Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.
2
votes
2answers
24 views
Let $f: \Bbb{R} \to [0; \infty)$ .Prove that $\forall n \in \Bbb{N}$ $\forall y \in \Bbb{R}$ $ \exists t=t(n;y)$ such that $\int_{y}^{t}f(x)dx=n$
The problem goes like this:
Let $f: \Bbb{R} \to [0; \infty)$ be a continuous function such that $\lim_{x \to \infty}f(x)=\infty$. Prove that $\forall n \in \Bbb{N^{*}}$ and $\forall y \in \Bbb{R}$ ...
6
votes
2answers
116 views
$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that
$$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
2
votes
2answers
55 views
Proving a sharp inequality
After experimenting, I've come to the conclusion that if $x\geq y\geq z\geq 0:$
$$\sum_{x,y,z}\frac{x}{\sqrt{x+y}}\geq \sum_{x,y,z}\frac{y}{\sqrt{x+y}}$$
(the sums are cyclic)
Does anyone know how ...
4
votes
1answer
93 views
find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $
This is a very hard functional equation.
the problem is this :
find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $
to solve it i have no idea! can we solve it ...
2
votes
0answers
38 views
Number of queries required to find the function.
this is a slight variation to question $3$ of the Nordic mathematical olympiad of 2010.(in short that one deals with bijections and this one deals with any kind of function).We have 2010 buttons and ...
1
vote
1answer
20 views
Get days on basis of Sum of values of
Lets suppose i have list of days :
Sun - 1
Mon - 2
Tue - 4
Wed - 8
Thu - 16
Fri - 32
Sat - 64
Now user can select one or more then one from checklist .My database just storing the sum of days ...
3
votes
0answers
53 views
A function on sets which is constant for all permutations
Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions:
...
3
votes
1answer
39 views
A bijection defined on the set of configuration of lamps
Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle.
Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A cool procedure consists in perform, ...
0
votes
2answers
28 views
Is my solution of this time and distance problem correct or wrong?
P and Q start running in opposite directions (towards each other) on a circular track starting at diametrically opposite points. They first meet after P has run for 75m and then they next meet after Q ...
1
vote
4answers
47 views
Possible arrangements of marbles in bags?
I've come across a question on a math test asking, "How many different ways can you put a dozen identical marbles into six bags so that each bag has atleat one marble in it?". I would imagine that ...
1
vote
1answer
54 views
Ratio of sum in black and white squares
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).
Let $S_1$ be the sum of the numbers put in the ...
0
votes
1answer
66 views
Cutting a chessboard into domino pieces!
A friend of mine gave me this problem from a european olympiad:
Suppose we have a $8\times8$ chessboard. Each edge has a number; the number of ways of dividing this chessboard into $1\times2$ and ...
0
votes
1answer
42 views
Algebra books for olympiad preparation
I was looking for some good books for algebra and number theory at the olympiad level. Does anybody have any suggestions? I specifically want books that work on techniques and concepts (not just ...
0
votes
0answers
57 views
Putnam 2013 B4 inequality
For any continuous real-valued function $f$ defined on the interval $[0,1],$ let $$\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|$$ Show that if $f$ ...
15
votes
2answers
215 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
47 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
58 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
100 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
$$a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 ...
2
votes
2answers
111 views
Geometry problem (Iran Olympiad)
Let $\triangle ABC$ be any triangle. Suppose the angle bisector of $\angle BAC$ intersects $BC$ at $D$. Let $\Gamma$ be a circle tangent to $BC$ at $D$ and so that $A$ belongs to the circumference ...
0
votes
0answers
20 views
Space variant of problem 5 from RMM 2011
We have a finite set of points $\{A_1, ... , A_n\}$ in $d$-dimensional space such that distances from $A_i$ to all other points are just a permutation of distances from $A_j$ to all other points. For ...
1
vote
0answers
26 views
Maximal number of inequivalent elements in $\mathbb{E}_p$
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let
$$ \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
91 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
63 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
86 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
42 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
98 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
61 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
213 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
105 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
876 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)!$$ has a prime divisor $P$ such that $P>k!$.
My idea:
Let us denote by ...
4
votes
2answers
95 views
Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]
I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
0
votes
1answer
52 views
Understanding 2012 AMC 12B #23
Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that
$P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros
at $x=-59$,$-57$,$-51$ and $-49$. What is ...
0
votes
1answer
20 views
HCF and LCM related [closed]
A certain number when succwssively divided by 2ˏ 5 and 7 leaves remainder 1ˏ2 and 3 respectively. When the same number divided by 70. What is the remainder?
1
vote
1answer
38 views
Parallelogram constructed through medians
Bdmo
In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ...
1
vote
2answers
60 views
Product identities
I need to use the following identities for poisson integral but i can't guz i don't know how to prove them.
$$\alpha^{2n}-1=\prod_{k=0}^{k=2n-1}(\alpha-e^{i\frac{2k\pi}{2n}})$$
...
7
votes
3answers
323 views
Math Algebra Question with Square Roots
For $a\ge \frac{1}{8}$, we define,
$$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}}}$$
Find the maximum value of $g(a)$.
I ...
8
votes
2answers
787 views
Math Olympiad Algebra Question
If $ax + by = 7$, $ax^2 + by^2 = 49$, $ax^3 + by^3 = 133$, and $ax^4 +$ $by^4 = 406$, find the value of $2014(x+y-xy) - 100(a+b)$.
I came across this question in a Math Olympiad Competition and I ...
0
votes
2answers
103 views
Math Olympiad Perfect Square Question
Let $N$ = $(\overline{abcd}) $ be a 4-digit perfect square that satisfies $(\overline{ab}) =3× (\overline{cd}) +1$ Find the sum of all possible values of $N$.
(The notation $n= ...
15
votes
2answers
2k views
Math Olympiad Prime Number Question
If $p$, $q$ and $r$ are prime numbers such that their product is $19$
times their sum, find $p^2$ + $q^2$ + $r^2$.
I came across this question in a Math Olympiad Competition and had no idea how ...
0
votes
1answer
35 views
Angle Manipulation Contest Math Problem
The problem is as follows:
In triangle $ABC$, $BC=2$. Point $D$ is on $\overline{AC}$ such that $AD=1$ and $CD=2$. If $m\angle BDC=2m\angle A$, compute $\sin A$.
I tried several ways of making ...
5
votes
4answers
735 views
Math Olympiad Divisor Problem
The sum of the two smallest positive divisors of an integer $N$ is $6$,
while the sum of the two largest positive divisors of $N$ is $1122$.
Find $N$.
I came across this question in a Math ...
0
votes
1answer
74 views
Math Olympiad Geometry Question: Similar Triangles
In the diagram below, △ABC and △CDE are two right-angled triangles
with AC = 24, CE =7 and ∠ ACB = ∠ CED. Find the length of the line
segment AE.
The above is the diagram.
I came ...
6
votes
4answers
961 views
Math Olympiad Algebraic Question Comprising Square Roots
If $m$ and $n$ are positive real numbers satisfying the equation $$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
53 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
711 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
5answers
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
142 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
77 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
45 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 ...
