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

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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
48 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
134 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
170 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
105 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
36 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
85 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
86 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
55 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( ...
-2
votes
2answers
118 views

Combinatorics Choosing Objects Under Condition

If 28 objects are arranged in a circle at equal distance from each other, in how many ways can 3 objects be chosen such that no two are adjacent or diametrically opposite.
2
votes
1answer
73 views

$\cos k\theta$ and $\cos(k+1)\theta$ are both rational only when $\theta=\pi/6$

Let $\theta$ be an angle in the interval $(0,\pi/2)$. Given that $\cos \theta$ is irrational and $\cos k\theta$ and $\cos (k+1)\theta$ are both rational for some positive integer $k$, show that ...
0
votes
2answers
41 views

Prove parallel line is tangent to second circle

Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents ...
0
votes
1answer
32 views

inequality with a sequence of real numbers

Given $n > 2$ and reals $x_1 \leqslant x_2 \leqslant \cdots \leqslant x_n$, show that $$ \left(\sum_{i, j} |x_i - x_j|\right)^2 \leqslant \frac {2} {3} (n^2 - 1) \sum_{i, j} (x_i - x_j)^2. $$ Show ...
0
votes
2answers
190 views

We have the recurrence relation $a_0 = 1$, $a_1 = 2$ and $a_n = 4a_{n-1} - a_{n-2}$. Could you find an odd integer factor of $a_{2015}$?

Question : We have the recurrence relation $a_0 = 1$, $a_1 = 2$ and $a_n = 4a_{n-1} - a_{n-2}$. Could you find an odd integer factor of $a_{2015}$? I tried to find an explicit formula for this ...
-2
votes
2answers
47 views

IMO 2014 problem 1

Let $a_0 < a_1 < a_2 < \cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geqslant 1$ such that $$ a_n < \frac {a_0 + a_1 + \cdots + a_n} ...
0
votes
0answers
10 views

Circumcircle of DEF passes through the Feuerbach point

Let $ABC$ be a triangle and let $N_a$ be its Nagel Point. Let $AN_a\cap BC=D$ and define $E$ and $F$ similarly so that $E$ lies on $AC$ and $F$ on $AB$. Prove that the $\odot (DEF)$ passes through ...
8
votes
1answer
125 views

Partitioning $\{1,\cdots,k\}$ into $p$ subsets with equal sums

Let $p$ be a prime. For which $k$ can the set $\{1,\cdots,k\}$ be partitioned into $p$ subsets with equal sums of elements ? Obviously, $p\mid k(k+1)$. Hence, $p\mid k$ or $p\mid k+1$. All we ...
4
votes
1answer
83 views

Quadratic permutations of $(1,2,\cdots{},n)$

Call a permutation $\left(p_1,p_2,\cdots{},p_n\right)$ of $\left(1,2,\cdots{},n\right)$ quadratic if there exists a perfect square among $$p_1,p_1+p_2,\cdots{},p_1+p_2+\cdots{}+p_n.$$ Find all natural ...
1
vote
1answer
57 views

Representing $m$ numbers using partial sums of at most $2^m$ numbers

Let $m$ positive integers $a_1, \cdots, a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \cdots, b_n$ such that all sums of distinct $b_k$s are distinct and all $a_i ...
1
vote
2answers
50 views

Triangle inequalities in proof

If we called the perimeter of some triangle $b$. Prove that if you added the lengths of any two of its medians (i) It would not be not larger than $\frac{3b}{4}$ (ii) It would not be smaller than ...
1
vote
1answer
65 views

Olympiad problem algebra inequality

I'm having trouble solving the following inequality problem: If $n$ is positive integer greater than $1$, and $x>y>1$, then show that: $\frac{x^{n+1}-1}{x(x^{n-1}-1)} > ...
1
vote
2answers
43 views

Expected value of number of moves on a chessboard

This is a problem from HMMT 2015. On an $8\times8$ chessboard, a rook starts at the lower left corner. Each minute, it moves to a square in the same row or same column with equal probability (however ...
2
votes
1answer
28 views

Condition in an inequality problem

I recently encountered an equality with the condition that $a,b,c$ are positive reals and $\displaystyle\sum_{\text{cyc}}\frac{1}{1+a}\le 1.$ The solution says that this condition is equivalent to ...
3
votes
0answers
42 views

Proving that the circumcenters are concyclic.

I was completely lost when handed this at a math competition a couple of weeks ago. I drew the diagram and was able to make sense of the question. My diagram also seemed to show that the ...
1
vote
1answer
38 views

Condition of touching of 2 circles in a triangle

Let $ABC$ be a triangle and let $D$ be a point on side $BC$. Show that the incircles of triangles $ABD$ and $ACD$ touch each other if and only if $D$ is the point of contact of the incircle of ...
0
votes
1answer
29 views

Determine $(n, p)$ such that $(p - 1)^n + 1$ is divisible by $n^{p - 1}$

Determine all pairs $(n, p)$ of positive integers such that $p$ is a prime, $n$ does not exceed $2p$, and $(p - 1)^n + 1$ is divisible by $n^{p - 1}$. IMO 1999/4
0
votes
1answer
27 views

Proving the points $P,O,N$ are collinear

In a triangle $ABC$, let $M$ be the midpoint of side $BC$ and $N$ be the midpoint of median $AM$. Let $O$ be the circumcentre of triangle $ABM$. If the circumcircle of triangle $BOM$ cuts the side ...
0
votes
1answer
31 views

Taking numbers away and then add the remaining to get 100

This question came up in a math competition a few weeks ago. My reasoning for (a) was that if we took away the 9 smallest numbers (1-9), the smallest 9 numbers that we would then be able to choose ...
4
votes
2answers
37 views

Polygons joining together to make similar polygons

I was given the below question in a math competition a few weeks ago. I was bit confused about the wording of the problem and what was meant by the word "similar" in the given context. I tried ...
2
votes
0answers
30 views

Adjacent pairs of numbers with the same prime divisors

For a positive integer $n$, let $P(n)$ be the set of distinct prime divisors of $n$. We are looking for pairs of numbers $A,B$ such that $P(A)=P(B)$ and $P(A+1)=P(B+1)$. The pair $A=2$, $B=8$ is a ...
3
votes
3answers
92 views

Prove that $k$ divides $a_k$

Define the sequence $a_k$ recursively by $\displaystyle\sum_{d|k}a_d=2^k$ with $d>0$. Prove that $a_k$ is a multiple of $k$.
0
votes
2answers
64 views

Find all triples (a,b,c) such that h(h(x))=x, and a,b and c are non-zero real numbers

Suppose that $a,b$ and $c$ are non-zero real numbers. Define $$h(x) = \frac{ax+b}{bx+c}$$ for $x\neq -\frac cb$. Determine all triples $(a,b,c)$ for which $h(h(x)) =x$ for every real number $x\neq ...
1
vote
1answer
68 views

IMO 1997 problem 6

For each positive integer $n$ , let $f (n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with non-negative integer exponents. Representations which differ only in the ...
4
votes
2answers
66 views

IMO 1997 problem 1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard). For any pair of positive integers $m$ and ...
0
votes
1answer
55 views

IMO 1996 problem 6

Let $p, q, n$ be three positive integers with $p + q < n$. Let $(x_0, x_1, \cdots, x_n)$ be an $(n + 1)$-tuple of integers satisfying the following conditions: (i) $x_0 = x_n = 0$; (ii) For each ...
4
votes
2answers
44 views

prove that, for some $p$ & $q, a_p, a_p+a_{p+1} + \cdots$ all are positive

Let $ a_1,a_2,\ldots ,a_{100}$ be real numbers, each less than one, satisfy $ a_1+a_2+\cdots+a_{100} > 1$ Show that there exist two integers $p$ and $q$ , $p<q$, such that the numbers $$a_q, ...
1
vote
1answer
32 views

How to measure the length of train $B$ given that trains $A,B$ are moving and that $A$ is $x$ feet long.

Train $A$ is $x$ feet long and is going east at $r_1$ mph. On a parallel track going west is train $B$ going at $r_2$ mph. f the trains take $y$ seconds to pass each other completely, how ...
1
vote
1answer
56 views

Putnam 2009 A4 clarification

Can anyone verify if this proof of the 2009 A4 Putnam problem is correct? Thanks! 2009 A4. Let $S$ be a set of rational numbers such that I. $0 \in S$. II. If $x \in S$, then $x \pm 1 \in S$. III. ...
0
votes
2answers
40 views

Finding the probability of the disease

Only 0.01% of people have triskaidekaphobia. The Dreizehn Club has developed a test for the phobia. If you have Triskadekaphobia, the test is 99% likely to identify that you have the disease. ...
1
vote
1answer
53 views

Using Lagrange Multipliers to find the maximum of a asymmetric value

This problem is from Korean Mathematical Olympiad 2015 P3. The problem asks to find, with proof, the maximum value of $$(ax+by)^2+(bx+cy)^2$$ with the constraint of $$a^2+b^2+c^2+x^2+y^2=1$$ Now, I ...
0
votes
1answer
37 views

Length of segment $PA$ in rectangle $ABCD$

In rectangle $ABCD$ ,$AB=10$ and $BC=15$. A point $P$ inside the rectangle such that $PB=12$ and $PC=9$.What is the length of $PA$ ? I've calculated that $PA=10$ by using the law of cosines applied ...
1
vote
0answers
53 views

An Elementary Solution to a Polynomial Problem?

The following problem is from Larson's problem solving through problems: If $a,b$ and $c$ are the roots of the equation $x^3-x^2-x-1=0$, show that $$ \frac{a^{1000}-b^{1000}}{a-b}+ ...
0
votes
1answer
67 views

Different ways to arrange a set of numbers, so X can be seen from the left and Y can be seen from the right

Given an set of unique integers of length N. What are number of different ways you can rearrange the array so that, you can only see X numbers of integers from the left and Y numbers of integers from ...
0
votes
5answers
38 views

How to determine the weight of a coin from each of the $4$ bags.

We are given $4$ bags of coins such that (a) all coins in a given bag weigh the same, and (b) the coins of a given bag weigh either $1,2, $ or $3$ ounces. Take $1$ coin from bag $1,3$ coins ...
7
votes
3answers
481 views

How does aptitude at solving Olympiad problems relate to success at further mathematical studies? [duplicate]

I spent last 6 years mostly practising my problem solving skills so I do well in my national Math Olympiad. Out of curiosity I did some reading on basics of what undergraduate students are taught - ...
0
votes
0answers
22 views

Using Affine Transformation to prove Concurrency

Let $ABCDE$ be a convex pentagon with $F=BC\cap DE, G=CD\cap EA, H=DE\cap AB, I=EA\cap BC, J=AB\cap CD$, Suppose that the areas of $\triangle AHI, \triangle BIJ, \triangle CJF, \triangle DFG, ...
2
votes
1answer
37 views

Lineal functions problem; interpreting $\;{g}^{-1}(x)=g(x)$

I'm preparing for a local math competition/olympiad, so I've been researching for past exams, and I've found this problem: Consider the lineal functions $f$ and $g$ such that ...