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

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0
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1answer
32 views

Find the solution to the system (not linear)

Find all $(x, y, z) \in \mathbb{R^3}$ satisfying: $$x^2 + 4y^2 = 4xz \tag1$$ $$y^2 + 4z^2 = 4xy \tag2$$ $$z^2 + 4x^2 = 4yz \tag3$$ This is a very difficult problem. I added $-4(1) + (3)$ to ...
0
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1answer
30 views

How to get real value after discount.

Suppose my item real value is 100. and i have given discount 10 % to my customer. Now the changed value of item id 90. If i set value 110 and give 10% discount then i got result 99. but i need result ...
2
votes
1answer
34 views

Would the powerset of $\mathbb{Z}$ also not denumerable?

Would the powerset of $\mathbb{Z}$ also be not denumerable?, Since Cantor's theorem says that the $\mathbb{N}$ is denumberable but the powerset of $\mathbb{N}$ is not denumberable because there does ...
0
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2answers
98 views

How would you prove that $2^{n-1} > n!$?

How do i prove that $2^{n-1} < n!$ for all $n \ge 1$ This is my proof: Base case: Let n=1 then $2^{1-1} =1$ is the same on the right side so it holds Inductive step: let $k \le 1$ we assume that ...
4
votes
0answers
110 views

An identity satisfying the divisors of a positive integer

I saw a hard competition problem with long and ugly proof in http://solmu.math.helsinki.fi/olympia/valmennus/2013/vt2013_12var.pdf ? The question is from Australian mathematical olympiad 1985. Is ...
1
vote
2answers
94 views

Posed in regional mathematics Olympiad 1995

Call a positive integer $n$ good if there are $n$ integers, positive or negative, and not necessarily distinct, such that their sum and product are both equal to $n$ For example, $8$ is good since ...
3
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2answers
74 views

2014 Putnam A1 Prime number factorial help

Question: Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number. ...
1
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1answer
31 views

Taylor Series for $e^x(x^2 -x + 1)$

Find the Taylor Series for $e^x(x^2 -x + 1)$ about $x=0$. More importantly, find the COEFFICIENT (for nonzero terms) of the taylor series. The answer says: $$e^x(x^2 -x + 1) = 1 + ...
0
votes
1answer
56 views

How to tell if a function and a composite function is onto or one to one

For each of the following, f : A → B, g : B → C. Which one are true and which ones are false? So far i have, f is onto but g ◦ f is not onto. (False) f is 1-1 but g ◦ f is not 1-1. (False) g is onto ...
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2answers
37 views

How to prove if $A \times C \subseteq B \times D \implies A \subseteq B$

My proof Given $(x,y) \in A \times C \implies x \in A$ and $y \in C$ since $A \times C \subseteq B \times D$ then $(x,y) \in B \times D$ then $x \in B$ and $y \in D$ since $x \in A$ and $x \in B$ ...
0
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2answers
46 views

How to prove intersections and subsets of sets

Simple proofs for these are pretty straight forward such as proving if two sets are equal then they are subsets of each other or if you want to show one set is a subset of the other just show that ...
0
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0answers
31 views

How can you prove that a collection of union set is equal to the set of natural numbers?

For example $\bigcup_{n\in \mathbb{N}}A_n=\mathbb{N}$ My proof, to prove that two sets are equal i must show that they are subsets of each other. I understand how to show $\bigcup_{n\in ...
2
votes
1answer
167 views

Book recommendation to prepare for geometry in the International Mathematical Olympiad

What is the best book for preparation for "Geometry" for IMO? I've been searching one for past many weeks, got loads of names but couldn't finalize one, please help me.
37
votes
2answers
1k views

Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$
5
votes
2answers
96 views

How to prove whether the equation set has a unique solution?

\begin{eqnarray} \begin{cases} \sin A \sin C-(\sin B)^2=0 \cr AC-B^2=0 \cr A+B+C-\pi=0 \cr A>0,B>0,C>0 \end{cases} \end{eqnarray} How to prove whether the equation set has a unique solution ...
2
votes
3answers
122 views

Putnam 2006 B1 Problem

Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle, and find its area. Yikes. Without ...
5
votes
1answer
60 views

Length from tangent circles

A circle $Γ_1$ of radius $25$ is externally tangent to a circle $Γ_2$ of radius $16$ at $C$. Let $AB$ be a common direct tangent, so that $A$ lies on $Γ_1$ and $B$ lies on $Γ_2$. Draw the tangent to ...
0
votes
1answer
83 views

How to prove that there are no integers a,b such that $b^2=4a+2$

How to prove that there are no integers a,b such that $b^2=4a+2$ This seems like a very simple prof but when i tried to work through it i keep on hitting walls. I tried to prove this by ...
4
votes
1answer
28 views

Sum of numbers in a grouping question

A person grouped numbers in the following way: $$\left \{ 1 \right \},\left \{ 3,5 \right \},\left \{ 7,9,11 \right \},\left \{ 13,15,17,19 \right \},...$$ What is the sum of the numbers in the $9$th ...
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votes
2answers
74 views

Volume and surface area of a drilled out cube (BM01 2010/11 Contest Question 2)

Let $s$ be an integer greater than $6$. A solid cube of side $s$ has a square hole of side $x < 6$ drilled directly through from one face to the opposite face (so the drill removes a cuboid). The ...
10
votes
2answers
638 views

1985 Putnam A1 Solution

I dont see what they mean by bijection of triples of subsets of $\{1, \ldots, 10\}$ and the $10\times3$ matrix with $0, 1$ entries? How is that created?
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0answers
60 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 ...
3
votes
2answers
42 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 ...
15
votes
1answer
527 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 ...
1
vote
1answer
29 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 ...
0
votes
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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vote
2answers
112 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
24 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
31 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
votes
0answers
68 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 ...
2
votes
1answer
95 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. ...
1
vote
2answers
74 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 ...
0
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2answers
42 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 ...
0
votes
1answer
25 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 ...
2
votes
1answer
39 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 ...
0
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1answer
35 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
75 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
votes
1answer
182 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
votes
2answers
79 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
272 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
114 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 ...
3
votes
2answers
77 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 ...
0
votes
1answer
54 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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votes
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 ...
0
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1answer
27 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
65 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
58 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 = ...
1
vote
1answer
34 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 ...
4
votes
1answer
89 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 $3$.
3
votes
5answers
38 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 ...