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

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Existence of a constant $C$ and a sequence $ x_1, x_2, …\in [0,1] $

a) Prove that for every sequence $x_1, x_2, ...\in [0,1]$ there exists some $C>0$ such that for every positive integer $r$ there are positive integers $m,n$ satisfying $ |n-m|\geq r $ and $ ...
4
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0answers
13 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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1answer
70 views

BMO Round 2 question [on hold]

I need help with this BMO question: The first term $x_1$ of a sequence is $2014$. Each subsequent term of the sequence is defined in terms of the previous term. The iterative formula is ...
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2answers
32 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 ...
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1answer
78 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 + ... ...
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2answers
111 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
61 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 ...
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1answer
18 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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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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2answers
144 views

Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example(s) 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 $f$ ...
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2answers
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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 ...
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2answers
60 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} = ...
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1answer
25 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 ...
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What are some interesting and useful math “tricks” or manipulations that are great for competitions and difficult questions in general [on hold]

An example of a "clever" trick includes componendo dividendo. The "tip" should be applicable for amc 10 type questions and more difficult contest type questions.
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1answer
49 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 ...
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2answers
31 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 ...
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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 ...
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2answers
48 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 $x$?
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1answer
32 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 ...
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Is Problem Solving Strategies by Engel sufficient?

Is a book like, Problem Solving Strategies by Arthur Engel sufficient for the Putnam Exam or should I consult something else? I asked a similar question asking for recommendation, no one discussed ...
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3answers
30 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 ...
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1answer
68 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 ...
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2answers
82 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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1answer
67 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
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2answers
82 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, ...
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3answers
25 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
56 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
53 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 + ...
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1answer
34 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
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1answer
153 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} + ...
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2answers
393 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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43 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
35 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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2answers
71 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 ...
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2answers
36 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 $x=0$ ??
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uva 12236 - In a Crazy City immediately solution please [closed]

I live in a crazy city full of crossings and bidirectional roads connecting them. On most of the days, there will be a celebration in one of the crossings, that's why I call this city crazy. Everyday, ...
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1answer
206 views
+50

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 ...
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2answers
53 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...
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1answer
75 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 ...
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1answer
26 views

2013th powered sequence

Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the 2013th power of the digits of $a_n$. Do there exist distinct positive ...
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1answer
14 views

Cyclic quadrilateral problem

In convex quadrilateral $ABCD$, $AB=2$, $AD=4$, and $2BC+CD=10$. If angle $DAC$ equals angle $DBC$, and the diagonals of $ABCD$ are perpindicular to each other, what is the area of $ABCD$? I have a ...
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1answer
95 views

Prove that for any positve real

Prove that for any positive real numbers $x,y,z$ such that $xyz \geq 1$ $$\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{y^5+z^2+x^2} +\frac{z^5-z^2}{z^5+x^2+y^2} \geq 0.$$ This problem is from the ...
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1answer
60 views

Biggest number of creatures in forest

In crazy forest there are 6 werewolf's,17 unicorns and 55 spiders. Werewolf can eat unicorn and spider,but can't eat another werewolf. Spider can eat unicorn,but can't eat werewolf or another spider ...
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59 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
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1answer
35 views

Permutations and Combinations Olympiad

Suppose that all positive integers which are relatively prime to 105 are arranged in an increasing sequence - a1 , a2 ,a3 ,.... Evaluate a1000.
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34 views

Is there a relation between product of digits of a number and perfect square?

I want to find all numbers less than N whose product of digits is a perfect square. for example if N is equal to 100 then some of possible numbers are 22 (2*2), 49 (4*9=36), 2*8, 8*2 etc. I was ...
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Let $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$

A monic polynomial $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$ One solution is: Let us write $P(x) = ...
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Prove that the maximum in absolute value of any monic real polynomial of n-th degree on [-1, 1] is not less than $\frac{1}{2^{n-1}}$

One solution is: Note that equality holds for a multiple of the n-th Chebyshev polynomial $T_{n}(X)$ The leading coefficient of $T_{n}$ equals $2^{n-1}$, so $C_{n}(X) = \frac{1}{2^{n-1}}T_{n}(X)$ is ...
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1answer
66 views

Given that $\sum\limits_{i=1}^{n}x_i=m+r$, show that $\sum\limits_{i=1}^{n}x_i^2\leq{m+r^2}$

The summation of real numbers $x_i\in (0,1)\, \text{for}\, i=1,\ldots ,n$ is equal to $m+r$, where $m$ is an integer and $r\in [0,1)$. Show that $$\sum_{i=1} ^n x_i^2\leq m+r^2.$$ I pick up this ...
2
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1answer
91 views

Interesting Olympiad Questions.

Rather than through research, I much prefer discovering new maths or interesting theories through doing problems and I also enjoy contest maths which has led me to this question: Which (high school) ...