Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that: a.) $N_a$ is odd; b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.

Click for solution We have that: $k=\sum b_ta^t=\sum b_t^2$ we also have: $b_ta^t>(a-1)^2+b_t^2$ for $t>1$, so we get that a good $k$ has at most $2$ digits in base $a$. So we have: case 1: $b_2a+b_1=b_2^2+b_1^2$ Now for a pair $b_2,b_1$, we verify that $a-b_2,b_1$ also works, and also when $b_2=a/2$ we get: $(2b_1-1)^2=a^2+1$ wich gives $a=0$, contradiction. so there is an even number of solutions in this case. case 2: $b_1=b_1^2$ wich gives $k=1$ So $N_a$ is odd. Now finally we show case 1 has as many solutions as we would like for some $a$. Pick an even $a$ such that $a^2+1$ has at least $t$ prime factors all of them $=1$ mod $4$. Then, $a^2+1$ can be expressed as a sum of two squares in at least $2^t$ ways. Since $b_2=a/2+-\sqrt{a^2-4b_1^2+4b}/2$, iff $a^2+1=c^2+d^2$ with $c$ odd, put $2b_1-1=c$ and $b_2=a/2+-d/2$ So we get $N_a>2^{t+1}$, because every $c$ and $d$ give different values for $b_1$ and $b_2$

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

a.) Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that \[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \] Prove that \[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\] where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$ b.) Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: a.) $f(x, y)$ takes non-negative integer value; b.) $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; c.) If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.