Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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2015 AMC 10B Problem 21

The problem and solutions I've attempted to solve another AMC 10 problem, and the problem is basically like this: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of ...
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0answers
38 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
2
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2answers
129 views

2015 AMC 10A Problems/Problem 14

The Clockblock Problem - problem and solutions I'm preparing myself for AMC 10 (which I'm sure a lot of other students would be doing too), but then I just don't know how to solve this problem (and ...
6
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0answers
56 views

What is the probability of a pen touching a bar given that the length of the pen is $10$ cm and the bars are regularly spaced at $15$ cm?

Problem: If a pen of length $10$ cm is thrown out of infinitely large window having vertical bars regularly spaced at $15$ cm, then find the probability that it will touch any of the bars. (Assume ...
5
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1answer
119 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
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0answers
71 views

Integer solutions to $y^2 = \frac{x^5-1}{x-1}$

$$y^2 = \frac{x^5-1}{x-1}$$ has integer solutions. How many pairs $(x,y)$ are there? My Work If $\sqrt{x^4+x^3+x^2+x+1}$ is an integer then there is a solution. But what to do now. Note: This ...
2
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1answer
45 views

Given that there is at least one error in the bit, what is the probability that it will be retransmitted?

A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message ...
2
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1answer
48 views

Probability of having at least one error in block of three bits?

A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message ...
2
votes
4answers
137 views

A Quadrilateral's area given four sides and a diagonal [closed]

Assume there exists a quadrilateral called ABCD and AB=5cm,BC=13cm,CD=16cm, DA=20cm and diagonal AC=12cm. The exercise now states that I should calculate the area of a quadrilateral. Thank you for ...
6
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4answers
135 views

# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?

How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
4
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2answers
106 views

How many different strings can be made from letters in CHICAGOLAND, subject to constraints? [closed]

How many different strings can be made from the letters in CHICAGOLAND, using all letters, and such that no two vowels are adjacent to each other?
3
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1answer
69 views

Good set with $n$ elements must have element $\ge {2\over n}\binom{n}{n\over2}$?

Let $n$ be even. A set $\{a_1, \dots, a_n\}$ consisting of positive integer s is good if for every two different disjoint subsets $S$, $T \subseteq [n]$ of the same cardinality we have$$\sum_{i \in S} ...
3
votes
1answer
61 views

Putnam: Show that $a(n)=b(n+2)$

Let $a(n)$ be the number of representations of positive integer $n$ as a sum of 1's and 2's taking order into account. $$ \text{Example $n=4$: } (1+1+1+1), (1+2+1),(1+1+2),(2+1+1),(2+2)\implies ...
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1answer
59 views

Dual program is wrong. Authors claim is right.

In a well respected book, I found the following. The authors claim that it is correct. But I think it is wrong. This is the primal Linear Problem: $$ \begin{array}{cccc} ...
3
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1answer
115 views

1995 USAMO Problems/Problem 2

I tried to solve this problem: A calculator is broken so that the only keys that still work are the $\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}$ buttons. The display initially shows $0$. ...
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1answer
85 views

math contest geometry problem

Consider a triangle $ABC$ with circumcircle $\omega$. Let $O$ be the center of $\omega$ and let $D, E, F$ be the midpoints of minor arcs $BC, CA, AB$ respectively. Let $DO$ intersect $\omega$ again at ...
6
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2answers
71 views

At least $P(m, n - 1) = {{m!}\over{(m - n+1)!}}$ surjective functions from $[m]$ to $[n]$?

How do I see that there are at least$$P(m, n - 1) = {{m!}\over{(m - n+1)!}}$$surjective functions from $[m]$ to $[n]$?
4
votes
1answer
128 views

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$? [closed]

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$?
2
votes
1answer
50 views

Find $N$ so that the sequence is the product of three consecutive numbers

Find the smallest natural number $N$ such that $13 \cdot 17 \cdot N$ is the product of three consecutive natural numbers. $x(x+1)(x+2) = 13 \cdot 17 \cdot N$. So let $x=N$, then, $N+1 = 13$ and ...
0
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2answers
34 views

How many cards need to be picked at least?

You have $50$ cards and you have the numbers from $1$ to $50$ written on them, and you randomly pick cards. How many cards do you need to pick out so you can ensure that at least $3$ cards with ...
5
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1answer
67 views

A number $n$ has $12$ divisors and $d_{d_4-1} = (d_1+d_2+d_4)d_8$.

Find a number $n$ which has - $1.$ $12$ divisors $(1 = d_1 < d_2 < \cdots <d_{12}=n )$ and $2.$ $d_{d_4-1}=(d_1+d_2+d_4)d_8$. Note: This is a problem from Russian Mathematical Olympiad ...
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2answers
61 views

How many three digit numbers exist such that the third digit is the geo mean

How many three digit numbers exist such that one of the digits is the geometric mean of the other two? A 12, B 18, C 24, D other So, $N = 100a + 10b + c$ let $c =\sqrt{ab}$. $ab$ must be a ...
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1answer
48 views

How to find the missing digit?

A student calculated the value of $1 \times 2\times 3\times \cdots \times 2015\times 2016=2016!$ Then he took the summation of all digits of that answer ! He got $24135$ , but later he realized ...
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1answer
41 views

How to prove that there is no infinite arithmetic progression of perfect squares

How to prove that there is no infinite arithmetic progression of perfect squares This question from a school Olympiad paper ! How can I prove this directly or using contradiction ? For example : 1 ...
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1answer
42 views

The $n-th$ term of a sequence is the LCM of the integers from $1$ to $n$

The $n-th$ term of a sequence is the least common multiple (LCM) of the integers from $1$ to $n$. Which term of the sequence is the first one that is divisible by $100$? How I'll solve this? ...
6
votes
2answers
74 views

Compute $\lim_{n \to +\infty} n^{-\frac12 \left(1+\frac{1}{n}\right)} \left(1^1 \cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\frac{1}{n^2}}$

How to compute $$\displaystyle \lim_{n \to +\infty} n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} \left(1^1\cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\dfrac{1}{n^2}}$$ I'm interested in more ways of ...
4
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3answers
55 views

Prove the triangle is equilateral

HINTS ONLY please. This is very confusing right off the bat. My guess was that we show the angle $C, M, N$ are all $60^{\text{o}}.$ But I am having difficulty doing as as none of the givens have ...
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1answer
22 views

Difference between the number of lucky numbers and medium numbers

Problem: Consider all the natural numbers from $000000$ to $999999$. Among these, those numbers with sum of first 3 digits equal to sum of last 3 digits are called lucky. And those with sum of all ...
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0answers
66 views

Mr.Smith commute word problem. Solved through logic, where is the argument unsound?

Mr. Smith commutes to the city regularly and invariably takes the same train home which arrives at the his home station at 5 pm. At this time, his chauffeur always just arrives, promptly picks him up ...
0
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0answers
44 views

What's the name of this problem? Interesting minimisation of a length.

There is a problem which has to do with minimising the length of a (possibly disjoint) barrier in a region of space (often a 2D circle) such that no straight line can pass through the particular ...
2
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4answers
36 views

How many lineups of 20 are possible where Sally is first, second or third, and Adam is somewhere in the line?

The line of 20 is created from 300 students. The next part of the question was to find how many ways there are where Sally is first, second or third. I did a permutation of 299 choose 19 for the ...
7
votes
3answers
281 views

Find the number of all subsets of $\{1, 2, \ldots,2015\}$ with $n$ elements such that the sum of the elements in the subset is divisible by 5

The problem is as in the question title. Only one addition - $n$ is not divisible by $5$. I already have a solution involving permutations, but recently I read about generating functions and I was ...
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0answers
23 views

Why is the diagonal a symmedian?

The problems asks: Let $ABCD$ be a cyclic quadrilatedral, and let $L$ and $N$ be the midpoints of its diagonals $AC$ and $BD$, respectively. Suppose that the line $BD$ bisects the angle $ANC$. Prove ...
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2answers
76 views

is $7^{101}+8^{101}$ divisible by 25? If not, what is $ 7^{101} + 8^{101}$ mod 25

What i derived is: $$\begin{align}7^{101}+8^{101} &\equiv (5+2)^{101}+ (5+3)^{101} \\ &\equiv 2^{101}+101\cdot5\cdot2^{100}+3^{101}+101\cdot 5\cdot 3^{100} \\ &\equiv ...
4
votes
1answer
57 views

Closed form expression for the number of ordered pairs $\{A, B\}$, where $A, B \subseteq \{1, 2, \dots, n\}$ such that $|A \cap B| = 1$?

What is a closed form expression for the number of ordered pairs $\{A, B\}$, where $A, B \subseteq \{1, 2, \dots, n\}$ such that $|A \cap B| = 1$?
3
votes
1answer
96 views

Combinatorics problem; counting in two ways, china 1993

I'm trying to solve the combinatorics problems provided in Yufei Zhao's blog. Can you help me with this one? China (1993): A group of $10$ people went to a bookstore. It is known that ...
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2answers
53 views

Proving that four lines (which are perpendicular bisectors of chords) meet a point

In the diagram above, each of the four lines is a perpendicular bisector of one of the circles' chord. There are two pairs of circles which touch each other, and of course, as shown in the diagram, ...
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2answers
37 views

Proving sum of product forms a pattern in n * nnnnnn…

I am consider a problem regarding numbers which are, in decimal, one digit repeated - for instance, $88888888$ is such a number. In particular, I am looking at the following problem: The sum of ...
0
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1answer
39 views

What is the largest prime number in the denominator of a fraction that creates a decimal that repeats every 4 digits?

I was studying a Target question for Math League competitions, and after a few hours of pondering, I finally came up with the following method of solving the mentioned problem: For any decimal, it is ...
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1answer
20 views

Showing there exists a sequence that majorizes another

The exact quantity of gas needed for a car to complete a single loop around a track is distrubuted among $n$ containers placed along the track. Show that there exists a point from which the car can ...
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1answer
53 views

How do you find the sum: $\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$

How do you find the sum: $$\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$$ I managed to solve this question using complex numbers so I thought I'd share the solution. If you know of any better ...
7
votes
2answers
110 views

How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$?

Problem: Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$ Note: The angles here are in degrees. I don't know how to solve this question because trigonometric simplifications didn't get me ...
3
votes
1answer
49 views

How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$?

Problem: Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$. I can't think of ...
3
votes
3answers
84 views

Computing $2016$ using basic operations on the fewest integers, in sequence

Using the operators $$+,-,\div,\times,\exp,(,),!$$ what is the least $n$ to come up with the number $2016$ using the sequence of numbers $1,2,3,\ldots,n$ in that order. You cannot combine numbers, so ...
3
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3answers
54 views

Finding the number of sequences with $0 \leq a_m \leq 3m$

Problem: Let $\alpha, \beta$ be non-negative numbers. Suppose the number of strictly increasing sequences $a_0, a_1, a_2 \cdots a_{2014}$ satisfying $0 \leq 3m$ is $2^{\alpha}(2\beta+1)$. Find ...
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vote
2answers
68 views

Labelling the edges of a cube with {1, 2, 3,…,12}

I did the following problem: a) Is it possible to label the edges of a cube by $1, 2, \cdots 12$ (using each number only once) so that at each vertex, the labels of the edges leaving that vertex ...
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1answer
51 views

Determine all positive rational numbers $r \neq 1$ such that $r^{\frac{1}{r-1}}$ is rational?

Here's what I've got so far: Let $r = \frac{a}{b}$, where $a$ and $b$ are integers. We then have $$r^{\frac{1}{r-1}} = \frac{a^{\frac{b}{a-b}}}{b^{\frac{b}{a-b}}}$$ Clearly, $a-b=1$ and $a-b=-1$ ...
1
vote
1answer
455 views

Powerball Mass Quickpick Odds

The odds of picking the right powerball numbers for the jackpot are 1:292,201,338. Right now, because the powerball has reached 1.4 billion dollars, many people are claiming that you could buy every ...
1
vote
1answer
42 views

Determine what when multiplied with $180$ gives a perfect cube

Recently, at a math competition, I was given the following question: Determine the smallest number that gives a perfect cube when multiplied by $180$ . I had thirty seconds to solve this question and ...
4
votes
1answer
88 views

Match off points into $N$ red/blue pairs with straight lines connecting pairs, so that none of lines we draw intersect

Suppose we are given $2N$ points in the plane (we may assume that no $3$ are collinear). Assume that $N$ of these points are colored red, and $N$ points are colored blue. Can we match off the points ...