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

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0
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1answer
102 views

If $\Gamma\cup\{\sim(A\land B)\}$ is consistent, what can be said about $\Gamma\cup\{\sim(A\lor B)\},\Gamma\cup\{\sim A\},\Gamma\cup\{\sim B\}$?

The following question arose in the NOI of India Section taken a few days back: Let $\Gamma$ be a set of predicate formulas, and let $A, B$ be two predicate formulas; if the theory $\Gamma \cup \{ ...
1
vote
2answers
317 views

How many 6 digit numbers can you write?

How many six-digit positive integers can you write, if each number must have strictly increasing digits from left to right I thought it was 6! Butt hat is wrong. How do you do these sort of problems ...
0
votes
1answer
81 views

AoPS putnam 2003 A1 solution issue

I am having problems transferring and copying the data here. Take a look at AOPS KENT MERRYFIELD ANSWER. I don't understand. He gets to: $$a(k-r) + (a+1)r = n$$ But how does this show that there ...
0
votes
2answers
56 views

Nice proof of a polynomial root $x \in [0, 1]$

Prove that if $$\sum_{k=0}^{n} \frac{a_k}{k+1} = 0$$ then $$\sum_{k=0}^{n} a_k\cdot x^n = 0$$ for some $x$ in $[0,1]$. (original image) So: $$\sum_{k=0}^{n} \frac{a_k}{k+1} = 0$$ We ...
3
votes
2answers
207 views

Determine whether $712! + 1$ is a prime number or not

Let $n = 712! + 1$ If $n$ was a prime number then, by Wilson's theorem: $ (712!)! \equiv -1 \pmod{712}$ The double factorial makes it seriously more difficult... But We can require: $$712!! + 1 ...
4
votes
1answer
688 views

1000 numbers on a blackboard

The numbers $1, 2, …,1000$ are written on a blackboard, in some order. Between every pair of consecutive terms, the absolute difference of the two terms is written between them, and then all the ...
0
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1answer
42 views

For how many distinct triangles ABC, with AB = $2011$, are both $\cos(2\angle{A} + 3\angle{C})$ and $\sin(2\angle{B} + \angle{C})$ integers?

I came across a tough geometry question: For how many distinct triangles $\text{ABC}$, with $\text{AB}$ = $2011$, are both $\cos(2\angle{A} + 3\angle{C})$ and $\sin(2\angle{B} + \angle{C})$ ...
1
vote
1answer
41 views

The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of k?

I was trying to solve the following question: The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of ...
3
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2answers
56 views

How many positive integers less than $2011$ cannot be expressed in the form $4a + 5b$, where $a$ and $b$ are positive integers?

How would I solve the following question: How many positive integers less than $2011$ cannot be expressed in the form $4a + 5b$, where $a$ and $b$ are positive integers? I was trying to apply ...
0
votes
1answer
63 views

Proof of an alternate form of the triangle inequality

Since it is all positive squaring does not change anything. So: $$ (a_1^2 + \cdots + a_n^2) + 2\sqrt{(a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots b_n^2)} + (b_1^2 + \cdots + b_n^2) \ge (a_1 + b_1)^2 + ...
0
votes
1answer
49 views

Trying to simplify the expression

Can anybody simplify it? Show me the way of simplification. The expression is as follows: $$F(x) = 1 *(1!+x)+2*(2!+x)+ ..+x*(x!+x)$$ for a positive integer $x$ I've tried but nothing got.
1
vote
1answer
44 views

Finding the rank of a particular number in a sequence of the sum of numbers and their highest prime factor

This question comes from a maths contest (infer no calculators or other electronic calculating aids) for 14-16 year olds (infer no use of complicated theorems, but those accessible to high-school ...
5
votes
1answer
69 views

prove the inequality with fractional parts

$$ \frac{n^k-n}{2} \leq \left\{\sqrt[k]{1}\right\} + \left\{\sqrt[k]{2}\right\} + \dots + \left\{\sqrt[k]{n^k}\right\} \leq \frac{n^k-1}{2} $$ how it can be proven?
3
votes
2answers
56 views

Numbers $a$ that are the sum of the fractional parts $\{x^2\} + \{x\}$ for some $x$

Are there infinitely many rational numbers $a\in\mathbb{Q}$ with the following property: $\{x^2\}+\{x\}=a$ for infinitely many $x\in\mathbb{Q}^+$
0
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1answer
56 views

Find all functions satisfying $(1+y)\,f(x) - (1+x)\,f(y) = y \, f(x/y) - x \, f(y/x)$

Find all functions which satisfy: $$(1+y)\,f(x) - (1+x)\,f(y) = y \, f(x/y) - x \, f(y/x)$$ for all real, $x,y \ne 0$ and which takes the values $f(1) = 32$ and $f(-1) = -4$ I am not sure, which ...
-3
votes
2answers
50 views

If a machine takes 3min to process 1 byte, how many machines are required to process 1000 bytes in 30min?

We have a machine that takes 3 minutes to process a byte. Now if I send 1000 bytes together the machine will take 3000 minutes to process them serially. If we want to do that in 3 minutes only we need ...
1
vote
1answer
292 views

Prove that if x is irrational, then sqrt(x) is irrational.

I believe the contrapositive method should be correct but i get, The contrapositive of this statement should be, (If $\sqrt{x}$ is rational, then $x$ is rational) Then I end up with ...
2
votes
1answer
103 views

Putnam A4 2010, proving an expression is not prime.

Prove that each positive integer $n$: $ x = \displaystyle 10^{10^{10^n}}+10^{10^n}+10^n-1 $ is not prime. This seems like a very difficult problem, any ideas at all? I would like to use modular ...
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1answer
77 views

what is required for a person to do well on imo

What kind of skill is required to solve IMO or Putnam sort of problems. Does one have to be a genius or just learn some tricks.
3
votes
2answers
58 views

A grandmother is giving out apples to her grandchildren.

A grandmother has 7 grandchildren, and 14 apples to give. How many ways can she give apples to her grandchildren so that each grandchild gets aT LEAST one? (but she has to get rid of hers). This ...
0
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1answer
60 views

Consider the set $Q=\{p+q \sqrt2 : p,q \in\Bbb Q\}$. Prove that if $a\in Q\setminus\{0\}$ then $1/a\in Q$

Given (For all $a,b\in Q$, $a+b\in Q$ and $ab\in Q$) This was a two part question. Part a) is to prove that $Q$ is closed under addition and multiplication. Part b) is prove that if $a\in Q$ and ...
2
votes
0answers
61 views

Number theory - equation

I´m preparing for math contests and found the following problem from this pdf: http://www.fmf.uni-lj.si/~lavric/Santos%20-%20Number%20Theory%20for%20Mathematical%20Contests.pdf Find all integers $a, ...
0
votes
1answer
74 views

In how many ways can we distribute 6 identical pears?

In how many ways can we distribute 6 identical pears between 3 children so that each child receives at least one pear? I am not too sure. I thought, 6 ways to distribute to first, 5 ways to second, ...
4
votes
1answer
67 views

Does performance in math competitions accurately reflect natural aptitude in mathematics? [closed]

Many great and respected mathematicians have won accolades in math (ex: IMO), does that necessarily mean that these competitions reflect one's potential to be a great mathematician?
7
votes
1answer
112 views

Using two coins to select a person fairly.

Good evening, I would like to know if the solution to this problem, I know it can be solved because it is from a Hungarian Olympiad. The problem is as follows: You need to fairly select a person ...
0
votes
2answers
37 views

Given $A \subseteq \mathbf{Z}$ and $x\in \mathbf{Z}$, we say that $x$ is $A$-mirrored if and only if $−x\in A$. We also define…

Sorry if this question seems kind of long but I am confused for part C. My proof for part C that $M_a$ is closed under addition is as follows: The set $M_a$ is closed. Let $x$ be in $M_a$ and ...
1
vote
1answer
32 views

Number of paths in 3D coordinates

A cute problem which is an extension of a well-known counting problem: Find the number of paths of length $12$ from $(0,0,0)$ to $(4,4,4)$ passing through adjacent lattice points (for two ajacent ...
3
votes
1answer
100 views

Functional equation defined over non-negative real numbers

I'm new to this forum and I don't know how to write mathematical symbols. I have the following functional equation: $f$ defined on $[0, +\infty)$ with values in $[0, +\infty)$ $f$ is bijective and ...
11
votes
5answers
1k views

How to derive this infinite product formula?

Show: $$\prod_{n=0}^{\infty}\left(1 + x^{2^n}\right) = \frac{1}{1-x}$$ I tried numerous things, multiplying by $x$, dividing, but none of that worked. Also, I realized that: $$\prod_{n=0}^{\infty} ...
3
votes
1answer
103 views

Length of the non-periodic portion of the decimal expansion of $\frac 1n$

The following question was asked in the Indian National Mathematics Olympiad (INMO) 2015. For any natural number $n>1$,write the infinite decimal expansion of $\frac 1n$. Determine the length ...
1
vote
4answers
257 views

Reversing the digits with a subtraction [closed]

How many 3-digits numbers possess the following property: After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.
0
votes
1answer
51 views

Solving the Sequence of this question on Putnam Exam

Problem: Solution: Solution for 2003 A1 Putnam $ka_1 = a_1 + a_1 ... a_1 \le n \le a_1 + (a_1 + 1) + (a_1 + 1) ... (a_1 + 1)$ $= ka_1 + k - 1$ I know these then: What should I do next? Without ...
9
votes
3answers
727 views

How many 0's are in the end of this expansion?

How many $0's$ are in the end of: $$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$ The answer is supposed to be $1100$ but I have absolutely NO clue how to get there. Any advice?
2
votes
1answer
79 views

Prove that $ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2\ge\left( \frac{x_1+\dots+x_n}{n} \right)\left(\frac{y_1+\dots+y_n}{n} \right). $

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that ...
4
votes
0answers
63 views

Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...
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votes
2answers
100 views

Arranging identical balls in a circle

In how many ways can 4 identical red balls and two identical white balls be arranged in a circle? This is an elementary problem, but many tries have not yet yielded results. I tried by taking the ...
0
votes
1answer
98 views

How does this person solve the Putnam problem?

Consider this: 2003 A1 Putnam Solution. I am only looking at A1 for Putnam 2003. The problem is here: Problem A1 2003 I would like to proceed step-by-step: I understand $ka_1 = a_1 + a_1 + ... ...
6
votes
2answers
249 views

Distance between four points

I have four points as shown in this figure: I want to calculate one vector for all these points. So, what would be the correct way: 1) I take the vector between $A-B, B-C, C-D$ and add them $(A-B ...
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votes
4answers
78 views

2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...
0
votes
1answer
25 views

How to use totient function here?

I have asked this before, but I had no idea how to use Totient, now I do here is the questions: How many positive integers $< 2013$ cannot be divided by $2, 3, 5$ ?? An advice given was find ...
1
vote
0answers
25 views

Find all points on the line 9x-21y=6

For this equation we are suppose to use the Euclidean Algorithm. But I run into a problem For the GCD (9,-21)= i tried 9=(-21)(0)+9 -21=9(3)+6 9=6(1)+3 6=3(2) +0 which gives a gcd of 3 and the ...
49
votes
10answers
2k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
0
votes
2answers
75 views

Probability of getting 6 letters right [duplicate]

A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 ...
2
votes
2answers
87 views

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$?

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$? This was an olympiad question. I thought of writing a number $x \le 2012$ in the form: $x = 2^{a}3^{b}4^{c}5^{d} = ...
0
votes
1answer
55 views

CHKMO 2015 and cubic equations

Let $a,b,c$ be distinct real numbers. If the equations $E_1: ax^3+bx+c=0, E_2: bx^3+cx+a=0$ and $E_3: cx^3+ax+b=0$ have a common root, prove that at least one of these equations has three real ...
0
votes
1answer
132 views

How many 10 digit numbers are there so the sum of the digits is $2$?

How many 10 digit numbers are there so the sum of the digits is $2$? $abcdefghij$ is the 10 digit number. By default, $a=1$ is a must. $= 1bcdefghij$ Now we need: $bcdefghij = 1$ How can I solve ...
0
votes
2answers
41 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
0
votes
1answer
30 views

Sum of divisor powers?

A given number is divisible by 2, 3, and 5, and has altogether 2013 divisors. The smallest such number is $2^N \cdot 3^M \cdot 5^p$ where $N + M + P=$? I would $N + M + P = 2012$ because by a ...
0
votes
2answers
51 views

Smallest integer $x$ for which 10 divides $2^{2013} - x$

Find the smallest integer $x$ for which 10 divides $2^{2013} - x$ Obviously, $x \equiv 2^{2013} \pmod{10}$ But how can I reduce $x$?
0
votes
1answer
52 views

Angle quadrisection in a triangle

In triangle ABC, AB=84, BC=112, and AC=98. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is ...