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

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4
votes
2answers
124 views

sum of 14 4th powers and sum of 14 cubes

Prove that $4(x_1^4 + x_2^4 + x_3^4 + \dots + x_{14}^4) = 7(x_1^3+ x_2^3 + x_3^3 + \dots + x_{14}^3)$ has no solution in positive integers. Hint : suppose on the contrary $\sum_{k=1}^{14} {(x_k^4 - ...
3
votes
0answers
48 views

Math contest geometry proof problem 2 [duplicate]

Could someone help me with this? Let A, B, C, D, L, M, N be distinct points in the plane such that A, B, C, D are the vertices of a square with sides AB, BC, CD, DA and L, M, N lie on the sides AB, ...
7
votes
2answers
634 views

Olympiad number theory problem

I found this problem in previous problems of the olympiads of my country If $t^2+n^2=r^2$, where $t$ has $3$ positive divisors, $n$ has $30$ positive divisors and $t,n,r$ are natural numbers, ...
13
votes
4answers
381 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
8
votes
1answer
162 views

An exponential “rearrangement” inequality: $x^x+y^y>x^y+y^x$

Let $x,y$ be distinct real numbers greater than $0$. Prove $$x^x+y^y>x^y+y^x .$$ Source: I think it comes from a Russian test given in 1991, but I haven't been able to verify this.
11
votes
5answers
551 views

$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.

Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$. I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying ...
6
votes
2answers
126 views

Arguing about a homework problem correctness

I've recently completely a homework in a problem solving class, I think my reasoning is correct but my teacher insisted that my answer is incorrect. I'm not sure if I'm correct or not. Question: ...
3
votes
1answer
140 views

Math contest proof equation problem

Could someone help me with this? If $m$ and $n$ are positive integers, then show that $$\frac{m}{ \sqrt n}+ \frac{m}{\sqrt[4]{n}} \neq 1$$.
4
votes
1answer
166 views

Straightedge-only construction of a perpendicular

There is a circle in the plane with a drawn diameter. Given a point inside the circle (not on the diameter or the circle), draw the perpendicular from the point to the diameter using only a ...
26
votes
7answers
4k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
1
vote
2answers
125 views

math contest geometry proof problem

Could someone help me with this? Suppose $A,B,C$ are vertices of a triangle and $D$ is a point on the side $BC$. Let $l$ be the line that contains $A$ and bisects $∠CAB$. Suppose there is a point $E$ ...
4
votes
4answers
225 views

Math contest geometry probability

Could someone help me with this? Suppose P is an 11-sided regular polygon and S is the set of all lines that contain two distinct vertices of P. If three lines are randomly chosen from S, what is the ...
12
votes
1answer
320 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
10
votes
3answers
265 views

Math contest proof problem fractions

Could someone help me with this? Let $x, y, z$ be positive integers with greatest common divisor $1$. If $\frac 1 x +\frac 1 y=\frac 1 z$, then show that $\sqrt{x + y}$ is an integer.
1
vote
1answer
50 views

For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
1
vote
3answers
127 views

Prove that there are infinitely many perfect cubes of the form $p^2+3q^2$

Prove that there are infinitely many perfect cubes of the form $p^2+3q^2$ where $p$ and $q$ are integers. Hint: one approach is to set $p^2+3q^2=(a^2+3b^2)^3$ and then find $(p,q)$ in terms of $a,b$. ...
14
votes
2answers
307 views

Question from Putnam '89: Primes of the form $101\ldots01$

I'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general input ...
0
votes
1answer
363 views

Prime Numbers And Perfect Squares

Find all primes $p$ and $q$ such that $p^2$+$7pq$+$q^2$ is a perfect square. One obvious solution is $p$=$q$ and under such a situation all primes p and q will satisfy. Further if $p\neq$$q$ then we ...
0
votes
1answer
271 views

Competition style mathematics

I'm currently 18 years old and only as of the part year taken a strong interest in maths. I'm working with my school curriculum (UK a-level) and receiving A grades so I am happy with this. However I ...
3
votes
1answer
242 views

Ramsey Type problem (variant of people at a party)

There is $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor n/2\rfloor-1$ of them, each of whom either knows both or else ...
4
votes
1answer
167 views

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
5
votes
3answers
143 views

Find $\lim_{n\to\infty}$ of this quotient.

Find, with proof, the value of this limit $$\lim_{n\to\infty}\frac{\sum^n_{r=0}\binom{2n}{2r}\cdot2^r}{\sum^{n-1}_{r=0}\binom{2n}{2r+1}\cdot2^r}$$ I have tried using binomial identities but two ...
9
votes
3answers
289 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove ...
4
votes
2answers
424 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
5
votes
1answer
270 views

Another olympiad problem

This problem is a problem in the last selection phase of the math olympiads in my country. If $\alpha, \beta,\gamma$ are angles $\in[0,\frac\pi2]$ such that ...
8
votes
2answers
217 views

How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
3
votes
1answer
292 views

block matrices problem

Let $A,B,C$ and $D$ be n by n matrics such that $AC=CA$. Prove that $\det \begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(AD-CB)$. The solution is to first assume that $A$ is invertible and ...
19
votes
3answers
1k views

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$ I just know $x^{x^x}$ is increasing in $x$ and hence the equation has a unique solution, ...
3
votes
2answers
61 views

Norms of eigenvalues bigger than 1 implies $|Ax|>x$ for all nonzero $x$?

If all the eigenvalues of $A$ (an n by n real matrix) have norms bigger than 1, is it true that $|Ax|>|x|$ for all nonzero $x\in\mathbb{R}^n$? (This is clearly true if $x$ is an eigenvector ...
3
votes
1answer
259 views

These two sequences have the same limit

Let $a_1$ and $b_1$ be any two positive numbers, and define $\{ a_n\}$ and $\{ b_n\}$ by $$a_n = \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}},$$ $$b_n = \sqrt{a_{n-1}b_{n-1} }.$$ Prove that the ...
2
votes
1answer
45 views

A $4$ variable inequality

If $a,b,c,d$ are positive numbers such that $c^2+d^2=(a^2+b^2)^3$, prove that $$\frac{a^3}{c} + \frac{b^3}{d} \ge 1,$$ with equality if and only if $ad=bc$. Source: Don Sokolowsky, Crux ...
4
votes
1answer
143 views

An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$

Prove that for any positive integer $n$, $$n^{1/n} < 1 + \sqrt{\frac{2}{n}}.$$ This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29. Hint:
7
votes
5answers
965 views

Mathematical reasoning question

I was giving a look at my country's mathematical olympiads, and I found this problem If I want to color a $4\times 4$ grid with black and white squares, in how many ways can I paint it such ...
11
votes
1answer
311 views

Finding $x^4 + y^4 + z^4$ using geometric series

This is a problem from the 2001 Stanford Math Tournament Algebra section. $$$$Given that $$x+y+z=3$$ $$x^2 + y^2 + z^2 = 5$$$$x^3+y^3+z^3=7$$Find $x^4+y^4+z^4$. $$$$My friend claimed that he was able ...
8
votes
4answers
221 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
35
votes
3answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
12
votes
5answers
845 views

Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7) $

The question is to simplify $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7)$ without using a calculator . My friend has given me ...
3
votes
2answers
91 views

For what $n$ is it true that $(1+\sum_{k=0}^{\infty}x^{2^k})^n+(\sum_{k=0}^{\infty}x^{2^k})^n\equiv1\mod2$

Let $A:=\sum_{k=0}^{\infty}x^{2^k}$. For what $n$ is it true that $(A+1)^n+A^n\equiv1\mod2$ (here we are basically working in $\mathbb{F}_2$.) The answer is all powers of 2, and it's fairly simple ...
2
votes
1answer
95 views

Determine all positive integers $n$ for which $B_n=\{0\}$.

Let $A_1,A_2,...,A_n,...$ and $B_1,B_2,...,B_n,...$ be sequences of sets defined by $a_1=\emptyset$, $B_1=\{0\}$, $A_{n+1}=\{x+1|x\in B_n\},B_{n+1}=(A_n\cup B_n)\setminus(A_n\cap B_n)$. Determine all ...
4
votes
0answers
84 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
1
vote
1answer
70 views

Minimum comparisons to identify the heaviest weights.

How many times I would have to make comparisons (between 2 weights) at minimum in order to identify the heaviest weight and also the second heaviest weight out of 128 weights? I'm not sure how to do ...
8
votes
1answer
223 views

How to prove there exists a polynomial with degree at most $100\sqrt{nk}$ satisfying this condition

Show that for arbitrary positive integers $n,k$, there exists a polynomial $p(x)$, with degree at most $100\sqrt{nk}$, such that ...
4
votes
1answer
151 views

Prove that $f$ is a linear combination of $f_1,f_2,\dots,f_n$.

Let $V$ be a vector space and let $f, f_1,f_2,\dots,f_n$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x)=0$ whenever $f_1(x)=f_2(x)=\cdots=f_n(x)=0$. Prove that $f$ is a linear combination ...
3
votes
2answers
233 views

Find the value of $\sin 2013^\circ$

How do I find the value of $\sin 2013^\circ$? A precise decimal is not required, but must be expressed with $\sin 30^\circ,$ $\sin 45^\circ,$ and $\sin 60^\circ$ (cosine is also fine). Hint: Use ...
0
votes
1answer
63 views

On a infinite series problem of IMC

In the solution 2 of problem of 2 of IMC 1999 I want to ask why $$\sum_{n=1}^{\infty}\frac{\pi (n)}{n^2}= \sum_{n=1}^{\infty}(\pi (1)+ \pi(2)+\cdots + \pi(n))\left( ...
0
votes
3answers
257 views

geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
2
votes
4answers
192 views

How do I show that $\gcd(a^2, b^2) = 1$ when $\gcd(a,b)=1$? [duplicate]

How do I show that $\gcd(a^2, b^2) = 1$ when $\gcd(a,b)=1$? I can show that $\gcd(a,b)=1$ implies $\gcd(a^2,b)=1$ and $\gcd(a,b^2)=1$. But what do I do here?
2
votes
3answers
254 views

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ [duplicate]

How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$? I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?
2
votes
3answers
139 views

Find the $\gcd(6, 14, 21)$ and express it in the form $6r+14s+21t$ for $r,s, t\in\mathbb{Z}$

Find the $\gcd(6, 14, 21)$ and express it in the form $6r+14s+21t$ for $r,s,t\in \mathbb{Z}$. I'm trying to learn some number theory, which starts with this gcd thing. But I ran into a problem: I ...
1
vote
3answers
88 views

Choosing $n$ objects in $2^{2n}$ ways

Of $3n+1$ objects, $n$ are indistinguishable, and the remaining ones are distinct. How that one can choose from them $n$ objects in $2^{2n}$ ways.