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

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4
votes
2answers
492 views

simple games with cute winning strategies?

Im thinking of games of two players ($A$ goes first and $B$ second) like the following: There are 35 chips in a table, during each turn a player can remove 1,2,3 or 4 chips. Prove player $B$ can ...
0
votes
2answers
60 views

Algebra contest problem

Suppose $x$ and $y$ are integers. Given $2xy+14y=-53-13x$, what does $xy$ equal? The answer is $-15$, but how do I get that? I feel like I should be able to find this.
0
votes
1answer
59 views

Complex Combinatorics Hexagon/Triangle Contest Problem

The problem is as follows: The six sides of convex hexagon $A_1A_2A_3A_4 A_5A_6 $ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings ...
2
votes
2answers
117 views

Analytical solution to a nonlinear ODE

How might I analytically solve the following differential equation? $$yy'' = y' + y^3$$ I've tried certain substitutions ($y = ux$ etc.) but none of them work.
1
vote
1answer
54 views

Logarithmic Contest Question

The Problem was as follows: Define $\log*(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to $1$. For example ...
2
votes
1answer
89 views

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$. This is a problem from a selection to IMO 2014.
1
vote
1answer
51 views

Power Factoring Contest Question

The question was as follows: Compute the smallest positive integer $n$ such that $n^n$ has at least $1,000,000$ positive divisors. I did some work, finding that if $n=2^a*3^b*5^c*7^d$ then the $n^n= ...
2
votes
1answer
47 views

Find radius of sphere

Imagine eight spheres of radius 1 that are at $(\pm1,\pm1,\pm1)$. Place sphere A with its center at the origin externally tangent to all of the other spheres. Then place sphere B externally tangent to ...
2
votes
1answer
52 views

Find roots of a function

$f$ is a function defined on the whole real line which has the property that $f(1+x)=f(2-x)$ for all $x$. Assume that the equation $f(x)=0$ has $8$ distinct real roots. Find the sum of these roots. I ...
6
votes
1answer
416 views

What is a simple way of computing the following fraction?

Compute the value of the expression: $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
4
votes
2answers
98 views

Probability that $5 \mid x^4 - y^4$ for random $x, y$

Two numbers $x$ and $y$ are chosen at random without replacement from the set $\{1,2,3,\cdots,100\}$. Find the probability that $x^4 - y^4$ is divisible by $5$. I don't know how to proceed with this ...
11
votes
3answers
227 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
161 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
2answers
66 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
56 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
27 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
103 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
79 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
51 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
205 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
53 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
78 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
151 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
105 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
68 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
64 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 ...
0
votes
1answer
142 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
60 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
61 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
66 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
100 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 ...
3
votes
7answers
15k 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
48 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
459 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
217 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
53 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
29 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
57 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
124 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
89 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
1k 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
105 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 ...
1
vote
1answer
81 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
135 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
1k 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
92 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
68 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 ...
2
votes
1answer
157 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
87 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.