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

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5
votes
1answer
102 views

Inequality olympiad

For all positive numbers $a,b,c$, prove that $$\frac{a^3}{b^2-bc+c^2}+\frac{b^3}{a^2-ac+c^2}+\frac{c^3}{a^2-ab+b^2}\geq 3 \frac{(ab+bc+ac)}{a+b+c}$$ Note that both side are homogeneous of degree 1, ...
0
votes
0answers
23 views

Minimum number of equal moves to reach the end of destination if there are some bricks in the path?

I have given some numbers $A_1, A_2, A_3, ..., A_n$. I have asked that if I have to reach from 0 to $A_n+1$/$\ge$$A_n+1$ , what is the minimum number of equal jumps I have to make in order reach my ...
2
votes
1answer
55 views

Prove that $f \left(\lambda x + (1- \lambda )x' , \lambda y +(1- \lambda )y' \right) > \min \{f(x,y), f(x',y')\}$

Let $f(x,y)=xy$ where $x,y\geq 0$. Prove that the function $f$ satisfies the following property: $$f \left(\lambda x + (1- \lambda )x' , \lambda y +(1- \lambda )y' \right) \geq \min \{f(x,y), ...
0
votes
1answer
45 views

Calculating the area of a triplet of circles.

I have an image of the problem which is quite self-explanatory. Any ideas?
18
votes
5answers
289 views

How to solve $\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256\ldots }}}}}$

How to solve this equation? $$x=\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256\ldots }}}}}.$$ Answer: $x=2$
3
votes
0answers
35 views

Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
4
votes
1answer
102 views

Inequality exercise (olympiad)

For positive $a$, $b$, $c$ such that $abc=1$. Show that $$(ab+bc+ca)(a+b+c)+6\geq 5(a+b+c).$$ From the LHS, using AM-GM, we see that $(ab+bc+ca)(a+b+c)+6\geq 3(abc)^{2/3}3(abc)^{1/3}+6=15$. But ...
4
votes
1answer
102 views

If $a,b,c>0$ and $a+b+c=1$, prove $\frac{a}{a+bc}+\frac{b}{b+ca}+\frac{\sqrt{abc}}{a+ba}\le 1+\frac{3\sqrt{3}}{4}$

If $a,b,c>0$ and $a+b+c=1$, then prove $$\frac{a}{a+bc}+\frac{b}{b+ca}+\frac{\sqrt{abc}}{a+ba}\le 1+\frac{3\sqrt{3}}{4}$$
0
votes
3answers
58 views

palindrome 4th grade

This is a 4th grade question. A palindrome is an integer number that does not change when read backwards. E.g. 123321 is a 6 digit palindrome. How many 9-digit palindromes are there that use only the ...
10
votes
2answers
444 views

Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no ...
4
votes
2answers
173 views

A contest math integral: $\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$

My school holds a math contest that has problems that vary level to level. Nobody managed to solve this particular one: $$\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$$ In terms of $n$ I was ...
4
votes
3answers
64 views

Proving $\cot { A+\cot { B+\cot { C=\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 4K } } } } $ [closed]

For any acute $\triangle ABC$, prove that $\cot { A+\cot { B+\cot { C=\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 4K } } } } $, where $K$ is the area of $\triangle ABC$. Unfortunately I'm ...
5
votes
1answer
31 views

Manipulation with strings riddle.

Starting with the "string" $PI$, can I or not transform it into the "string" $PK$ by applying the following rules (each rule can be used any number of times, in any order, and $x$ and $y$ represents a ...
0
votes
0answers
34 views

Number of ways to choose colored points equality

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x +y<n$. The points of $S$ are colored in red and blue so that ...
6
votes
1answer
163 views

Number of solutions to exceedingly contrived congruence.

Let $a$ be the number of solutions to$$x^{2011}-96x^{728}-x^{24}+67 \equiv y^{2011}+12718253987182795172957215781251235234235y \pmod{2^{57885161}-1}$$where $x$ and $y$ are integers in-between $0$ and ...
0
votes
1answer
60 views

How do you evaluate this summation: $S=\sum\limits_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$

Find S: $$S=\sum_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$$ My attempt: I tried writing the summation as: $$S=3!(15!)\sum_{r=0}^{15} (-1)^r \frac{1}{(15-r)!(r+3)!}$$ and tried to ...
7
votes
2answers
98 views

Finding the maximum value of $ab+ac+ad+bc+bd+3cd$

If $a,b,c,d>0$ satisfy the condition ${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 }=1$, find the maximum value of $ab+ac+ad+bc+bd+3cd$. I'm not progress in this inequality problem. Please ...
1
vote
0answers
89 views

Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with ...
1
vote
1answer
53 views

Writing a summation as the ratio of polynomial with integer coefficients

Write the sum $\sum _{ k=0 }^{ n }{ \frac { { (-1) }^{ k }\left( \begin{matrix} n \\ k \end{matrix} \right) }{ { k }^{ 3 }+9{ k }^{ 2 }+26k+24 } } $ in the form $\frac { p(n) }{ q(n) }$, where ...
2
votes
1answer
66 views

Coloring diagonals in a regular polygon

Each side and diagonal of a regular $n$-gon ($n\geq 3$) is colored blue or green. A move consists of choosing a vertex and switching the color of each segment incident to that vertex (from blue to ...
3
votes
1answer
128 views

Olympiad Trigonometric Inequality

Let $R$ and $r$ be the circumradius and inradius of $\triangle ABC$. Prove that $$\frac { \cos { A } }{ { \sin }^{ 2 }A } +\frac { \cos { B } }{ { \sin }^{ 2 }B } +\frac { \cos { C } }{ { ...
0
votes
1answer
32 views

Prove that $CDEF$ is a rectangle

Two circle $\Sigma_1$ and $\Sigma_2$ having centres $C_1$ and $C_2$ intersect at $A$ and $B$. Let $P$ be a point on the segment $AB$ and let $AP\ne{}BP$. The line through $P$ perpendicular to $C_1P$ ...
8
votes
2answers
174 views

If $\small {x+\sqrt { (x+1)(x+2) } +\sqrt { (x+2)(x+3) } +\sqrt { (x+3)(x+1) } = 4}$, solve for $x$.

I came across this olympiad algebra problem, asking to solve for $x$: $x\ +\ \sqrt { (x+1)(x+2) } \ +\ \sqrt { (x+2)(x+3) } +\ \sqrt { (x+3)(x+1) } =\ 4$ Here was my try: If $$x\ +\ \sqrt { ...
1
vote
1answer
55 views

Prove that the circumcenter of a triangle lies on an angle bisector

Let $\triangle$ ABC be a triangle and let $\ell$ be the A-angle bisector. Denote by B' the reflection of B over $\ell$. Prove that the circumcenter of $\triangle$ CIB' lies on $\ell$. My work: Let D ...
14
votes
6answers
385 views

Prove that $e>2$ geometrically.

Q: Prove that $e>2$ geometrically. Attempt: I only know one formal definition of $e$ that is $\lim_\limits{n\to\infty} (1+\frac{1}{n})^n=e$. I could somehow understand that this is somehow related ...
4
votes
1answer
42 views

Polynomial interpolating sequence mod p has small degree

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that ...
3
votes
4answers
72 views

show that $a_1+a_2+a_3+a_4=8$ and that $64a_1+27a_2+8a_3+a_4=729$ given the following

Consider the sistem of equations: $$\begin{cases} a_1+8a_2+27a_3+64a_4=1 \\ 8a_1+27a_2+64a_3+125a_4=27 \\ 27a_1+64a_2+125a_3+216a_4=125\\ 64a_1+125a_2+216a_3+343a_4=343\\ \end{cases} $$ These ...
13
votes
1answer
262 views

Find the maximum of the $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|$

Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that $a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and ...
4
votes
2answers
50 views

Prove that it is not possible to assign the integers $1,2,3,\cdots,20$ to the twenty vertices of a dodecahedron so that each face have constant sum

Prove that it is not possible to assign the integers $1,2,3,\cdots,20$ to the twenty vertices of a regular dodecahedron so that the five numbers at the vertices of each of the twelve pentagonal ...
9
votes
2answers
181 views

Tiling of a $9\times 7$ rectangle

Can a rectangle $9\times 7$ be tiled by "L-blocks" (an L-block consists of $3$ unit squares)? Although the problem seems to be easy, coloring didn't help me. The general theory is interesting, but ...
1
vote
0answers
67 views

Finding all functions: $f(x(2y+1))=f(x(y+1))+f(x)f(y)$

Need to find all functions from integers to complex that satisfy for all $x,y \in \mathbb{Z}$: $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ Any help would be great.
1
vote
1answer
36 views

The functional equation $f(f(x) + y) = f(f(x) - y) + 4f(x)y$

I found the following functional equation: $f(f(x) + y) = f(f(x) - y) + 4f(x)y$ Up to now I tried setting $x = 0$ and $f(0) = c$ to get $f(c + y) = f(c - y) + 4cy$ If we define $g(x) = f(x) - x^2 ...
2
votes
2answers
62 views

How do you evaluate this sum of multiplied binomial coefficients: $\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} $?

We have to find the value of x+y in: $$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$ My approach: I figured that the required summation is nothing but the coefficient of $x^3$ is the ...
0
votes
2answers
86 views

two questions involving $x^3+y^3+z^3-3xyz$ factorization

(1) Given that $x^3+y^3+z^3=3xyz+1$, determine the minimum of $x^2+y^2+z^2$. I know that Lagrange multiplier can solve this but I believe there is a way out using the factorisation: ...
5
votes
1answer
91 views

Putnam Problem, Pigeonhole Principle

I have never attempted or considered any contest math problems, but I recently found a page of Putnam Prep problems in a recycling bin on campus and decided to give some a try since I am home for ...
2
votes
2answers
37 views

Find $\tan C$ in a triangle satisfying the constraint

Given a triangle with angles $A,B, C$ and sides $a, b, c$ opposite to their respective angles, how can I find $\tan C$ such that $$c^2={a^3+b^3+c^3\over a+b+c}$$ I used the law of Cosines on the LHS ...
0
votes
2answers
77 views

Probability to pass an exam, generalized competition problem

In an exam there are 10 questions. If you answer correctly to a question, you get $1$ point. If you answer incorrectly to a question, you get $-1$ point, or lose a point. If you don't answer to a ...
4
votes
1answer
44 views

circles, power of point, cross ratios

Let $w$ be a circle, and let $P$ be a point outside $w$. Let $X, Y$ be the tangents from $P$ to $w$. A line from $P$ intersects $w$ in two points $B, D$. Let $C$ be the intesection of $\overline{XY}$ ...
0
votes
0answers
53 views

The functional equation $ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$

I came across the functional equation: $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ So far I tried plugging $x=f(y)$ and got $f(x)=\frac{f(0)-x^2+1}{2}$ which holds for every $x = f(y)$. I suppose that $f(0)=1$ ...
9
votes
2answers
529 views

Infinitude of primes in 10 consecutive integers

Do there exist infinitely many sets of 10 consecutive positive integers where exactly one is a prime? By Dirichlet's Theorem, if $a$ and $d$ are relatively prime, then there infinitely many primes ...
4
votes
1answer
95 views

Prove the triangle is equilateral given that a quadrilateral related to its circumcircle is a kite

Let $\triangle ABC$ be a triangle. Let $Γ$ be its circumcircle, and let $I$ be it’s incenter. Let the internal angle bisectors of $∠A,∠B,∠C$ meet $Γ$ in $A',B',C'$ respectively. Let $B'C'$ intersect ...
1
vote
1answer
31 views

Product of the Radii

$A_1$ and $A_2$ are two circles in a plane. The common external tangent to $A_1$ and $A_2$ consists of length $2017$. The common internal tangent consists of length $2009$. Find $r_1 \cdot r_2$ the ...
5
votes
2answers
50 views

How can we calculate the degree of angle made by the matches?

I was playing a game on my phone when a question pop up on my screen coming from one of my best mathematics masters: If we know that all of the matches are in the same size, what would be the ...
5
votes
2answers
86 views

Counting the number of numbers

Problem In each of the following 6 digit numbers: 333333, 201102, 123123; every digit appears at least twice. Find the number of such 6-digit natural numbers. I have done this problem using ...
2
votes
1answer
80 views

Is this a sufficient proof of a math contest problem?

Problem: If a,b,c,d are real, prove that $$a^2+b^2=2$$ $$c^2+d^2=2$$ $$ac=bd$$ Is true if and only if $$a^2+c^2=2$$ $$b^2+d^2=2$$ $$ab=cd$$ My proof is as follows: Note that each of the ...
0
votes
3answers
47 views

How many triangles with ∠ABC = 90° and AB= 20 exist such that all sides have integer lengths? (A) 1 (B) 2 (C) 3 (D) 4 (E) 6

How many triangles with $\angle ABC = 90°$ and $\overline{AB}= 20$ exist such that all sides have integer lengths? $(A)\; 1 ,\;\;(B) \;2 ,\;\;(C)\; 3 ,\;\;(D)\; 4 ,\;\;(E)\; 6$ I know the answer ...
2
votes
2answers
65 views

Given $2n$ points in the plane, prove we can connect them with nonintersecting segments

Given $2n$ points on the plane such that no three points lie on one line. Prove that it is possible to draw n segments such that each segment connects a pair of these points and no two segments ...
2
votes
1answer
43 views

How do you solve this recurrence relation/use it in a sequence to find it's GIF value?

The sequence {$x_k$} is defined by $x_{k+1} = x_k^2 + x_k$ and $x_1=\frac{1}{2}$. Now, if [.] denotes the greatest integer function, then which of the following options is correct: A) $[\frac{1}{x_1 ...
1
vote
3answers
80 views

Inequality with $a,b,c\in{}\mathbb{R}$.

Prove that for every positive real numbers $a,b$ and $c$ we have $$(a+b+c)^5\ge 81(a^2+b^2+c^2)abc.$$ I tried using the u,v,w method by substituting $$a+b+c=3u$$ $$ab+bc+ca=3v^2$$ $$abc=w^3$$ ...
1
vote
0answers
47 views

Olympiad problem similar to Sperner's theorem, inspired by OMM 2 ( unproven conjecture of mine)

This problem is inspired by problem 2 here. Consider a set of cubes $F$, such that each corner $(x,y)$ of any given cube of $F$ satisfies $0\leq x,y \leq n$, and each cube has a corner with ...