Olympiad problems

2020 Canada National Olympiad, 3

Tags: combinatorics , Canada

There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?

2021 Canadian Mathematical Olympiad Qualification, 1

Tags: Canada , Functional Equations , Polynomials , algebra , polynomial

Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.

1999 Canada National Olympiad, 5

Tags: algebra , inequalities , three variable inequality , calculus , Canada

Let $ x$, $ y$, and $ z$ be non-negative real numbers satisfying $ x \plus{} y \plus{} z \equal{} 1$. Show that \[ x^2 y \plus{} y^2 z \plus{} z^2 x \leq \frac {4}{27} \] and find when equality occurs.

2002 Canada National Olympiad, 3

Tags: inequalities , Canada , AM-GM , am-gm inequality , algebra

Prove that for all positive real numbers $a$, $b$, and $c$, \[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \] and determine when equality occurs.

2020 Candian MO, 4#

Tags: number theory , Canada

$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$

2020 Canada National Olympiad, 2

Tags: geometry , rhombus , Canada

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

2015 Canada National Olympiad, 4

Tags: geometry , circumcircle , Canada , 2014 , National Olympiads

Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that \[BP\cdot CQ = AP\cdot AQ.\] Prove that $I$ is tangent to the circumcircle of triangle $BOC$.