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

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15
votes
2answers
232 views

A game on a graph

Alice and Bob play a game on a complete graph ${G}$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses ...
0
votes
1answer
53 views

Function Combination on Computer Science

I read some material on Computational Function, every one could describe the result of following combination? suppose $g_1(x)=3x$, $g_2(x)=4x$, $f(x,y)=x+y$, how we compute combination of $f$ with ...
2
votes
2answers
65 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
2
votes
1answer
31 views

Existence of sets with four common elements among 16000 sets.

In the Duma there are $1600$ delegates, who have formed $16000$ committees of $80$ persons each. Prove that one can find two committees having no fewer than four common members. Source: All-Russian ...
2
votes
1answer
108 views

A term of the sequence

Let $a_0$ and $a_n$ be diff erent divisors of a natural number $m$, and $a_0, a_1, a_2,\cdots, a_n$ be a sequence of natural numbers such that it satisfies $$a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 ...
2
votes
2answers
131 views

Geometry problem (Iran Olympiad)

Let $\triangle ABC$ be any triangle. Suppose the angle bisector of $\angle BAC$ intersects $BC$ at $D$. Let $\Gamma$ be a circle tangent to $BC$ at $D$ and so that $A$ belongs to the circumference ...
0
votes
0answers
22 views

Space variant of problem 5 from RMM 2011

We have a finite set of points $\{A_1, ... , A_n\}$ in $d$-dimensional space such that distances from $A_i$ to all other points are just a permutation of distances from $A_j$ to all other points. For ...
1
vote
0answers
28 views

Maximal number of inequivalent elements in $\mathbb{E}_p$

For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let $$ \displaystyle \mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}$$ For $(a,b), (a',b') \in ...
5
votes
3answers
95 views

Proving $n^2(n^2+16)$ is divisible by 720

Given that $n+1$ and $n-1$ are prime, we need to show that $n^2(n^2+16)$ is divisible by 720 for $n>6$. My attempt: We know that neither $n-1$ nor $n+1$ is divisible by $2$ or by $3$, therefore ...
2
votes
0answers
64 views

Find all pair(s) of positive integer $(a,b)$ such that $\frac{a^2}{2ab^2 -b^3+1}$ is also positive integer too?

Another number theory problem. I can find the small value of $b$ such that 0,1,2. But, I cannot find the upper limit of $b$, such that the value of $b$ is limited. How can I find the solution ...
4
votes
3answers
92 views

Find all positive integer solutions $(x,y,z)$ that satisfy $5^x \cdot 7^y +4= 3^z$?

This is another contest math-problem. The only problem that I cannot find the way to tackle this problem. Can anybody try to provide the solution to solve this problem? Thanks
3
votes
2answers
44 views

A number theory question

For integers from $1$ to $4n-1$, we pick two integers and replace them with their difference. That is for $n=1$ we have $1,2,3$ if we pick $1,3$ our new numbers will be $2,2$. Show that at $4n-2$ step ...
5
votes
4answers
103 views

Integer solutions to $a^{2014} +2015\cdot b! = 2014^{2015}$

How many solutions are there for $a^{2014} +2015\cdot b! = 2014^{2015}$, with $a,b$ positive integers? This is another contest problem that I got from my friend. Can anybody help me find the ...
1
vote
1answer
69 views

Geometry Math Olympiad Question [closed]

In the diagram below, AD is perpendicular to AC and $ ∠BAD = ∠DAE = 12^\circ$. If $AB + AE = BC$, find $∠ABC$. The above is the diagram. I came across this question in a Math Olympiad ...
8
votes
2answers
228 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
6
votes
2answers
109 views

Problem about functional equation (Bulgarian selection team test)

This problem was taken from the bulgarian selection team test for the 47th IMO and appeared in a chinese magazine, I came across it in my own training. http://www.math.ust.hk/excalibur/v10_n4.pdf ...
35
votes
4answers
925 views

How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$

Question: Let $k$ be a positive integer. Show that there exist $n$ such that $$I=k!+(2k)!+(3k)!+\cdots+(nk)!$$ has a prime divisor $P$ such that $P>k!$. My idea: Let us denote by ...
4
votes
2answers
99 views

Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
0
votes
1answer
55 views

Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...
0
votes
1answer
23 views

HCF and LCM related [closed]

A certain number when succwssively divided by 2ˏ 5 and 7 leaves remainder 1ˏ2 and 3 respectively. When the same number divided by 70. What is the remainder?
1
vote
1answer
39 views

Parallelogram constructed through medians

Bdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ...
1
vote
2answers
61 views

Product identities

I need to use the following identities for poisson integral but i can't guz i don't know how to prove them. $$\alpha^{2n}-1=\prod_{k=0}^{k=2n-1}(\alpha-e^{i\frac{2k\pi}{2n}})$$ ...
7
votes
3answers
336 views

Math Algebra Question with Square Roots

For $a\ge \frac{1}{8}$, we define, $$g(a)=\sqrt[\Large3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[\Large3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}$$ Find the maximum value of $g(a)$. I ...
8
votes
2answers
804 views

Math Olympiad Algebra Question

If $ax + by = 7$, $ax^2 + by^2 = 49$, $ax^3 + by^3 = 133$, and $ax^4 +$ $by^4 = 406$, find the value of $2014(x+y-xy) - 100(a+b)$. I came across this question in a Math Olympiad Competition and I ...
0
votes
2answers
109 views

Math Olympiad Perfect Square Question

Let $N$ = $(\overline{abcd}) $ be a 4-digit perfect square that satisfies $(\overline{ab}) =3× (\overline{cd}) +1$ Find the sum of all possible values of $N$. (The notation $n= ...
16
votes
2answers
2k views

Math Olympiad Prime Number Question

If $p$, $q$ and $r$ are prime numbers such that their product is $19$ times their sum, find $p^2$ + $q^2$ + $r^2$. I came across this question in a Math Olympiad Competition and had no idea how ...
0
votes
1answer
36 views

Angle Manipulation Contest Math Problem

The problem is as follows: In triangle $ABC$, $BC=2$. Point $D$ is on $\overline{AC}$ such that $AD=1$ and $CD=2$. If $m\angle BDC=2m\angle A$, compute $\sin A$. I tried several ways of making ...
5
votes
4answers
744 views

Math Olympiad Divisor Problem

The sum of the two smallest positive divisors of an integer $N$ is $6$, while the sum of the two largest positive divisors of $N$ is $1122$. Find $N$. I came across this question in a Math ...
0
votes
1answer
87 views

Math Olympiad Geometry Question: Similar Triangles

In the diagram below, △ABC and △CDE are two right-angled triangles with AC = 24, CE =7 and ∠ ACB = ∠ CED. Find the length of the line segment AE. The above is the diagram. I came ...
6
votes
4answers
984 views

Math Olympiad Algebraic Question Comprising Square Roots

If $m$ and $n$ are positive real numbers satisfying the equation $$m+4\sqrt{mn}-2\sqrt{m}-4\sqrt{n}+4n=3$$ find the value of $$\frac{\sqrt{m}+2\sqrt{n}+2014}{4-\sqrt{m}-2\sqrt{n}}$$ I came ...
2
votes
1answer
54 views

Alphametics Question

In the figure below, each distinct letter represents a unique digit such that the arithmetic sum holds. If the letter L represents 9, what is the digit represented by the letter T? ...
6
votes
3answers
728 views

Math Olympiad Algebraic Question

If both $n$ and $ \sqrt{n^2+204n} $ are positive integers, find the maximum value of $n$. I came across this question during a Math Olympiad Competition. I need help with solving the question. ...
13
votes
4answers
1k views

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing?

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing? I came across this question in a math competition and I am looking for how to solve this question without working it ...
1
vote
3answers
153 views

Algebra Difference in Roots Question.

Let D be the absolute value of the difference of the 2 roots of the equation 3x^2-10x-201=0. Find [D]. [x] denotes the greatest integer less than or equal to x. I came across this question in a ...
2
votes
3answers
79 views

Logarithm Fraction Contest Math Question

The question is as follows: If $\dfrac{\log_ba}{\log_ca}=\dfrac{19}{99}$ then $\dfrac{b}{c}=c^k$. Compute $k$.
0
votes
1answer
49 views

3D Geometry Contest Math Problem

The problem is as follows: Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon H with side length 1, and so that the vertices are not lying in the ...
2
votes
2answers
75 views

Algebra Manipulation Contest Math Problem

The question was as follows: The equations $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ have two roots in common. Compute the product of these common roots. Because $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ it means ...
2
votes
2answers
89 views

Number of non-decreasing sequences

How do I find the number of non-decreasing sequences of length $N$, such that all number in the sequences lie in the range $[a, b]$. Also, the frequency of the most frequently occurring element should ...
4
votes
2answers
419 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
53 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
32 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
94 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
40 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
68 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. I like thinking about this problem, it is ...
0
votes
0answers
160 views

Math Competitions such as Intel Science Talent Search for High School Freshmen

I am extremely interested in mathematical competitions. I was wondering if there was something like the Intel Science Talent Search, where one can present his/her research, for freshmen in high ...
1
vote
1answer
46 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
37 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
47 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
401 views

What is a simple way of computing the following fraction?

Compute the value of the expression: $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
1
vote
2answers
72 views

Tough probability problem

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 ...