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

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0
votes
3answers
66 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
507 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
184 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
68 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
33 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
174 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 ...
1
vote
1answer
64 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
108 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
97 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
54 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
74 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
149 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
36 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
182 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
61 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 ...
15
votes
6answers
433 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
45 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
75 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 ...
14
votes
1answer
268 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
53 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 ...
10
votes
2answers
210 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
68 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
38 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
63 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
89 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
115 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
39 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
81 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
54 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
535 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
104 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
53 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
91 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
88 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
57 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
95 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
44 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
86 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
52 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 ...
7
votes
5answers
143 views

Find all functions $f$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$. So far, I've managed to prove that if $f$ is linear, then either $f(x) = x + 1$ or $f(x) = -1$ must be ...
12
votes
1answer
176 views

Putnam 2015 B6, sum involving number of odd divisors on an interval.

For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. What is$$\sum_{k=1}^\infty (-1)^{k-1} {{A(k)}\over{k}}?$$
7
votes
1answer
110 views

Is there some number in this sequence whose base $10$ representation ends with $2015$?

Given a list of the positive integers $1$, $2$, $3$, $4$, $...$, take the first three numbers $1$, $2$, $3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest ...
0
votes
0answers
37 views

Permutation on a strange string

There is a strange string of 10 characters ether '0' or '1'. I have n filter strings each having 10 characters ether '0' or '1'. A '1' at the i-th position in a filter string means that if I applies ...
6
votes
1answer
98 views

Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$

How do I see that$$\sum_{k \ge 0} \binom{n}{3k} = (1 + 1)^n + (\omega + 1)^n + (\omega^2 + 1)^n,$$where $\omega = \text{exp}\left({2\over3}\pi i\right)$? What is the underlying intuition behind this ...
1
vote
0answers
88 views

How to prove the identity with matrix exponential?

How can this equality be proved? $$ \exp{\left( \begin{matrix} 0 & a & c \\ -a & b & b \\ -c & -b & 0 \end{matrix} \right)} = \cos{r \left( \begin{matrix} 1 & 0 ...
1
vote
0answers
56 views

How can we prove the following inequality?

How can this inequality be proved? $$ \max_{i \leq j \leq p} \sqrt{\sum\limits_{i=1}^{q} a_{ij}^{2}} \leq \left\Vert \left( \begin{matrix} a_{11} & \dots & a_{1p} \\ \vdots & ...
3
votes
1answer
45 views

How can we evaluate the following limit?

How can this problem be solved? $$ \lim_{(n,r) \rightarrow (\infty, \infty)} \frac{\prod\limits_{k=1}^{r} \left( \sum\limits_{i=1}^{n} i^{2k-1} \right)}{n^{r+1} \prod\limits_{k=1}^{r-1} \left( ...