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

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56 views

Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions

Let $p$ be a prime number and $a, b, c$ integers such that $a$ and $b$ are not divisible by $p$. Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions Well, this problem can be ...
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2answers
40 views

Quadratic Residues $\pmod {2^n}$

I'd imagine this is a duplicate question, but I can't find it: How many quadratic residues are there $\pmod{2^n}$. I tried small $n$: $n=1: 2, n=2:2, n=3: 3, n=4: 4, n=5: $not 5: 0, 1, 4, 9, 16, 25 ...
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1answer
516 views

2005 Putnam B1: Find a Polynomial

Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$ (Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$) I ...
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1answer
28 views

Angle of $x$ dependant on a kite in a rhombus

$ABCD$ is a rhombus. $E$ is the midpoint of $BC$. If $BAE = x$, then $FECG$ is a kite if $x$ is equal to what? Can someone please help me with this question? I just want a hint to help me because I ...
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1answer
33 views

How to prove that A $\subseteq$B $\implies$ |A|$\le$|B|?

How to prove that A $\subseteq$B $\implies$ |A|$\le$|B|? I know that for |A|$\le$|B| there has to be a function f:A $\mapsto$B which is an injective function. But i get stuck because the sets A and B ...
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2answers
105 views

How many perfect squares?

How many perfect squares are there between $2013$ and $3602$ WITHOUT CALCULATING, can I get some hints on how this can be done. the number will take the form $\sqrt{x}$ is an integer?
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1answer
22 views

How to prove by induction with a set of equivalence sets?

For example Prove by induction that the operation of raising to the power m$\in$ $\mathbb{N}$ is well defined in $\mathbb{Z}_n$ $\forall$m$\in$ $\mathbb{N}$ $\forall$[x]$\in$ ...
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2answers
30 views

how to prove that $x^2 + y^2 =1$ is injective and surjective depending on the restrictions?

Suppose we have $S=\{(x,y) \in [-1,1]\times[0,1]: x^2 + y^2 = 1\}$ I know this is a function since the domain(s)= $[-1,1]$ and I know this should be surjective and injective since the restriction ...
2
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0answers
67 views

Solve: $xyz$ divides $(x+y+z)^2$ [duplicate]

Find the number of positive integers less than $1000$ of the form $$\frac{(x+y+z)^2}{xyz}$$ where $x,y,z$ are positive integers. Usually I have lots of ideas on how to solve a problem, and I ...
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1answer
84 views

BM01 2011/12 Question 6 Geometry Problem

Let $ABC$ be an acute-angled triangle. The feet of the altitudes from $A,B$ and $C$ are $D, E$ and $F$ respectively. Prove that $DE +DF \le BC$ and determine the triangles for which equality holds. ...
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2answers
68 views

Prove that the midpoint lies on the right angle bisector.

2 equal circles are located inside a right-angled triangle so that they touch each other and each circle also touches one leg and the hypotenuse. Let M and N be points of tangency of the 2 circles ...
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2answers
38 views

AMC 12A, problem with days

In year N, the $300th$ day of the year is a Tuesday. In year $N+1$, the $200th$ day is also a Tuesday. On what day of the week did the $100th$ day of year $N - 1$ occur? (2000 AMC 10 #25) The ...
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1answer
19 views

Converting a complicated congruence equation

From: $$5991x \equiv -289 \pmod{2014}$$ I saw people converted this to: $$3x \equiv 17 \pmod{2014}$$ But how? My attempt: $$5991x \equiv -289 \pmod{2014} \equiv 1725$$ $$1997x \equiv 575 ...
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1answer
38 views

Definition of Probability

An example question is: John and Jayne each choose a number (not necessarily different) from 1 to 10 inclusive. What is the probability that they each pick a number greater than $7$? The obvious ...
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1answer
33 views

2 Forms of Probability?

Probability is defined as: $$P(A) = \frac{\text{Chance of Objective}}{\text{Number of Possible Outcomes}}$$ But some books defined it as: $$P(A) = \frac{\text{Number of ways ...
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1answer
57 views

Variant on classic geometric probability: 3 people meeting during the day

Zeus, Athena, and Poseidon arrive at Mount Olympus at a random time between 12:00 pm and 12:00 am, and stay for 3 hours. All three hours does not need to fall within 12 pm to 12 am. If any of the 2 ...
8
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1answer
180 views

Points “seeing” each other in a loop

For two points $P,Q$ with integer coordinates in $2$ dimensions, we say that $P$ "sees" $Q$ iff the segment $PQ$ contains no other points with integer coordinates. Do there exist points ...
2
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2answers
74 views

Odd number of students in odd number of classes

In a school there are an odd number of classes, and each class has an odd number of students. We want to choose a school council consisting of one student from each class. Prove that the following are ...
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3answers
254 views

$\lim_{x\to0}\frac{e^x-1-x}{x^2}$ using only rules of algebra of limits.

I would like to solve that limit solved using only rules of algebra of limits. $$\lim_{x\to0}\frac{e^x-1-x}{x^2}$$ All the answers in How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without ...
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1answer
98 views

Difficult triangle/circle geometry problem

Let $AB$ be a segment of length $12$. $ω1$ is a circle centered at $A$ with radius $13$, $ω2$ is a circle centered at $B$ with radius $7$. Let $l$ be a common tangent of the circles and $l$ intersects ...
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2answers
70 views

Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$

Let $a_1, a_2, \ldots, a_n \in \mathbb{R}$. Prove that the following inequality holds: $$\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$$ I first tried to restrict the ...
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1answer
40 views

How to find a relation when given the distinct equivalence classes?

For example I am not sure how to approach this type of problem. I know that the equivalence classes partition $A$. Suppose $[a]= \{1,4,5\}$, $[b]=\{2,6\}$ and $[c]= \{3\}$. $[a]\bigcap[b]= ...
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1answer
29 views

If $n \in \mathbb{N}$ , not divisible by 3 show there $\exists t \implies 3^t < n < 3^{t+1}$

If $n \in \mathbb{N}$ , not divisible by 3 show there $\exists t \implies 3^t < n < 3^{t+1}$ By the division algorithm: $$n = 3a + r \implies 0 < r < 3$$ For some $a$ But I cannot do ...
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1answer
24 views

How to show that a relation is an equivalence relation given a defined relation

I understand for the most part the conceptual aspects of an equivalence relation. A relation is considered a equivalence relation if it satisfies reflexive, symmetric and transitive properties but Im ...
2
votes
1answer
64 views

2014A AMC solution question

From: AMC 10 Q25 Solution I get everything besides the last part. How in the world does he get: $$3k + 2(867 - k) = 2013$$ I don't understand how he got this? What does this mean? Literally ...
2
votes
2answers
56 views

How to use Induction with Sequences?

I have posted this similar question here, but with no hopes. I would just like to know: Most of the solution I have no issue with. Look at where they say: "Choose a representation $(n - 3^m)/2 = ...
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1answer
32 views

How many combinations can be made for the fraction?

Rational numbers, $a, b$ are chosen from the set of rational numbers. The condition is: $a, b \in [0, 2)$. $a, b$ can be written as: $a, b = \frac{n}{d}$, where $n, d$ are integers with: $1 \le d \le ...
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1answer
82 views

Find all integral solutions of $y^2=x^3+7$

Find all integral solutions of $y^2 = x^3+7$. I tried to use many different moduli, but it never works. With modulo $9$, you can get $x$ is divisible by $3$.
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votes
5answers
35 views

Prove that $m$ is an integer

Suppose $n$ is a odd integer. It satisfies: $$3^{s} < n < 3^{s+1}$$ For some integer $s \ge 0.$ Show that: $$m = \frac{n - 3^{s}}{2}$$ Is an integer. So, $$2m = n - 3^{s}$$ But that wont ...
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6answers
99 views

If $n$ is an odd integer prove that $n - 2^k$ is divisible by $3$

So let $n$ be a odd integer. Show that $n - 2^k$ is divisible by $3$ if $k$ is SOME SPECIFIC positive integer. $k \ge 0$. So there only has to exist one. For example: $$7 - 2^2 = 3$$ is divisible by ...
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1answer
79 views

Putnam 2005 A1 Solution [duplicate]

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. (For example, $23=9+8+6.)$ ...
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1answer
67 views

Sum of the roots of an equation.

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. I used ...
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3answers
52 views

AIME 2013 Solutions (divisiblity)

Problem 2 Find the number of five-digit positive integers, $n$, that satisfy the following conditions: (a) the number $n$ is divisible by $5,$ (b) the first and last digits of $n$ are equal, and (c) ...
13
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1answer
141 views

Positive integer solutions of $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}=\frac{a_1+a_2+a_3+\cdots+a_n}{2}$

Find all ordered tuples of positive integers $(a_1,a_2,a_3,\ldots,a_n)$ such that $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}=\frac{a_1+a_2+a_3+\cdots+a_n}{2}$ The only ...
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1answer
47 views

Prove that at least on of the numbers is positive

Prove that for $a,b,c \in \mathbb{R}$ at least of the the following numbers is non-negative: $$(a+b+c)^2 - 9ab; \quad (a+b+c)^2 - 9ac; \quad (a+b+c)^2-9bc$$ If not all of $a,b,c$ are negative or ...
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1answer
22 views

How do the sets have similar properties?

Where they say: "Assume that we have sets $S_k$ with the desired properties for all $k < n$ (line 4 in solution)" What properties are they talking about? They said: "Let $S_n = \{2a ...
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1answer
63 views

How to prove this form of $n$?

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. From: AOPS Putnam A1 Solution I ...
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1answer
39 views

In how many ways can the rooks be arranged? [duplicate]

In how many ways can 9 black and 9 white rooks be placed on a 6 × 6 chess board, so that no white rook can capture a black one? A rook can capture another piece if it is in the same rank (row) ...
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2answers
33 views

Probability using combinatorics

If Sapphira randomly chooses a 4-digit number (not beginning with zero) what is the probability that all four digits will be distinct? Let $$x = abcd$$ where they are digits. Lets see first how ...
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1answer
49 views

Geometrical Combinatorics About Rectangles

Part of a olympiad problem The answer is $$441 = 21^2$$ I fail to understand why. How do you solve this? I actually dont see why there are $9$ rectangles there either? Can someone give me a hand?
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1answer
30 views

Minimizing certain integral

Define $$F(a)=\int_0^{\pi/2}|\sin x-a\cos x|dx.$$ Find $a$ such that $F(a)$ is minimum. My attempt is to use differentiation under integral sign. Namely, we have $$F(a)=\int_0^{\arctan a}(a\cos ...
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3answers
72 views

If $2^n$ balls are divided into piles, they can always be brought into a single pile by a finite number of operations

$64$ balls are separated into several piles. At each step, one takes two different piles $A$ and $B$, having $p$ and $q$ balls respectively. Suppose $p\ge q$. Then one takes $q$ balls from pile $A$ ...
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1answer
51 views

Hazel chooses a die, rolls it and wins. Find the probability that she chose the biased die.

The probabilities of the scores on a biased dice are shown in the table below : ...
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1answer
162 views

Olympiad inequality (Cauchy/AM-GM sort)

Given $n$ positive numbers $x_1,\ldots,x_n$ ($n\ge 3$) such that the product $x_1x_2\cdots x_n=1$, show that ...
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3answers
79 views

Integral Solutions of $x+y=x^2-xy+y^2$

Find all integral solutions of $x+y=x^2-xy+y^2$ A modulo 2 analysis does not work here, only says cannot both be odd.
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2answers
54 views

Integer Solutions to Equation

Find the integer solutions to $x^2+xy+y^2=x^2y^2$ I have tried doing a modulo 2 analysis, which only says that $x, y$ are congruent modulo 2. But I cannot continue from here.
5
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1answer
310 views

Sum of the squares of numbers

Let $x$ and $y$ be the two numbers so that: $$x^2 + y^2 = A^2$$ $$xy = A^2 + 2A + 2$$ $$xy - x^2 - y^2 = 2A + 2$$ $$\frac{xy - x^2 - y^2}{2} -1 = A$$ So what can I do?
2
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1answer
40 views

Finding polynomial Coefficients

Let $f(x) = x^5 - x^4 + ax^3 + bx^2 + 8x + 4$ The root will make sure, $f(2) = 0$ Which shows: $$2^5 - 2^4 + a2^3 + 4b + 16 + 4 = 0$$ $$16 + 8a + 4b + 20 = 0 \implies 8a + 4b = -36 \implies 2a + ...
2
votes
1answer
50 views

How to solve this sum problem?

For the first radical section. $$\sqrt{1\times 2\times 3\times 4 + 1} - 1 = 1 + 3 + 1 - 1= 5 - 1 = 4$$ The second radical section. $$(\sqrt{2\times 3\times 4\times 5 + 1}) = 4 = 4 + 6 + 1 - 4 = ...
0
votes
1answer
23 views

Modulus after X concatenation

Given a number $N$ we can form another number $Y$ by concatenating $N$, $X$ times towards right. How to compute $Y \mod \space M$ efficiently? For example: if $N = 456$ and $X = 3$ then $Y = ...