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

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10
votes
3answers
174 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
6
votes
4answers
154 views

Prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's.

This was taken from an old Brazilian Mathematical Olympiad (1992). As the title says, we're supposed to prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's (in the ...
0
votes
0answers
72 views

math student looking to do better in math competitions.

I am currently in my summer vacations. Next year I will star my undergraduate studies in mathematics. I used to be in mathematics competitions. Last year I got a silver medal in my countries national ...
0
votes
2answers
31 views

Inscribed Hexagon Geometry Contest Problem

The problem was as follows: Regular hexagon $HEXAGN$ is inscribed in the circle $O$, and $R$ is a point on minor arc $HN$ of circle $O$. If $RE=10$ and $RG=8$, then $RN$ can be expressed in the form ...
3
votes
1answer
38 views

Solve for “lucky” numbers

A rational number is called "lucky" if it equals both $a+\frac{b}{c}$ and $a\times\frac{b}{c}$ for some positive integers $a,b,c$. How many lucky numbers are there between $5$ and $10$? Here's what I ...
0
votes
2answers
16 views

Equivalent Planes?

The three planes $x=y$, $y=z$, $x=z$ cut the unit cube $0\le x\le1$, $0\le y\le1$, $0\le z\le1$ into $n$ pieces. Find $n$. My question is this: what does $x=y$, $y=z$, $x=z$ mean? If all of the ...
2
votes
4answers
88 views

Algebraic Solving Contest Problem

The problem is as follows If $x^2+x-1=0$, compute all possible values of $\frac{x^2}{x^4-1}$ This was a no-calculator 10 min for 2 problem format contest. I started by using quadratic formula, but ...
1
vote
2answers
67 views

From any list of $131$ positive integers with prime factor at most $41$, $4$ can always be chosen such that their product is a perfect square

Author's note:I don't want the whole answer,but a guide as to how I should think about this problem. BdMO 2010 In a set of $131$ natural numbers, no number has a prime factor greater than 42. ...
1
vote
0answers
41 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
10
votes
1answer
204 views

A finite group containing an element with some property is a $p$-group

Let $G$ be a finite group. Suppose there exists a non-trivial element $g \in G$ such that $gxg^{-1}=x^{p+1}$ for all $x\in G$. Prove that $G$ is a $p$-group.
3
votes
1answer
42 views

Prove the sequences $\lfloor \alpha n\rfloor $ and $\lfloor \beta n\rfloor $ are disjoint

Here is another problem from a problem set that I can't solve. Let $\alpha$ and $\beta$ be irrational positive numbers such that $\frac{1}{\alpha}+\frac{1}{\beta}=1$ Prove that the sets $\{ ...
0
votes
3answers
55 views

Logic Question with who has a key and truth

Four people are standing infront of a treasure chest, each makes a statement. One statement is false, the other three are true. Ann: "I do not have the key and Cal does not have the key." Ben: "I do ...
2
votes
3answers
81 views

Bisector of angle formed at the orthocentre passes through the circumcentre

BdMO 2012 In an acute angled triangle $ABC$, $\angle A= 60$. We have to prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the ...
1
vote
1answer
41 views

Inscribed Angles in Two Cyclic Quadrilaterals

This problem is driving me crazy. It's from Andreescu's Mathematical Olympiad Challenges: Let $AB$ be a chord in a circle and $P$ a point on the circle. Let $Q$ be the projection of $P$ onto $AB$ ...
3
votes
3answers
53 views

$\operatorname{lcm}(n,m,p)\times \gcd(m,n) \times \gcd(n,p) \times \gcd(n,p)= nmp \times \gcd(n,m,p)$, solve for $n,m,p$?

$\newcommand{\lcm}{\operatorname{lcm}}$ I saw this in the first Moscow Olympiad of Mathematics (1935), the equation was : $$\lcm(n,m,p)\times \gcd(m,n) \times \gcd(n,p)^2 = nmp \times \gcd(n,m,p)$$ ...
0
votes
2answers
52 views

Polynomial $P(x)$ with degree $1998$

$P(x)$ is a polynomial of degree 1998 such that $P(k) = \frac{1}{k} $ for all values of $k = 1,2,3,...,1999$. What is the value of $P(2000)$? I did try to substitute as $k = 2000$ but the highest ...
-1
votes
1answer
93 views

2014 USAMO Problem :with Points Collinear iff Sum is Constant

Prove that there exists an infinite set of points $$ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots $$ in the plane with the following property: For any three distinct ...
2
votes
1answer
41 views

Interesting Base summation contest math problem

The problem is as follows: Let $N_b=1_b+2_b+\cdots+100_b$ where $b$ is an integer greater than $2$. Compute the number of values of $b$ for which the sum of the squares of the digits of $N_b$ is at ...
2
votes
1answer
58 views

Given $|f(x) - f(y)| \le \frac{1}{2}|x-y|$ what are the points of intersection of the graph of $y = f(x)$ and the line $y = x$?

Let $f(x)$ be a real-valued function, defined for all real numbers $x$ such that $$|f(x) - f(y)| \le \frac{1}{2}|x-y|$$ for all $x,y$. Then the number of points of intersection of the graph of $y = ...
0
votes
0answers
35 views

Ratio of area of triangle to that formed by its medians

What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$? I see an obvious method of brute-force wherein I can ...
1
vote
4answers
88 views

Areas in a rectangle

Suppose $P,Q, R$, and $S$ are the midpoints of the sides $AB, BC, CD$, and $DA$, respectively of rectangle $ABCD$. If the area of the rectangle is $\delta$, then prove that the area of the figure ...
1
vote
5answers
2k views

A hammer and a nail cost $1.10, and the hammer costs one dollar more than the nail. How much does the nail cost?

The answer is 0.05. I used algebra. But my friends say, why not 0.10, and they also say, it can be that the hammer is 1.04 and the nail 0.06. How do I tell them that 0.05 is the definite answer, ...
1
vote
1answer
30 views

Orthocentre of triangle and related ratio

$ABC$ is a triangle with $AB = 13$, $BC = 14$ and $CA = 15$. $AD$ and $BE$ are the altitudes from $A$ to $B$ to $BC$ and $AC$ respectively. $H$ is the point of intersection of $AD$ and $BE$. Then the ...
4
votes
3answers
450 views

A group theoretical game: Is it possible to reach a state when only blue marbles are left?

I've found this problem in a math contest. Apparently it's solved by group theory but I have no idea how. We're playing a game with a set of red and blue marbles arranged in a line. Here are the ...
4
votes
1answer
189 views

Binomial Summation

The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then ...
2
votes
1answer
49 views

Algebraic maximum and minimum based on a constraint

Suppose $a,b,c$ are real numbers such that $a^2b^2 + b^2c^2 + c^2a^2 = k$, where $k$ is a constant. Then the set of all possible values of $abc(a+b+c)$ is? I attempted writing the constraint in the ...
0
votes
1answer
28 views

Algebra Value based on condition provided

Let $a, b, c$ be distinct real numbers such that $a^2 - b = b^2 - c = c^2 - a$ Then $(a+b)(b+c)(c+a)$ equals? I attempted manipulations with that condition provided, but then I'm unable to go ...
1
vote
1answer
43 views

How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
1
vote
1answer
51 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
1
vote
3answers
72 views

Infinite Sum of products

What is the infinite sum $$S = {1 + \frac{1}{3} + \frac{1\cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+ ....}?$$ I attempted messing around with the $n$ th term in the series but didnt ...
1
vote
2answers
84 views

Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
0
votes
1answer
65 views

Series Summation involving factorials, and powers.

What is the value of $\dfrac{1.2}{3!} + \dfrac{2.2^2}{4!} + \dfrac{3.2^3}{5!} + ...... + \dfrac{15.2^{15}}{17!}$ How would you proceed with this? I attempted writing the general term and tried some ...
-3
votes
1answer
57 views

Why there are two different values of θ for same quadrant?

Let Sin θ = 1/2 is function. Let us find its solution set. sine is +ve in I and II quadrant with reference angle π/6 θ = π/6 (I quadrant) Now here is my problem. We can use π-θ and (π/2)+θ to find ...
6
votes
1answer
102 views

2014 USAMO #6, analytic number theory

Prove that there is a constant $c > 0$ with the following property: If $a, b, n$ are positive integers such that $ \gcd(a+i, b+j)>1 $ for all $ i, j\in\{0, 1,\ldots, n\} $ then $$ \min\{a, ...
10
votes
3answers
969 views

Algebra Iranian Olympiad Problem

If: $x^2+y^2+z^2=2(xy+xz+zy)$ and $x,y,z \in R^+$ Prove: $\frac{x+y+z}{3} \ge \sqrt[3]{2xyz}$ I tried my best to solve this thing but no use. Hope you guys can help me.Thanks in advance.
3
votes
1answer
77 views

Proof that infinitely many $f$ exist if $f(f(x))=f(x)^{2013}$

Suppose $f(x)$ is function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(f(x))=f(x)^{2013}$. Show that there are infinitely many such functions, of which exactly four are polynomials. If $f$ is ...
0
votes
1answer
53 views

Squares constructed externally on the sides of a triangle and concurrent lines

On the sides $BC, CA$ and $AB$ of the triangle $ABC$ we construct externally the squares $BCDE, ACFG $ and $ABHI$. Denote $A', B'$ and $C'$ the intersectiond points of the lines $BF$ and $CH$, $AD$ ...
11
votes
5answers
2k views

Tricky Triangle Area Problem

This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that ...
1
vote
1answer
86 views

What are some of the more efficient ways of studying for an Olympiad?

This September I am participating in a competition called the Australian Intermediate Mathematics olympiad, and you may not have heard of it but it's very similar to the AIME. Could you please tell me ...
1
vote
2answers
59 views

Knot theory: Genus of a surface

Use Euler characteristic to determine the genus of the surface in Figure 4.24 in picture below. I am stuck with this question 4.10 from Colin Adams, the Knot Book.
3
votes
3answers
138 views

Find the maximum value of $abc$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
1
vote
2answers
79 views

Difficult infinite trigonometric series

Evaluate the sum of the following infinite series. $$\left(\sin{\frac{\pi}{3}}\right) + \left(\frac{1}{2}\sin{\frac{2\pi}{3}}\right) + \left(\frac{1}{3}\sin{\frac{3\pi}{3}}\right) + \ldots$$
1
vote
1answer
45 views

Number theory problem, trigonometry

Suppose $p$ and $q$ are relatively prime positive integers, and that $x$ is a positive rational number. Given that $x \in [-\frac{1}{2}, \frac{1}{2}]$ and $$q\sin{\pi x} = p$$ how can we compute $p, ...
2
votes
3answers
110 views

Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$

Given three non-negative (as pointed out by Calvin Lin) real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$. (Source : Singapore Math Olympiad ...
1
vote
1answer
68 views

Euclidean Geometry problem: prove that $C'$ is the midpoint of $A'B'$.

The tangents to a circumference centered at $O$, passing through an exterior point $C$, meet the circumference at the points $A$ and $B$. Let $S$ be an arbitrary point on the circumference. The ...
2
votes
0answers
112 views

Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $\mathbb ...
1
vote
0answers
45 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
4
votes
1answer
49 views

Lines covering points on napkin

Suppose we place a $100\times 100$ napkin on an infinite lattice plane. What is the minimum number of lines that can always cover all the lattice points lying inside or on the border of the napkin, no ...
1
vote
2answers
54 views

Let $n$ be a positive integer. Show that if $2^n -1$ is a prime number, then $n$ is a prime number.

Let $n$ be a positive integer. Show that if $2^n -1$ is a prime number, then $n$ is a prime number. This is how I started to tackle this question: Assume that instead of $n$ being a prime number, it ...
5
votes
2answers
99 views

A different type binomial expansion problem

Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$ What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$? The answer is $a_n$, but I can't solve it. ...