# Tagged Questions

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

58 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 ...
46 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 ...
56 views

72 views

### How can this sum be maximized?

Suppose that $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ are distinct integers from $1$ to $7$. What is, then, the maximum value of the sum $$|a_1-a_2|+|a_2-a_3|+|a_3-a_4|+|a_4-a_5|+|a_5-a_6|+|a_6-a_7|+a_7$$?
50 views

### Find all real polynomials $P(x)$ which satisfy the equation$P(x)P(-x)=P(x^2-1)$

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation $$P(x)P(-x)=P(x^2-1)$$ Please explain me the process and refer some books to learn polynomials. Thanks ...
55 views

### Two polynomials that differ by polynomial expansion of $e$

Let $h(n)=\sum_{k=1}^{n}\frac{1}{k!}$. Does there exist real polynomials $f(x)$ and $g(x)$ such that $f(n)=h(n)g(n)$ for every positive integer $n$? So far, I got that $f(x)$ and $g(x)$ needs to be ...
96 views

### How many numbers have unit digit $1$?

Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of $5$. Determine, with proof, how many of the $2014$ integers $f(1), f(2), . . . , f(2014)$ ...
43 views

### Arithmetic sequence to geometric sequence.

The numbers $a_1, a_2, a_3, . . .$ form an arithmetic sequence with $a_1 \ne a_2$. The three numbers $a_1, a_2, a_6$ form a geometric sequence in that order. Determine all possible positive ...
41 views

### Finding the maximum cardinality of a set

Let $B$ be a subset of $A$ such that for any two elements $b_1$ and $b_2$ in $B$, we always have $2b_1\not \equiv{0}\pmod{b_2}$ if $2b_1\ge b_2$. If $A=\{1,2,...,n\}$ then find the maximum possible ...
49 views

### Maximum possible number of elements in a subset given a condition

Let $B$ be a subset of $A$ such that no element in $B$ is twice the other. Find the maximum number of elements possible in $B$ if $A=\{1,2,...,n\}$.
### Let $a_1, a_2, a_3…$ be the sequence of all positive integers relatively prime to 75. Find the value of $a_{2008}$.
Let $a_1, a_2, a_3...$ be the sequence of all positive integers relatively prime to 75, where $a_1<a_2<a_3...$ with $a_1=1, a_2=2, a_3=4, a_4=7$. Find the value of $a_{2008}$. What I have done: ...