Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA + AX = CB + BX$ and $ BA + AY = BC + CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.

Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is $ 46$, there is a set of $ 10$ students in which no group is properly contained.

Let $ \Gamma$ be the circumcircle of a triangle $ ABC$. A circle passing through points $ A$ and $ C$ meets the sides $ BC$ and $ BA$ at $ D$ and $ E$, respectively. The lines $ AD$ and $ CE$ meet $ \Gamma$ again at $ G$ and $ H$, respectively. The tangent lines of $ \Gamma$ at $ A$ and $ C$ meet the line $ DE$ at $ L$ and $ M$, respectively. Prove that the lines $ LH$ and $ MG$ meet at $ \Gamma$.

Consider the function $ f: \mathbb{N}_0\to\mathbb{N}_0$, where $ \mathbb{N}_0$ is the set of all non-negative integers, defined by the following conditions : $ (i)$ $ f(0) = 0$; $ (ii)$ $ f(2n) = 2f(n)$ and $ (iii)$ $ f(2n + 1) = n + 2f(n)$ for all $ n\geq 0$. $ (a)$ Determine the three sets $ L = \{ n | f(n) < f(n + 1) \}$, $ E = \{n | f(n) = f(n + 1) \}$, and $ G = \{n | f(n) > f(n + 1) \}$. $ (b)$ For each $ k \geq 0$, find a formula for $ a_k = \max\{f(n) : 0 \leq n \leq 2^k\}$ in terms of $ k$.

Let $ a, b, c$ be integers satisfying $ 0 < a < c - 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$ be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.