Found problems: 14842
1990 ITAMO, 6
Some marbles are distributed over $2n + 1$ bags. Suppose that, whichever bag is removed, it is possible to divide the remaining bags into two groups of $n$ bags such that the number of marbles in each group is the same. Prove that all the bags contain the same number of marbles.
2016 Japan MO Preliminary, 6
Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.
2015 JBMO Shortlist, C5
An L-shape is one of the following four pieces, each consisting of three unit squares:
[asy]
size(300);
defaultpen(linewidth(0.8));
path P=(1,2)--(0,2)--origin--(1,0)--(1,2)--(2,2)--(2,1)--(0,1);
draw(P);
draw(shift((2.7,0))*rotate(90,(1,1))*P);
draw(shift((5.4,0))*rotate(180,(1,1))*P);
draw(shift((8.1,0))*rotate(270,(1,1))*P);
[/asy]
A $5\times 5$ board, consisting of $25$ unit squares, a positive integer $k\leq 25$ and an unlimited supply of L-shapes are given. Two players A and B, play the following game: starting with A they play alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares.
We say that a placement of L-shapes on unmarked unit squares is called $\textit{good}$ if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board.
B wins if every $\textit{good}$ placement of L-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which B has a winning strategy.
2010 Tournament Of Towns, 6
Each cell of a $1000\times 1000$ table contains $0$ or $1$. Prove that one can either cut out $990$ rows so that at least one $1$ remains in each column, or cut out $990$ columns so that at least one $0$ remains in each row.
V Soros Olympiad 1998 - 99 (Russia), 9.4
There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than $100$ different ones. What is the largest possible value for $n$?
1999 IMO Shortlist, 3
A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that:
[list=1][*]the first fly is caught after a resting period of one minute;
[*]the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly;
[*]when the chameleon stops resting, he catches a fly instantly.[/list]
[list=a][*]How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row?
[*]After how many minutes will the chameleon catch his $98^\text{th}$ fly?
[*]How many flies were caught by the chameleon after 1999 minutes have passed?[/list]
2007 Chile National Olympiad, 1
On a chessboard of $16 \times 16$ squares, a "horse" moves making only movements of two types: from each square you can move either two squares to the right and one up, or two boxes up and one to the right. Determine in how many ways different the horse can move from the lower left square of the board to the top right box.
2013 Balkan MO Shortlist, C4
A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.
India EGMO 2022 TST, 2
Let $a,b$ be arbitrary co-prime natural numbers. Alice writes the natural number $t < b$ on a blackboard. Every second she replaces the number on the blackboard, say $x$, with the smallest natural number in $\{x \pm a, x \pm b \}$ that she has not yet ever written. She keeps doing this as long as possible. Prove that this process goes on indefinitely and that Alice will write down every natural number.
[i]~Pranjal Srivastava and Rohan Goyal[/i]
2009 India IMO Training Camp, 6
Prove The Following identity:
$ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$.
The Second term on left hand side is to be regarded zero for j=0.
Russian TST 2021, P3
Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied:
[list]
[*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$;
[*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$.
[/list]
A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.
2011 Costa Rica - Final Round, 3
The archipelago Barrantes - $n$ is a group of islands connected by bridges as follows: there are a main island (Humberto), in the first step I place an island below Humberto and one above from Humberto and I connect these 2 islands to Humberto. I put $2$ islands to the left of these $2$ new islands and I connect them with a bridge to the island that they have on their right. In the second step I take the last $2$ islands and I apply the same process that I applied to Humberto. In the third step I apply the same process to the $4$ new islands. We repeat this step n times we reflect the archipelago that we have on a vertical line to the right of Humberto. We connect Humberto with his reflection and so we have the archipelago Barrantes -$n$. However, the archipelago Barrantes -$n$ exists on a small planet cylindrical, so that the islands to the left of the archipelago are in fact the islands that are connected to the islands on the right. The figure shows the Barrantes archipelago -$2$, The islands at the edges are still numbered to show how the archipelago connects around the cylindrical world, the island numbered $1$ on the left is the same as the island numbered $1$ on the right.
[img]https://cdn.artofproblemsolving.com/attachments/e/c/803d95ce742c2739729fdb4d74af59d4d0652f.png[/img]
One day two bands of pirates arrive at the archipelago Barrantes - $n$: The pirates Black Beard and the Straw Hat Pirates. Blackbeard proposes a game to Straw Hat: The first player conquers an island, the next player must conquer an island connected to the island that was conquered in the previous turn (clearly not conquered on a previous shift). The one who cannot conquer any island in his turn loses. Straw Hat decides to give the first turn to Blackbeard. Prove that Straw Hat has a winning strategy for every $n$.
2012 Romanian Masters In Mathematics, 5
Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours.
[i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]
2012 Lusophon Mathematical Olympiad, 2
Maria has a board of size $n \times n$, initially with all the houses painted white. Maria decides to paint black some houses on the board, forming a mosaic, as shown in the figure below, as follows: she paints black all the houses from the edge of the board, and then leaves white the houses that have not yet been painted. Then she paints the houses on the edge of the next remaining board again black, and so on.
a) Determine a value of $n$ so that the number of black houses equals $200$.
b) Determine the smallest value of $n$ so that the number of black houses is greater than $2012$.
2018 Tuymaada Olympiad, 5
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
[i]Proposed by N. Vlasova, S. Berlov[/i]
2015 Caucasus Mathematical Olympiad, 1
At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar.
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?
(A statement that is at least partially false is considered false.)
2020 Purple Comet Problems, 17
The following diagram shows four vertices connected by six edges. Suppose that each of the edges is randomly painted either red, white, or blue. The probability that the three edges adjacent to at least one vertex are colored with all three colors is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/6/4/de0a2a1a659011a30de1859052284c696822bb.png[/img]
2007 Federal Competition For Advanced Students, Part 1, 3
Let $ M(n )\equal{}\{\minus{}1,\minus{}2,\ldots,\minus{}n\}$. For every non-empty subset of $ M(n )$ we consider the product of its elements. How big is the sum over all these products?
2024 IFYM, Sozopol, 6
Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).
2014 Turkey MO (2nd round), 6
$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is [i]properly-connected[/i]. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.
2020 BMT Fall, 2
There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL can start playing games? The CBL may not revoke membership of the $38$ people already in the CBL.
2023 ISL, C3
Let $n$ be a positive integer. A [i]Japanese triangle[/i] consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\dots$, $n$, the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured red. A [i]ninja path[/i] in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles.
[asy]
// credit to vEnhance for the diagram (which was better than my original asy):
size(4cm);
pair X = dir(240); pair Y = dir(0);
path c = scale(0.5)*unitcircle;
int[] t = {0,0,2,2,3,0};
for (int i=0; i<=5; ++i) {
for (int j=0; j<=i; ++j) {
filldraw(shift(i*X+j*Y)*c, (t[i]==j) ? lightred : white);
draw(shift(i*X+j*Y)*c);
}
}
draw((0,0)--(X+Y)--(2*X+Y)--(3*X+2*Y)--(4*X+2*Y)--(5*X+2*Y),linewidth(1.5));
path q = (3,-3sqrt(3))--(-3,-3sqrt(3));
draw(q,Arrows(TeXHead, 1));
label("$n = 6$", q, S);
label("$n = 6$", q, S);
[/asy]
In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.
2012 BmMT, Ind. Round
[b]p1.[/b] What is the slope of the line perpendicular to the the graph $\frac{x}{4}+\frac{y}{9}= 1$ at $(0, 9)$?
[b]p2.[/b] A boy is standing in the middle of a very very long staircase and he has two pogo sticks. One pogo stick allows him to jump $220$ steps up the staircase. The second pogo stick allows him to jump $125$ steps down the staircase. What is the smallest positive number of steps that he can reach from his original position by a series of jumps?
[b]p3.[/b] If you roll three fair six-sided dice, what is the probability that the product of the results will be a multiple of $3$?
[b]p4.[/b] Right triangle $ABC$ has squares $ABXY$ and $ACWZ$ drawn externally to its legs and a semicircle drawn externally to its hypotenuse $BC$. If the area of the semicircle is $18\pi$ and the area of triangle $ABC$ is $30$, what is the sum of the areas of squares $ABXY$ and $ACWZ$?
[img]https://cdn.artofproblemsolving.com/attachments/5/1/c9717e7731af84e5286335420b73b974da2643.png[/img]
[b]p5.[/b] You have a bag containing $3$ types of pens: red, green, and blue. $30\%$ of the pens are red pens, and $20\%$ are green pens. If, after you add $10$ blue pens, $60\%$ of the pens are blue pens, how many green pens did you start with?
[b]p6.[/b] Canada gained partial independence from the United Kingdom in $1867$, beginning its long role as the headgear of the United States. It gained its full independence in $1982$. What is the last digit of $1867^{1982}$?
[b]p7.[/b] Bacon, Meat, and Tomato are dealing with paperwork. Bacon can fill out $5$ forms in $3$ minutes, Meat can fill out $7$ forms in $5$ minutes, and Tomato can staple $3$ forms in $1$ minute. A form must be filled out and stapled together (in either order) to complete it. How long will it take them to complete $105$ forms?
[b]p8.[/b] Nice numbers are defined to be $7$-digit palindromes that have no $3$ identical digits (e.g., $1234321$ or $5610165$ but not $7427247$). A pretty number is a nice number with a $7$ in its decimal representation (e.g., $3781873$). What is the $7^{th}$ pretty number?
[b]p9.[/b] Let $O$ be the center of a semicircle with diameter $AD$ and area $2\pi$. Given square $ABCD$ drawn externally to the semicircle, construct a new circle with center $B$ and radius $BO$. If we extend $BC$, this new circle intersects $BC$ at $P$. What is the length of $CP$?
[img]https://cdn.artofproblemsolving.com/attachments/b/1/be15e45cd6465c7d9b45529b6547c0010b8029.png[/img]
[b]p10.[/b] Derek has $10$ American coins in his pocket, summing to a total of $53$ cents. If he randomly grabs $3$ coins from his pocket, what is the probability that they're all different?
[b]p11.[/b] What is the sum of the whole numbers between $6\sqrt{10}$ and $7\pi$ ?
[b]p12.[/b] What is the volume of a cylinder whose radius is equal to its height and whose surface area is numerically equal to its volume?
[b]p13.[/b] $15$ people, including Luke and Matt, attend a Berkeley Math meeting. If Luke and Matt sit next to each other, a fight will break out. If they sit around a circular table, all positions equally likely, what is the probability that a fight doesn't break out?
[b]p14.[/b] A non-degenerate square has sides of length $s$, and a circle has radius $r$. Let the area of the square be equal to that of the circle. If we have a rectangle with sides of lengths $r$, $s$, and its area has an integer value, what is the smallest possible value for $s$?
[b]p15.[/b] How many ways can you arrange the letters of the word "$BERKELEY$" such that no two $E$'s are next to each other?
[b]p16.[/b] Kim, who has a tragic allergy to cake, is having a birthday party. She invites $12$ people but isn't sure if $11$ or $12$ will show up. However, she needs to cut the cake before the party starts. What is the least number of slices (not necessarily of equal size) that she will need to cut to ensure that the cake can be equally split among either $11$ or $12$ guests with no excess?
[b]p17.[/b] Tom has $2012$ blue cards, $2012$ red cards, and $2012$ boxes. He distributes the cards in such a way such that each box has at least $1$ card. Sam chooses a box randomly, then chooses a card randomly. Suppose that Tom arranges the cards such that the probability of Sam choosing a blue card is maximized. What is this maximum probability?
[b]p18.[/b] Allison wants to bake a pie, so she goes to the discount market with a handful of dollar bills. The discount market sells different fruit each for some integer number of cents and does not add tax to purchases. She buys $22$ apples and $7$ boxes of blueberries, using all of her dollar bills. When she arrives back home, she realizes she needs more fruit, though, so she grabs her checkbook and heads back to the market. This time, she buys $31$ apples and $4$ boxes of blueberries, for a total of $60$ cents more than her last visit. Given she spent less than $100$ dollars over the two trips, how much (in dollars) did she spend on her first trip to the market?
[b]p19.[/b] Consider a parallelogram $ABCD$. Let $k$ be the line passing through A and parallel to the bisector of $\angle ABC$, and let $\ell$ be the bisector of $\angle BAD$. Let $k$ intersect line $CD$ at $E$ and $\ell$ intersect line $CD$ at $F$. If $AB = 13$ and $BC = 37$, find the length $EF$.
[b]p20.[/b] Given for some real $a, b, c, d,$ $$P(x) = ax^4 + bx^3 + cx^2 + dx$$ $$P(-5) = P(-2) = P(2) = P(5) = 1$$
Find $P(10).$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1994 Canada National Olympiad, 3
$25$ men sit around a circular table. Every hour there is a vote, and each must respond [i]yes [/i]or [i]no[/i]. Each man behaves as follows: on the $n^{\text{th}}$, vote if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.
2021 LMT Fall, 10
There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.