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

1999 Bosnia and Herzegovina Team Selection Test, 5

For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that $a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$ $b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$ where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$

2013 Greece Junior Math Olympiad, 3

Let $A=\overline{abcd}$ be a four-digit positive integer with digits $a, b, c, d$, such that $a\ge7$ and $a>b>c>d>0$. Consider the positive integer $B=\overline{dcba}$ , that comes from number $A$ by reverting the order of it's digits. Given that the number $A+B$ has all it's digits odd, find all possible values of number $A$.

1975 Bulgaria National Olympiad, Problem 2

Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$, and $\ell$ is the count of the knots of the network lying inside $F$, prove that if the surface of every square from the network is $1$, then the surface $S$ of $F$ is calculated with the formulae: $$S=\frac k2+\ell-1$$ [i]V. Chukanov[/i]

2004 Greece National Olympiad, 4

Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$ If no element of $M$ is equal to the sum of any two elements of $M,$ find the least value that the greatest element of $M$ can take.

2001 Junior Balkan Team Selection Tests - Moldova, 1

On a circle we consider a set $M$ consisting of $n$ ($n \ge 3$) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set $M$ are more: those that contain the red dot or those that do not contain those points? How many more are there than others?

2013 CHMMC (Fall), 9

A $ 7 \times 7$ grid of unit-length squares is given. Twenty-four $1 \times 2$ dominoes are placed in the grid, each covering two whole squares and in total leaving one empty space. It is allowed to take a domino adjacent to the empty square and slide it lengthwise to fill the whole square, leaving a new one empty and resulting in a different configuration of dominoes. Given an initial configuration of dominoes for which the maximum possible number of distinct configurations can be reached through any number of slides, compute the maximum number of distinct configurations.

2018 Moscow Mathematical Olympiad, 4

We call the arrangement of $n$ ones and $m$ zeros around the circle as good, if we can swap neighboring zero and one in such a way that we get an arrangement, that differs from the original by rotation. For what natural $m$ and $n$ does a good arrangement exist?

MBMT Guts Rounds, 2018

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$. [b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers. [b]G.18 / C.23[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint? [b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u] Set 5[/u] [b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$? [b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up? [b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$. [b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible? [b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]

2024 CMIMC Combinatorics and Computer Science, 10

Suppose 100 people are gathered around at a park, each with an envelope with their name on it (all their names are distinct). Then, the envelopes are uniformly and randomly permuted between the people. If $N$ is the number of people who end up with their original envelope, find the expected value of $N^5$. [i]Proposed by Michael Duncan[/i]

2019 Federal Competition For Advanced Students, P2, 3

In Oddland there are stamps with values of $1$ cent, $3$ cents, $5$ cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value. In Squareland, on the other hand, there are stamps with values of $1$ cent, $4$ cents, $9$ cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations. Prove for every positive integer $n$: there are the same number in the two countries possibilities to send a letter with stamps worth a total of $n$ cents. It makes no difference if you have the same stamps on arrange a letter differently. (Stephan Wagner)

2011 Indonesia TST, 2

Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$

1994 India National Olympiad, 4

Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers.

2023 Hong Kong Team Selection Test, Problem 6

(a) Find the smallest number of lines drawn on the plane so that they produce exactly 2022 points of intersection. (Note: For 1 point of intersection, the minimum is 2; for 2 points, minimum is 3; for 3 points, minimum is 3; for 4 points, minimum is 4; for 5 points, the minimum is 4, etc.) (b) What happens if the lines produce exactly 2023 intersections?

2022 Lusophon Mathematical Olympiad, 6

A necklace contains 2024 pearls, each one of them having one of the following colours: black, green and yellow. Each moment, we will switch each one of all pearls simultaneously to a new one following the following rules: i) If its two neighbours are of the same colour, then it'll be switched to that same colour. ii) If its two neighbours are of different colours, then it'll be switched to the third colour. a) Does there exist any necklace that can be transformed into a necklace that consists of only yellow pearls if initially half of the pearls are black and the other half is green? b) Does there exist a necklace that can be transformed into a necklace that consists of only yellow pearls if initially 998 pearls are black and the rest 1026 pearls are green?

2022 Durer Math Competition (First Round), 5

a) A game master divides a group of $12$ players into two teams of six. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.) b) On the next occasion, the game master writes the names of $3$ teammates and $1$ opposing player on each card (possibly in a mixed up order). Now he wants to write the names in such away that the players together cannot determine the two teams. Is it possible for him to achieve this? c) Can he write the names in such a way that the players together cannot determine the two teams, if now each card contains the names of $4$ teammates and $1$ opposing player (possibly in a mixed up order)?

2023 Korea Summer Program Practice Test, P4

In a country there are infinitely many towns and for every pair of towns there is one road connecting them. Initially there are $n$ coin in each city. Every day traveller Hong starts from one town and moves on to another, but if Hong goes from town $A$ to $B$ on the $k$-th day, he has to send $k$ coins from $B$ to $A$, and he can no longer use the road connecting $A$ and $B$. Prove that Hong can't travel for more than $n+2n^\frac{2}{3}$ days.

2006 Mid-Michigan MO, 7-9

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2020 BMT Fall, 24

For positive integers $N$ and $m$, where $m \le N$, define $$a_{m,N} =\frac{1}{{N+1 \choose m}} \sum_{i=m-1}^{N-1} \frac{ {i \choose m-1}}{N - i}$$ Compute the smallest positive integer $N$ such that $$\sum^N_{m=1}a_{m,N} >\frac{2020N}{N +1}$$

2012 India Regional Mathematical Olympiad, 4

Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.

2024 Belarusian National Olympiad, 9.4

In some company, consisting of $n$ people, any two have at most $k \geq 2$ common friends. Lets call group of people working in the company unsocial if everyone in the group has at most one friend from the group. Prove that there exists an unsocial group consisting of at least $\sqrt{\frac{2n}{k}}$ people [i]M. Zorka[/i]

2002 Iran Team Selection Test, 5

A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.

2021 Bolivia Ibero TST, 1

Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. [b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid. [b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.

2007 Vietnam National Olympiad, 1

Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.

2013 AMC 12/AHSME, 21

Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas? ${ \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D}}\ 840\qquad\textbf{(E)}\ 870 $

VII Soros Olympiad 2000 - 01, 10.7

The President of the Bank "Glavny Central" Gerasim Shchenkov announced that from January $2$, $2001$ until January $31$ of the same year, the dollar exchange rate would not go beyond the boundaries of the corridor of $27$ rubles $50$ kopecks. and $28$ rubles $30$ kopecks for the dollar. On January $2$, the rate will be a multiple of $5$ kopecks, and starting from January 3, each day will differ from the rate of the previous day by exactly $5$ kopecks. Mr. Shchenkov suggested that citizens try to guess what the dollar exchange rate will be during the specified period. Anyone who can give an accurate forecast for at least one day, he promised to give a cash prize. One interesting person lives in our house, a tireless arguer. For his passion for arguments and constant winnings, he was even nicknamed Zhora Sporos. Zhora claims that he can give such a forecast of the dollar exchange rate for every day from January 424 to January 4314, which he will surely guess at least once, if, of course, the banker strictly acts in accordance with the announced rules. Is Zhora right? Note: 1 ruble =100 kopecks [hide=original wording]10-I-7. Президент банка "Главный централ" Герасим Щенков объявил, что со 2-го января 2001 года и до 31-го января этого же года курс доллара не будет выходить за границы коридора 27 руб. 50 коп. и 28 руб. 30 коп. за доллар. 2-го января курс будет кратен 5 копейкам, а, начиная с 3-го января, каждый день будет отличаться от курса предыдущего дня ровно на 5 копеек. Господин Щенков предложил гражданам попробовать угадать, каким будет курс доллара в течение указанного периода. Тому, кто сумеет дать точный прогноз хотя бы на один день, он обещал выдать денежный приз. В нашем доме живет один интересный человек, неутомимый спорщик. За страсть к спорам и постоянные выигрыши его даже прозвали Жора Спорос. Жора утверждает, что может дать такой прогноз курса доллара на каждый день со 2-го по 31-е января, что обязательно хотя бы один раз угадает, если, конечно, банкир будет строго действовать в соответствии с объявленными правилами. Прав ли Жора? [/hide]