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

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1answer
68 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
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2answers
57 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
55 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
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0answers
55 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
47 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
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1answer
64 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
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1answer
108 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
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2answers
33 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
30 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
97 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 ...
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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
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1answer
99 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 ...
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4answers
231 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
48 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
722 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
77 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
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0answers
55 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 ...
0
votes
2answers
74 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
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1answer
92 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
236 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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4answers
77 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
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1answer
23 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 ...
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0answers
24 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 ...
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10answers
1k 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
60 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
74 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
41 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
87 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
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2answers
37 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
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2answers
50 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
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1answer
48 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 ...
0
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3answers
33 views

Find the Inverse Modulus using Euclid's algorithm

I asked this before, but unfortunately, I didnt know the methods, nor was the questions phrased properly. Find the inverse of $4258 \pmod{147}$ Using Euclidean Extended Algorithm. Begin By Stating ...
0
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1answer
93 views

Book recommendation for Putnam/Olympiads

I have been concentrating on olympiad questions, and PUTNAM exams, Putnam is my main focus. Can you suggest a book from one of these: Problem Solving Strategies By Arthur Engel Putnam and Beyond by ...
2
votes
2answers
103 views

How do I solve this Olympiad question with floor functions?

Emmy is playing with a calculator. She enters an integer, and takes its square root. Then she repeats the process with the integer part of the answer. After the third repetition, the integer part ...
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vote
1answer
85 views

Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$

Four different positive integers $a, b, c$, and $d$ are such that $a^2 + b^2 = c^2 + d^2$ What is the smallest possible value of $abcd$? I just need a few hints, nothing else. How should I begin? ...
4
votes
2answers
86 views

$A\subseteq \{1,2,3, \ldots 2000\}, $ and for any $a,b\in A,\; |a-b|$ is not equal to 4 or 7,

$A\subseteq \{1,2,3,\ldots2000\}$, and for any $a,b\in A,$ $|a-b|$ is not equal to 4 or 7. Then, at most, how many element does $A$ contain? For general condition,$|a-b|$ is not equal to $i$ or $j, ...
0
votes
3answers
26 views

Solving for mod indirectly

How many positive integers $n$ exist such that $\frac{680}{n}$ is an integer? So, this is quite obvious, $680 \equiv 0 \pmod{n}$ How should I solve for $n$? There will be multiple $n$?
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2answers
70 views

A game where starting with 3 boxes, with 10 balls in each, the goal is to remove as many balls as possible following the rules

This is a Norwegian olympiad problem: Peter has three boxes, with ten balls in each. He plays a game where the goal is to end up with as few balls as possible in the boxes. The boxes are each ...
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2answers
56 views

Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$

For $x, y \ge 0$ prove that: $$x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$$ What I think would apply is the AM-GM Inequality, so first, $$(x^2 + y^2 + 1)^2 \le (x^3 + y + 1)(y^3 + x + ...
0
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1answer
47 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
3
votes
1answer
228 views

Solve an inequality using Cauchy-Schwarz Inequality

Le $a,b,c,d \in \mathbb{R^{+}}$. Using Cauchy-Schwarz Inequality prove that the following inequality holds: $$\frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \ge \frac{1}{\frac 1a + \frac 1b} + ...
4
votes
2answers
395 views

Where can the knight be?

The answer is 33. I get $24$. Because of $8 \cdot 3 = 24$? How can I do this using combinatorics?
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2answers
50 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
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1answer
42 views

How many possible paths?

The answer is $32$. Its supposed to be $2^5$ but I do not see how you get that? The way I see it, there are $5$ ways to go up and $5$ ways to go right, total ways = $5x5= 25$
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vote
2answers
75 views

We write all the positive integers run together as follows: $123456789101112131415 . . .$

We write all the positive integers run together as follows: $123456789101112131415 . . .$ What three digit number begins at the $2014th$ digit? I was thinking number theory here. Modulus. Can ...
0
votes
2answers
37 views

Sum of the coefficients of the expansion

Find the sum of the coefficients of the expansion: $$\frac{(1+x)\cdot(2+x^2)\cdot(3+x^3)...(103 + x^{103})}{103!}$$ The answer says let $x=1$, is this the way to go? Why not let $x=0$ ??
11
votes
1answer
354 views

A set of integers whose elements all divide $2015^{200}$ but do not divide each other

Let $S$ be a set of natural numbers,such that each element divides $2015^{200}$ but for no two elements $a$ and $b$, $a|b$. Find the maximum number of elements in $S$ . $2015^{200}=(5\cdot ...
2
votes
2answers
55 views

Find the least number b for divisibility

What is the smallest positive integer $b$ so that 2014 divides $5991b + 289$? I just need hints--I am thinking modular arithmetic? This question was supposed to be solvable in 10 minutes...
2
votes
1answer
80 views

How many ordered triples $(a, b, c)$ exist?

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Breaking it up, $500 = 2^2\cdot5^3$ $abc = 2^2 \cdot 5^3 = 2\cdot 2 \cdot 5 \cdot 5 \cdot ...