Found problems: 14842
2020 International Zhautykov Olympiad, 2
Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$.
2022 Novosibirsk Oral Olympiad in Geometry, 7
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
2021 Serbia National Math Olympiad, 2
In the country of Graphia there are $100$ towns, each numbered from $1$ to $100$. Some pairs of towns may be connected by a (direct) road and we call such pairs of towns [i]adjacent[/i]. No two roads connect the same pair of towns.
Peter, a foreign tourist, plans to visit Graphia $100$ times. For each $i$, $i=1,2,\dots, 100$, Peter starts his $i$-th trip by arriving in the town numbered $i$ and then each following day Peter travels from the town he is currently in to an adjacent town with the lowest assigned number, assuming such that a town exists and that he hasn't visited it already on the $i$-th trip. Otherwise, Peter deems his $i$-th trip to be complete and returns home.
It turns out that after all $100$ trips, Peter has visited each town in Graphia the same number of times. Find the largest possible number of roads in Graphia.
2019 China Team Selection Test, 6
Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time.
Determine all $k$ such that $A$ can always win the game.
2011 Estonia Team Selection Test, 6
On a square board with $m$ rows and $n$ columns, where $m\le n$, some squares are colored black in such a way that no two rows are alike. Find tha biggest integer $k$ such that, for every possible coloring to start with, one can always color $k$ columns entirely red in such a way that still no two rows are alike.
2016 Brazil National Olympiad, 4
What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?
2024 Israel National Olympiad (Gillis), P7
A rook stands in one cell of an infinite square grid. A different cell was colored blue and mines were placed in $n$ additional cells: the rook cannot stand on or pass through them. It is known that the rook can reach the blue cell in finitely many moves. Can it do so (for every $n$ and such a choice of mines, starting point, and blue cell) in at most
[b](a)[/b] $1.99n+100$ moves?
[b](b)[/b] $2n-2\sqrt{3n}+100$ moves?
[b]Remark.[/b] In each move, the rook goes in a vertical or horizontal line.
1997 Baltic Way, 20
Twelve cards lie in a row. The cards are of three kinds: with both sides white, both sides black, or with a white and a black side. Initially, nine of the twelve cards have a black side up. The cards $1-6$ are turned, and subsequently four of the twelve cards have a black side up. Now cards $4-9$ are turned, and six cards have a black side up. Finally, the cards $1-3$ and $10-12$ are turned, after which five cards have a black side up. How many cards of each kind were there?
2021 AMC 12/AHSME Fall, 25
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2023 Lusophon Mathematical Olympiad, 1
A long time ago, there existed Martians with $3$ different colours: red, green and blue. As Mars was devastated by an intergalactic war, only $2$ Martians of each colours survived. In order to reconstruct the Martian population, they decided to use a machine that transforms two Martians of distinct colours into four Martians of colour different to the two initial ones. For example, if a red Martian and a blue Martian use the machine, they'll be transformed into four green Martians.
a) Is it possible that, after using that machine finitely many times, we have $2022$ red Martians, $2022$ green Martians and $2022$ blue Martians?
b) Is it possible that, after using that machine finitely many times, we have $2021$ red Martians, $2022$ green Martians and $2023$ blue Martians?
2019 Belarusian National Olympiad, 10.4
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.
Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums not greater than $9$.
[i](I. Gorodnin)[/i]
2006 Pre-Preparation Course Examination, 5
Express the sum $S_m(n)=1^m+2^m+\ldots +(n-1)^m$ with Bernolli numbers.
ICMC 4, 4
Does there exist a set $\mathcal{R}$ of positive rational numbers such that every positive rational number is the sum of the elements of a unique finite subset of $\mathcal{R}$?
[i]Proposed by Tony Wang[/i]
1985 Bundeswettbewerb Mathematik, 1
Sixty-four dice with the numbers ”one” to ”six” are placed on one table and formed into a square with eight horizontal and eight vertical rows of cubes pushed together. By rotating the dice, while maintaining their place, we want to finally have all sixty-four dice the "one" points upwards. Each dice however, may not be turned individually, but only every eight dice in a horizontal or vertical row together by $90^o$ to the longitudinal axis of this row may turn. Prove that it is always possible to solve the dice by repeatedly applying the permitted type of rotation to the required end position.
2022 CMIMC, 2.6 1.2
A sequence of pairwise distinct positive integers is called averaging if each term after the first two is the average of the previous two terms. Let $M$ be the maximum possible number of terms in an averaging sequence in which every term is less than or equal to $2022$ and let $N$ be the number of such distinct sequences (every term less than or equal to $2022$) with exactly $M$ terms. What is $M+N?$ (Two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are said to be distinct if $a_i \neq b_i$ for some integer $1 \leq i \leq n$).
[i]Proposed by Kyle Lee[/i]
1998 Portugal MO, 3
Could the set $\{1,2,3,...,3000\}$ contain a subset of $2000$ elements such that none of them is twice the size of another?
Kettering MO, 2016
[b]p1.[/b] Solve the equation $3^x + 9^x = 27^x$.
[b]p2.[/b] An equilateral triangle in inscribed in a circle of area $1$ m$^2$. Then the second circle is inscribed in the triangle. Find the radius of the second circle.
[b]p3.[/b] Solve the inequality: $2\sqrt{x^2 - 5x + 4} + 3\sqrt{x^2 + 2x - 3} \le 5\sqrt{6 - x - x^2}$
[b]p4.[/b] Peter and John played a game. Peter wrote on a blackboard all integers from $1$ to $18$ and offered John to choose $8$ different integers from this list. To win the game John had to choose 8 integers such that among them the difference between any two is either less than $7$ or greater than $11$. Can John win the game? Justify your answer.
[b]p5.[/b] Prove that given $100$ different positive integers such that none of them is a multiple of $100$, it is always possible to choose several of them such that the last two digits of their sum are zeros.
[b]p6.[/b] Given $100$ different squares such that the sum of their areas equals $1/2$ m$^2$ , is it possible to place them on a square board with area $1$ m$^2$ without overlays? Justify your answer.
PS. You should use hide for answers.
2006 Romania Team Selection Test, 2
Let $m$ and $n$ be positive integers and $S$ be a subset with $(2^m-1)n+1$ elements of the set $\{1,2,3,\ldots, 2^mn\}$. Prove that $S$ contains $m+1$ distinct numbers $a_0,a_1,\ldots, a_m$ such that $a_{k-1} \mid a_{k}$ for all $k=1,2,\ldots, m$.
Russian TST 2017, P2
A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On them, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). Prove that if we color the triangles black and white such that no two adjacent triangles have the same color, there will be as many rooks on the black triangles as on the white ones.
2019 Belarusian National Olympiad, 9.8
Andrey and Sasha play the game, making moves alternate. On his turn, Andrey marks on the plane an arbitrary point that has not yet been marked. After that, Sasha colors this point in one of two colors: white and black. Sasha wins if after his move it is impossible to draw a line such that all white points lie in one half-plane, while all black points lie in another half-plane with respect to this line.
[b]a)[/b] Prove that Andrey can make moves in such a way that Sasha will never win.
[b]b)[/b] Suppose that Andrey can mark only integer points on the Cartesian plane. Can Sasha guarantee himself a win regardless of Andrey's moves?
[i](N. Naradzetski)[/i]
2012 BMT Spring, round 1
[b]p1.[/b] Find all prime factors of $8051$.
[b]p2.[/b] Simplify $$[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],$$ where $x = 628$, $y = 233$, $z = 340$.
[b]p3.[/b] In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G.
Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s?
[b]p4.[/b] What is the smallest positive $n$ so that $17^n + n$ is divisible by $29$?
[b]p5.[/b] The legs of the right triangle shown below have length $a = 255$ and $b = 32$. Find the area of the smaller rectangle (the one labeled $R$).
[img]https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png[/img]
[b]p6.[/b] A $3$ dimensional cube contains ”cubes” of smaller dimensions, ie: faces ($2$-cubes),edges ($1$-cubes), and vertices ($0$-cubes). How many 3-cubes are in a $5$-cube?
PS. You had better use hide for answers.
1997 Czech And Slovak Olympiad IIIA, 2
Each side and diagonal of a regular $n$-gon ($n \ge 3$) for odd $n$ is colored red or blue. One may choose a vertex and change the color of all segments emanating from that vertex. Prove that, no matter how the edges were colored initially, one can achieve that the number of blue segments at each vertex is even. Prove also that the resulting coloring depends only on the initial coloring.
1995 Tournament Of Towns, (461) 6
Does there exist a nonconvex polyhedron such that not one of its vertices is visible from a point $M$ outside it? (The polyhedron is made out of an opaque material.)
(AY Belov, S Markelov)
2024 Polish MO Finals, 2
Let $n$ be a positive integer. Bolek draws $2n$ points in the plane, no two of them defining a vertical or a horizontal line. Then Lolek draws for each of these $2n$ points two rays emanating from them, one of them vertically and the other one horizontally.
Lolek wants to maximize the number of regions in which these rays divide the plane. Determine the largest number $k$ such that Lolek can obtain at least $k$ regions independent of the points chosen by Bolek.
2015 Postal Coaching, Problem 5
For each point $X$ in the plane, a real number $r_X > 0$ is assigned such that $2|r_X - r_Y | \le |XY |$, for any two points $X, Y$ . (Here $|XY |$ denotes the distance between $X$ and $Y$) A frog can jump from $X$ to $Y$ if $r_X = |XY |$. Show that for any two points $X$ and $Y$ , the frog can jump from $X$ to $Y$ in a finite number of steps.