This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 14842

2013 Junior Balkan Team Selection Tests - Romania, 3

The three-element subsets of a seven-element set are colored. If the intersection of two sets is empty then they have different colors. What is the minimum number of colors needed?

Mid-Michigan MO, Grades 10-12, 2009

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2019-IMOC, C2

For $2n$ numbers in a row, Bob could perform the following operation: $$S_i=(a_1,a_2,\ldots,a_{2n})\mapsto S_{i+1}=(a_1,a_3,\ldots,a_{2n-1},a_2,a_4,\ldots,a_{2n}).$$ Let $T$ be the order of this operation. In other words, $T$ is the smallest positive integer such that $S_i=S_{i+T}$. Prove that $T<2n$.

2017 IOM, 4

Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$. Proposed by Mikhail Evdokimov

2017 Czech And Slovak Olympiad III A, 1

There are $100$ diamonds on the pile, $50$ of which are genuine and $50$ false. We invited a peculiar expert who alone can recognize which are which. Every time we show him some three diamonds, he would pick two and tell (truthfully) how many of them are genuine . Decide whether we can surely detect all genuine diamonds regardless how the expert chooses the pairs to be considered.

1995 North Macedonia National Olympiad, 4

On a $ 30 \times30 $ square board or placed figures of shape 1 (of 5 squares) (in all four possible positions) and shaped figures of shape 2 (of 4 squares) . The figures do not overlap, they do not pass through the edges of the board and the squares of which they are drawn lie exactly through the squares of the board. a) Prove that the board can be fully covered using $100$ figures of both shapes. b) Prove that if there are already $50$ shaped figures on the board of shape 1, then at least one more figure can be placed on the board. c) Prove that if there are already $28$ figures of both shapes on the board then at least one more figure of both shapes can be placed on the board. [img]https://cdn.artofproblemsolving.com/attachments/3/f/f20d5a91d61557156edf203ff43acac461d9df.png[/img]

2009 Korea Junior Math Olympiad, 4

There are $n$ clubs composed of $4$ students out of all $9$ students. For two arbitrary clubs, there are no more than $2$ students who are a member of both clubs. Prove that $n\le 18$. Translator’s Note. We can prove $n\le 12$, and we can prove that the bound is tight. (Credits to rkm0959 for translation and document)

2006 QEDMO 2nd, 11

On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.

2024 OMpD, 1

We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties: 1. \( T \) has at least two distinct elements; 2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \). For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \). What is the largest possible number of elements that a [b]kawaii [/b]subset can have?

2024 ISI Entrance UGB, P8

In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and losing team gets $0$ points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows: \[x_1 \ge x_2 \ge \cdots \ge x_N . \] Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

2021 Israel TST, 2

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

2019 China Team Selection Test, 6

Given positive integers $d \ge 3$, $r>2$ and $l$, with $2d \le l <rd$. Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$, such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$, and no more than $l-d$. A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$, such that for $G$'s any proper coloring with $r-1$ colors, and for an arbitrary color $C$, either all numbers in color $C$ appear in $A$, or none of the numbers in color $C$ appear in $A$. Show that $G$ has a proper coloring within $r-1$ colors.

2022 Saudi Arabia IMO TST, 1

There are a) $2022$, b) $2023$ plates placed around a round table and on each of them there is one coin. Alice and Bob are playing a game that proceeds in rounds indefinitely as follows. In each round, Alice first chooses a plate on which there is at least one coin. Then Bob moves one coin from this plate to one of the two adjacent plates, chosen by him. Determine whether it is possible for Bob to select his moves so that, no matter how Alice selects her moves, there are never more than two coins on any plate.

1998 Nordic, 4

Let $n$ be a positive integer. Count the number of numbers $k \in \{0, 1, 2, . . . , n\}$ such that $\binom{n}{k}$ is odd. Show that this number is a power of two, i.e. of the form $2^p$ for some nonnegative integer $p$.

2021 LMT Spring, B21

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Take five good haikus Scramble their lines randomly What are the chances That you end up with Five completely good haikus (With five, seven, five)? Your answer will be m over n where m,n Are numbers such that m,n positive Integers where gcd Of m,n is 1. Take this answer and Add the numerator and Denominator. [i]Proposed by Jeff Lin[/i]

2001 Estonia Team Selection Test, 4

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

1998 Korea - Final Round, 3

Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions: 1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$ 2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$ Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.

2025 JBMO TST - Turkey, 2

Let $n$ be a positive integer. Aslı and Zehra are playing a game on an $n\times n$ grid. Initially, $10n^2$ stones are placed on some of the unit squares of this grid. On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square. For which values of $n$, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?

2004 IberoAmerican, 1

It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that: 1: if 2 squares are adjacent then one of them is marked. 2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked. Find the minimun number of squares we most mark.

JOM 2015 Shortlist, C7

Navi and Ozna are playing a game where Ozna starts first and the two take turn making moves. A positive integer is written on the waord. A move is to (i) subtract any positive integer at most 2015 from it or (ii) given that the integer on the board is divisible by $2014$, divide by $2014$. The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$, $a\ge 2015$ such that all numbers of the form $an+b$ will not be the starting integer, where $n$ is any positive integer. Find the minimum number of starting integer where Navi wins.

2003 China Team Selection Test, 3

Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.

1967 IMO Shortlist, 1

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

2005 MOP Homework, 5

A group consists of $n$ tourists. Among every three of them there are at least two that are not familiar. For any partition of the group into two busses, there are at least two familiar tourists in one bus. Prove that there is a tourist who is familiar with at most two fifth of all the tourists.

2009 IMO Shortlist, 7

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

2000 Finnish National High School Mathematics Competition, 5

Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?