.
Find the minimal value of $k$ such that the goal is attainable for every possible configuration of 4027 points. | 2013 |
5,976 | A sphere with radius 1 is drawn through vertex \( D \) of a tetrahedron \( ABCD \). This sphere is tangent to the circumscribed sphere of the tetrahedron \( ABCD \) at point \( D \) and is also tangent to the plane \( ABC \). Given that \( AD = 2\sqrt{3} \), \( \angle BAC = 60^\circ \), and \( \angle BAD = \angle CAD = 45^\circ \), find the radius of the circumscribed sphere of tetrahedron \( ABCD \). | 3 |
34,588 | Example 3. Integrate the homogeneous equation
$$
\left(y^{4}-2 x^{3} y\right) d x+\left(x^{4}-2 x y^{3}\right) d y=0
$$ | x^{3}+y^{3}=Cxy |
56,863 | 3. A point in a triangle is connected to the vertices by three segments. What is the maximum number of these segments that can equal the opposite side?
# | 1 |
28,103 | 7. Given the ellipse $\Gamma: \frac{x^{2}}{4}+y^{2}=1$, a line with a non-zero slope is drawn through the right focus $F$, intersecting the ellipse $\Gamma$ at points $A$ and $B$. If $A O^{2}=A B$, then the coordinates of point $A$ are $\qquad$ . | (\frac{\sqrt{3}}{2},\\frac{\sqrt{13}}{4}) |
6,782 | Consider two circles \(\Gamma_{1}\) and \(\Gamma_{2}\) intersecting at points \(A\) and \(D\). Consider two lines \(d_{1}\) and \(d_{2}\) passing through \(A\) and \(B\) respectively. Let \(C\) and \(E\) be the points of intersection of \(\Gamma_{1}\) and \(\Gamma_{2}\) with \(d_{1}\), and \(D\) and \(F\) the points of intersection with \(d_{2}\). Show that \((CD)\) and \((EF)\) are parallel. | (CD) \parallel (EF) |
63,588 | 8. Proportion. The numbers $A, B, C, p, q, r$ are related by the following ratios:
$$
A: B=p, \quad B: C=q, \quad C: A=r
$$
Write the proportion
$$
A: B: C=\square: \square: \square
$$
such that the expressions in the blanks consist of $p, q, r$, and these expressions can be transformed into one another by cyclic permutations of the letters $p, q, r$. (We understand this as follows: if we replace $p$ with $q$, $q$ with $r$, and $r$ with $p$, then the first expression will transform into the second, the second into the third, and the third into the first).
8a. Symmetric Expressions. Expressions such as $x+y+z$ or $x y z$ are symmetric. By this, we mean that their value does not change when the variables $x, y, z$ are permuted in any way. The symmetry of the examples given above is obvious; however, there are symmetric expressions whose symmetry is not obvious, for example:
$$
|| x-y|+x+y-2 z|+|x-y|+x+y+2 z
$$
Prove the symmetry of this expression and determine its value in such a way that the symmetry becomes obvious. | 4\max{x,y,z} |
60,377 | 5. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, the angle between $A B_{1}$ and $A_{1} D$ is $\alpha$, the angle between $A C$ and $B C_{1}$ is $\beta$, and the angle between $A_{1} C_{1}$ and $C D_{1}$ is $\gamma$. Therefore, $\alpha+\beta+\gamma=$ $\qquad$ . | 180 |
54,146 | In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }24\qquad\textbf{(E) }36$ | 24 |
28,506 | 8.3. On the side $AC$ of triangle $ABC$, a point $M$ is taken. It turns out that $AM = BM + MC$ and $\angle BMA = \angle MBC + \angle BAC$. Find $\angle BMA$. | 60 |
14,509 | Let \(ABCD\) be a trapezoid such that \(AB \parallel CD\), \(\angle BAC = 25^\circ\), \(\angle ABC = 125^\circ\), and \(AB + AD = CD\). Compute \(\angle ADC\). | 70^\circ |
54,747 | Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$). | ABC |
64,756 | A merchant has a two-pan balance and weights of 1, 3, 9, 27, and 81 decagrams, one of each. What masses can he measure? | 1to121 |
63,696 | 6 Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in[a, a+2]$, the inequality $f(x+a) \geqslant 2 f(x)$ always holds, then the range of the real number $a$ is $\qquad$. | [\sqrt{2},+\infty) |
9,348 | Real numbers \( r, s, t \) satisfy \( 1 \leq r \leq s \leq t \leq 4 \). Find the minimum value of \( (r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2} \). | 12 - 8\sqrt{2} |
22,719 | 5. In a certain month, there are more Mondays than Tuesdays, and more Sundays than Saturdays. What day of the week is the 5th of this month? Could this month be December? | Thursday |
50,872 | We call a path Valid if
i. It only comprises of the following kind of steps:
A. $(x, y) \rightarrow (x + 1, y + 1)$
B. $(x, y) \rightarrow (x + 1, y - 1)$
ii. It never goes below the x-axis.
Let $M(n)$ = set of all valid paths from $(0,0) $, to $(2n,0)$, where $n$ is a natural number.
Consider a Valid path $T \in M(n)$.
Denote $\phi(T) = \prod_{i=1}^{2n} \mu_i$,
where $\mu_i$=
a) $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$
b) $y$, if the $i^{th} $ step is $(x, y) \rightarrow (x + 1, y - 1)$
Now Let $f(n) =\sum _{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$ | 0 |
8,108 | By definition, \( n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n \). Which factor should be crossed out from the product \( 1! \cdot 2! \cdot 3! \cdot \ldots \cdot 20! \) so that the remaining product becomes the square of some natural number? | 10! |
23,036 | A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $ 67$ ?
[asy]size(200);
defaultpen(fontsize(10));
label("1", origin);
label("3", (2,0));
label("5", (4,0));
label(" $\cdots$ ", (6,0));
label("97", (8,0));
label("99", (10,0));
label("4", (1,-1));
label("8", (3,-1));
label("12", (5,-1));
label("196", (9,-1));
label(rotate(90)*" $\cdots$ ", (6,-2));[/asy] | 17 |
6,762 | Let vectors $\overrightarrow{a_{1}}=(1,5)$, $\overrightarrow{a_{2}}=(4,-1)$, $\overrightarrow{a_{3}}=(2,1)$, and let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be non-negative real numbers such that $\lambda_{1}+\frac{\lambda_{2}}{2}+\frac{\lambda_{3}}{3}=1$. Find the minimum value of $\left|\lambda_{1} \overrightarrow{a_{1}}+\lambda_{2} \overrightarrow{a_{2}}+\lambda_{3} \overrightarrow{a_{3}}\right|$. | 3\sqrt{2} |
23,601 | In Figure 18-1, let $A^{\prime}, B^{\prime}, C^{\prime}$ be the midpoints of the sides $BC, CA, AB$ of $\triangle ABC$, respectively. Let $O_{1}, O_{2}, O_{3}, I_{1}, I_{2}, I_{3}$ be the circumcenters and incenters of $\triangle AB^{\prime}C^{\prime}, \triangle A^{\prime}BC^{\prime}, \triangle A^{\prime}B^{\prime}C$, respectively. Prove that $\triangle O_{1}O_{2}O_{3} \cong \triangle I_{1}I_{2}I_{3}$. | \triangle O_{1}O_{2}O_{3} \cong \triangle I_{1}I_{2}I_{3} |
5,069 | A rectangular prism has three distinct faces of area $24$ , $30$ , and $32$ . The diagonals of each distinct face of the prism form sides of a triangle. What is the triangle’s area? | 25 |
55,644 | 1614. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$. | 0.034 |
69,099 |
Problem 8.1. The graph of a linear function is parallel to the graph of $y=\frac{5}{4} x+\frac{95}{4}$, passing through $M(-1 ;-25)$, and intersects the coordinate axes $O x$ and $O y$ in $A$ and $B$ correspondingly.
(a) Find the coordinates of $A$ and $B$.
(b) Consider the unity grid in the plane. Find the number of squares containing points of $A B$ (in their interiors).
| 38 |
52,851 | 3. Ivan Semenov is taking a Unified State Exam (USE) in mathematics. The exam consists of three types of tasks: A, B, and C. For each task of type A, there are four answer options, only one of which is correct. There are 10 such tasks in total. Tasks of types B and C require detailed answers. Since Ivan constantly skipped classes, his knowledge of mathematics is shallow. He answers tasks of type A by guessing. The first task of type B, Ivan solves with a probability of $\frac{1}{3}$. He cannot solve anything else. For a correct answer to one task of type A, 1 point is awarded, and for a task of type B, 2 points. What is the probability that Ivan will score more than 5 points?
Take the tasks of type A from the 2008 USE practice version. (http://ege.edu.ru/demo/math.zip) and conduct the experiment of randomly selecting answers 10 times. Compare the result with the theoretical one (for 5 correct answers). Ensure that the results do not differ significantly.
# | 0.088 |
7,741 | Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$ a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1} $$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ *Proposed by Krit Boonsiriseth* | 2^m - 1 |
51,041 | 11.1. Paramon set out from point A to point B. At $12^{00}$, when he had walked half the way to B, Agafon ran out from A to B, and at the same time, Solomon set out from B to A. At $13^{20}$, Agafon met Solomon, and at $14^{00}$, he caught up with Paramon. At what time did Paramon and Solomon meet? | 13 |
23,286 | On the side \( BC \) of triangle \( ABC \), points \( A_1 \) and \( A_2 \) are marked such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On the side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) lie on the same line. Find \( B_1B_2 \). | 12 |
53,806 | 1. Given real numbers $a, b, c$ satisfy $abc=1$, then the number of numbers greater than 1 among $2a-\frac{1}{b}, 2b-\frac{1}{c}, 2c-\frac{1}{a}$ is at most $\qquad$. | 2 |
16,013 | Given positive numbers \(a, b, c, x, y, z\) satisfying the equations \(cy + bz = a\), \(az + cx = b\), \(bx + ay = c\), find the minimum value of the function
\[ f(x, y, z) = \frac{x^{2}}{1+x} + \frac{y^{2}}{1+y} + \frac{z^{2}}{1+z}. \] | \dfrac{1}{2} |
31,143 | 6. The altitudes of an acute-angled, non-isosceles triangle \(ABC\) intersect at point \(H\). \(O\) is the center of the circumcircle of triangle \(BHC\). The center \(I\) of the inscribed circle of triangle \(ABC\) lies on the segment \(OA\). Find the angle \(BAC\). | 60 |
59,464 | ## 7. Light Bulbs
In the room, there are two light bulbs. When the switch of the first light bulb is turned on, it lights up after 6 seconds and stays on for 5 seconds, then it is off for 6 seconds and on for 5 seconds, and this repeats continuously. When the switch of the second light bulb is turned on, it lights up after 4 seconds and stays on for 3 seconds, then it is off for 4 seconds and on for 3 seconds, and this repeats continuously. Linda turned on both switches at the same time and turned them off after 2021 seconds. How many seconds did both light bulbs shine simultaneously during this time? | 392 |
34,766 | 13. $A B C D$ is a convex quadrilateral in which $A C$ and $B D$ meet at $P$. Given $P A=1$, $P B=2, P C=6$ and $P D=3$. Let $O$ be the circumcentre of $\triangle P B C$. If $O A$ is perpendicular to $A D$, find the circumradius of $\triangle P B C$.
(2 marks)
$A B C D$ is a convex quadrilateral, in which $A C$ and $B D$ intersect at $P$. Given $P A=1$, $P B=2$, $P C=6$, and $P D=3$. Let $O$ be the circumcenter of $\triangle P B C$. If $O A$ is perpendicular to $A D$, find the circumradius of $\triangle P B C$. | 3 |
58,897 | 3. (7 points) On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$ on the circle, points $B$ and $C$ are chosen such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is $15$, and $\sin \alpha = \frac{\sqrt{21}}{5}$? | 12 |
67,210 | Exercise 12. A circle with diameter $2 n-1$ is drawn at the center of a chessboard $2 n \times 2 n$. How many squares are crossed by an arc of the circle? | 8n-4 |
33,918 | 14. In the triangle $A B C, \angle B=90^{\circ}, \angle C=20^{\circ}, D$ and $E$ are points on $B C$ such that $\angle A D C=$ $140^{\circ}$ and $\angle A E C=150^{\circ}$. Suppose $A D=10$. Find $B D \cdot C E$. | 50 |
32,367 | 6. A circle of radius 4 touches line $P$ at point $A$ and line $Q$ at point $B$ such that the chord $A B$ subtends an arc of $60^{\circ}$. Lines $P$ and $Q$ intersect at point $F$. Point $C$ is located on ray $F A$, and point $D$ is on ray $B F$ such that $A C = B D = 5$. Find the length of the median of triangle $C A D$ drawn from vertex $A$. | \frac{5\sqrt{3}}{2}-2 |
28,243 | Example 8 Find the smallest real number $A$, such that for every quadratic trinomial $f(x)$ satisfying the condition $|f(x)| \leqslant 1(0 \leqslant x \leqslant 1)$, the inequality $f^{\prime}(0) \leqslant A$ holds. | 8 |
16,224 | In an orthocentric tetrahedron \(ABCD\), the angle \(ADC\) is a right angle. Prove that \(\frac{1}{h^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\), where \(h\) is the length of the altitude of the tetrahedron drawn from vertex \(D\), \(a=DA\), \(b=DB\), and \(c=DC\). | \frac{1}{h^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}} |
58,488 | 6. The password setting for a certain lock involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex either red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How many different password settings are there for this lock?
(2010, National High School Mathematics League Competition) | 3^{n}+(-1)^{n}+2 |
1,444 | Find the smallest real number \( a \) such that for any non-negative real numbers \( x \), \( y \), and \( z \) that sum to 1, the following inequality holds:
$$
a(x^{2} + y^{2} + z^{2}) + xyz \geq \frac{a}{3} + \frac{1}{27}.
$$ | \dfrac{2}{9} |
17,977 | Let $A,B,C$ be angles of an acute triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} \end{align*} There are positive integers $p$ , $q$ , $r$ , and $s$ for which \[\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s},\] where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$ .
Please give the answer directly without any intermediate steps. | 222 |
3,674 | In a $10 \times 10$ grid square, the centers of all unit squares (a total of 100 points) are marked. What is the minimum number of lines, not parallel to the sides of the square, needed to pass through all the marked points? | 18 |
28,114 | 39. The little trains on多多Island are busily working. Thomas and Percy travel from the shed to the harbor, while Edward travels from the harbor to the station. They all immediately return to their starting points after reaching their destinations, completing one round trip before finishing their work. The three trains start at the same time. When Edward arrives at the station, Thomas also happens to be passing the station, while Percy has only traveled $45 \mathrm{~km}$; when Edward returns to the harbor, Thomas is passing the station again. At this point, Thomas's steam engine malfunctions, and he continues at half speed, meeting Percy face-to-face at the midpoint of the line between the shed and the harbor. What is the length of the railway line between the shed and the harbor in $\mathrm{km}$? | 225 |
58,532 | ## Task 4 - 300734
Someone wants to select as many different natural numbers from 1 to 1000 as possible according to the following rules:
The first number is to be chosen randomly from the numbers 1 to 6 by rolling a die and selecting the number shown on the die. The subsequent numbers are to be chosen such that the following applies:
When the selection of numbers is complete, any two of the overall selected numbers always have a sum that is divisible by 3.
Determine (depending on all possibilities of the first number) the maximum number of numbers that can be selected according to these rules! | 2 |
15,175 | In right triangle \( \triangle ABC \), \( AD \) is the altitude on hypotenuse \( BC \). The line connecting the incenters of \( \triangle ABD \) and \( \triangle ACD \) intersects side \( AB \) at \( K \) and side \( AC \) at \( L \). Line \( KL \) intersects \( AD \) at \( E \). Prove that \( \frac{1}{AB} + \frac{1}{AC} = \frac{1}{AE} \). | \frac{1}{AB} + \frac{1}{AC} = \frac{1}{AE} |
53,405 | ## Task A-1.3.
Two congruent squares with side lengths of $1+\sqrt{2}$ have the same center, and their intersection is a regular octagon. What is the area of this octagon? | 2+2\sqrt{2} |
30,038 | 7.1. How many values of the parameter $a$ exist for which the equation
$$
4 a^{2}+3 x \lg x+3 \lg ^{2} x=13 a \lg x+a x
$$
has a unique solution | 2 |
29,789 | Tokorev S.i.
In each cell of a $4 \times 4$ table, there is a sign "+" or "-". It is allowed to simultaneously change the signs to the opposite in any cell and in all cells that share a side with it. How many different tables can be obtained by repeatedly applying such operations? | 2^{12} |
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