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

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0
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40 views

Show that there exists no integer $x$ such that $3x$ is congruent to 5 (modulo 6)

So far my approach was to rewrite the congruency to $5-3x=6t$ for some integer $t$. My problem is I get stuck in trying to show how $5-3x$ is never divisible by $6$.
0
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2answers
28 views

Probability of getting 6 letters right

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
52 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
23 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 ...
1
vote
0answers
22 views

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.
0
votes
1answer
48 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
29 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
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1answer
29 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
47 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
31 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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vote
0answers
27 views

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 ...
0
votes
3answers
28 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
votes
1answer
63 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
80 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 ...
1
vote
1answer
63 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
77 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
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 $n$?
2
votes
2answers
55 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 ...
3
votes
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
31 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
0answers
110 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
392 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
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
34 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
69 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
35 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 $x=0$ ??
-10
votes
0answers
29 views

uva 12236 - In a Crazy City immediately solution please [on hold]

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, ...
8
votes
1answer
143 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 ...
2
votes
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...
2
votes
1answer
74 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 ...
1
vote
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 ...
0
votes
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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0answers
19 views

What quadratic function is created? [closed]

Directrix of $y = 2$ and a focus of $(3, -4)$ What quadratic equation is created? $f(x) = \frac{1}{12}(x - 3)^2 - 1$ $f(x) = - \frac{1}{6} (x + 3)^2 + 1$ $f(x) = \frac{1}{6}(x - 3)^2 + 1$ ...
3
votes
1answer
91 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 ...
2
votes
1answer
58 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 ...
2
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0answers
57 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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1answer
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 ...
0
votes
0answers
25 views

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) = ...
1
vote
0answers
27 views

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 ...
0
votes
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
votes
1answer
90 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) ...
4
votes
2answers
53 views

Math Contest Question with Polynomials

Prove that there does not exist a polynomial f(x) with integer coefficients for which f(2008) = 0 and f(2010) = 1867. This is a question from CMOQR (Qualifier for Canadian Math Olympiad , not the ...
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vote
1answer
47 views

Graph Theory Contest Maths

I have never covered Graph Theory so I've been put into a bit of a quandary over how to do these two questions (I am assuming the second is graph theory, if not I will edit it out of the question). ...
0
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1answer
64 views

Show the integral $\lim_{B\rightarrow\infty}\int_0^B \sin(x)\sin(x^2)\,dx$ converges

Show the integral $$\lim_{B\rightarrow\infty}\int_0^B \sin(x)\sin(x^2)\,dx$$ converges. I guess we should use the equality $$\sin(x)\sin(x^2)=\dfrac{1}{2}[-\cos(x+x^2)+\cos(x-x^2)],$$ so we have ...
8
votes
1answer
124 views

Rational matrix having roots of every degree

As the result of another question, now deleted, I am interested in the following problem. Problem. Let $A\in M_n(\mathbb Q)$ be an invertible matrix with the property that the equation $X^k=A$ has ...
1
vote
1answer
64 views

How prove this idenity this $mv-3nu=m-3u$ with unit circle

Assmue the $m,n,u,v$ be real numbers,and such $$m^2+n^2=1,u^2+v^2=1,nv>0,m>0,u>0$$ and $$5mu=3(1-nv)$$ show that $$mv-3nu=m-3u$$ Following is My methods: let ...
3
votes
2answers
39 views

BMO preparatory question

Q) Let $3\leq n$ be an odd integer and let $a_1,a_2,...a_n$ be fixed positive integers. For each of the $n!$ permutations $\pi=(\pi_1,\pi_2,...,\pi_n)$ of $(1,2,...,n)$, define $$f(\pi) = a_1\pi_1 + ...
4
votes
1answer
163 views

Problem from Iran Olympiad?

Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer. I gussed the answer is no. Let $\overline{a_n ,...,a_1 ,a_0}$ be ...
3
votes
2answers
81 views

How prove find this value $|AD|+|DF|+|FA|=2$

Question: if $ADB$ and $ACE$ are straight lines with $D,E$ and $B,C$ intersecting at $F$. if $$|AB|=|AC|=1,|AD|+|DE|+|EA|=4$$ show that: $$|AD|+|DF|+|FA|=2$$ I have read this ...