matematik-5:problem och uppgifter

Blandande och inte så enkla problem på kombinatorik och talteori

Chevalier de Méré's problem. Detta är historiskt intressant men går enkelt att googla upp.


a) Du satsar 10 kr och slår fyra sexsidiga tärningar. Du kan välja på att satsa på att minst en 6:a kommer upp, eller att ingen 6:a kommer upp. Om du har rätt dubblas din insats (dvs. du får tillbaka dina satsade 10 kr och ytterligare 10 kr), har du fel förlorar du insatsen. Vad ska du satsa på, och hur stor är chansen att du vinner?

b) Du satsar 10 kr och slår två sexsidiga tärningar 24 gånger i rad. Du kan välja på att satsa på att minst en dubbelsexa kommer upp, eller att detta inte inträffar. Om du har rätt dubblas din insats (dvs. du får tillbaka dina satsade 10 kr och ytterligare 10 kr), har du fel förlorar du insatsen. Vad ska du satsa på, och hur stor är chansen att du vinner?


På hur många sätt kan man måla 10 identiska hästar i en karusell så att tre av dem är bruna, tre är blå och fyra är svarta? Eftersom karusellen roterar räknas uppställningar som kan överföras till varandra genom rotation som lika.


På hur många sätt kan en lärare dela ut 12 olika läroböcker bland 16 elever om

a) ingen student får mer än en bok
b) den äldste studenten får två böcker och ingen annan student får mer än en bok


En slant singlas 60 gånger med resultatet 45 krona och 15 klave. På hur många sätt kan detta ha inträffat om det inte förekommit två på varandra följande klavar.


På hur många sätt kan ett dussin (identiska) äpplen fördelas på fem barn så att inget barn får mer än 7 äpplen?


Hur många femsiffriga tal finns det så att

a) siffrorna är avtagande om man läser från vänster till höger. T.ex. är 85320 ok.
b) talet är palindromt, vilket innebär att det blir samma om man läser från vänster till höger som från höger till vänster. T.ex. är 12321 ok.
c) om man slumpar om placeringen av siffrorna i talet så kommer det nya talet säkert att ha en siffra som är kvar på samma plats. T.ex. är 12345 inte ok ty kan slumpas om till 23451, medan 11111 är ok ty kan bara slumpas om till 11111 där t.o.m. alla siffror är intakta.


I bokens uppgift 1160 skulle man förflytta sig mellan två punkter på Manhattan. Vi såg att det vara samma som att ange en sekvens av tio förflyttningar varav tre höger och sju uppåt. Antag nu att en korsning, säg Park Avenue/38th street, är blockerad och inte kan passeras. Hur många sätt kan man då ta sig kortaste vägen mellan punkterna?


En partikel ska flyttas från punkten $(1,2)$ till punkten $(5,9)$ i ett koordinatsystem. På hur många sätt kan denna förflyttning göras om de tillåtna förflyttningarna i varje steg är: $(x,y) \mapsto (x+1,y)$, $(x,y) \mapsto (x,y+1)$ eller $(x,y) \mapsto (x+1,y+1)$.



At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?


Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?


Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?


A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?


A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?


A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?

  • There must be an even number of odd scores.
  • There must be an even number of even scores.
  • There cannot be two scores of 0.
  • The sum of the scores must be at least
  • The highest score must be at least 12.

What is the greatest number of consecutive integers whose sum is $45$?


A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and

(1)
\begin{align} a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}} \end{align}

for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q$


Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?


Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?


For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by 12?


For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50) = 12$ because the prime factorization of $50$ is $2 \times 5^{2}$, and $2 + 5 + 5 = 12$. What is the range of the function $r$, $\{r(n): n \text{ is a composite positive integer}\}$ ?

  • the set of positive integers
  • the set of composite positive integers
  • the set of even positive integers
  • the set of integers greater than 3
  • the set of integers greater than 4

A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?


Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?


For some positive integer $n$, the number $110n^3$ has 110 positive integer divisors, including 1 and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?


How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?


Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than 1. What is $f(2017)$?


The number $21!=51,090,942,171,709,440,000$ has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd?


Let $N=123456789101112 \dots 4344$ be the 79-digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when $N$ is divided by 45?


Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?


Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?


How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?


What is

(2)
\begin{align} \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \end{align}

How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?


Mary chose an even 4-digit number n. She wrote down all the divisors of n in increasing order from left to right: $1,2,...,\dfrac{n}{2},n$. At some moment Mary wrote 323 as a divisor of n. What is the smallest possible value of the next divisor written to the right of 323?


Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and the denominator?

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