Found problems: 32
2021 Canadian Mathematical Olympiad Qualification, 4
Let $O$ be the centre of the circumcircle of triangle $ABC$ and let $I$ be the centre of the incircle of triangle $ABC$. A line passing through the point $I$ is perpendicular to the line $IO$ and passes through the incircle at points $P$ and $Q$. Prove that the diameter of the circumcircle is equal to the perimeter of triangle $OPQ$.
2020 Canadian Junior Mathematical Olympiad, 3
There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.
2021 Canadian Mathematical Olympiad Qualification, 8
King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular table with $2021$ chairs around it. At The King's birthday celebration, he is sitting in his throne (one of the $2021$ chairs) and the other $2020$ chairs are filled with guests, with the shortest guest sitting to the King's left and the remaining guests seated in increasing order of height from there around the table. The King announces that everybody else must get up from their chairs, run around the table, and sit back down in some chair. After doing this, The King notices that the person seated to his left is different from the person who was previously seated to his left. Each other person at the table also notices that the person sitting to their left is different.
Find a closed form expression for the number of ways the people could be sitting around the table at the end. You may use the notation $D_{n},$ the number of derangements of a set of size $n$, as part of your expression.
2016 Canadian Mathematical Olympiad Qualification, 5
Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.
2021 Canadian Mathematical Olympiad Qualification, 7
If $A, B$ and $C$ are real angles such that
$$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$
find
$$\cos (A)+\cos (B)+\cos (C)$$
2002 China Girls Math Olympiad, 7
An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
2020 Canadian Junior Mathematical Olympiad, 4
$ABCD$ 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.
2017 Canada National Olympiad, 1
For pairwise distinct nonnegative reals $a,b,c$, prove that
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.
2008 Canada National Olympiad, 2
Determine all functions $ f$ defined on the set of rational numbers that take rational values for which
\[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y,
\]
for each $ x$ and $ y$.
2020 Canadian Junior Mathematical Olympiad, 5
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 $2020$?
2004 Canada National Olympiad, 2
How many ways can $ 8$ mutually non-attacking rooks be placed on the $ 9\times9$ chessboard (shown here) so that all $ 8$ rooks are on squares of the same color?
(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)
[asy]unitsize(3mm);
defaultpen(white);
fill(scale(9)*unitsquare,black);
fill(shift(1,0)*unitsquare);
fill(shift(3,0)*unitsquare);
fill(shift(5,0)*unitsquare);
fill(shift(7,0)*unitsquare);
fill(shift(0,1)*unitsquare);
fill(shift(2,1)*unitsquare);
fill(shift(4,1)*unitsquare);
fill(shift(6,1)*unitsquare);
fill(shift(8,1)*unitsquare);
fill(shift(1,2)*unitsquare);
fill(shift(3,2)*unitsquare);
fill(shift(5,2)*unitsquare);
fill(shift(7,2)*unitsquare);
fill(shift(0,3)*unitsquare);
fill(shift(2,3)*unitsquare);
fill(shift(4,3)*unitsquare);
fill(shift(6,3)*unitsquare);
fill(shift(8,3)*unitsquare);
fill(shift(1,4)*unitsquare);
fill(shift(3,4)*unitsquare);
fill(shift(5,4)*unitsquare);
fill(shift(7,4)*unitsquare);
fill(shift(0,5)*unitsquare);
fill(shift(2,5)*unitsquare);
fill(shift(4,5)*unitsquare);
fill(shift(6,5)*unitsquare);
fill(shift(8,5)*unitsquare);
fill(shift(1,6)*unitsquare);
fill(shift(3,6)*unitsquare);
fill(shift(5,6)*unitsquare);
fill(shift(7,6)*unitsquare);
fill(shift(0,7)*unitsquare);
fill(shift(2,7)*unitsquare);
fill(shift(4,7)*unitsquare);
fill(shift(6,7)*unitsquare);
fill(shift(8,7)*unitsquare);
fill(shift(1,8)*unitsquare);
fill(shift(3,8)*unitsquare);
fill(shift(5,8)*unitsquare);
fill(shift(7,8)*unitsquare);
draw(scale(9)*unitsquare,black);[/asy]
2022 Canadian Mathematical Olympiad Qualification, 1
Let $n \geq 2$ be a positive integer. On a spaceship, there are $n$ crewmates. At most one accusation of
being an imposter can occur from one crewmate to another crewmate. Multiple accusations are
thrown, with the following properties:
• Each crewmate made a different number of accusations.
• Each crewmate received a different number of accusations.
• A crewmate does not accuse themself.
Prove that no two crewmates made accusations at each other.
2021 Canadian Mathematical Olympiad Qualification, 6
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation
$$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$
2021 Canadian Mathematical Olympiad Qualification, 2
Determine all integer solutions to the system of equations:
\begin{align*}
xy + yz + zx &= -4 \\
x^2 + y^2 + z^2 &= 24 \\
x^{3} + y^3 + z^3 + 3xyz &= 16
\end{align*}
2022 Canadian Mathematical Olympiad Qualification, 7
Let $ABC$ be a triangle with $|AB| < |AC|$, where $| · |$ denotes length. Suppose $D, E, F$ are points on side $BC$ such that $D$ is the foot of the perpendicular on $BC$ from $A$, $AE$ is the angle bisector of $\angle BAC$, and $F$ is the midpoint of $BC$. Further suppose that $\angle BAD = \angle DAE = \angle EAF = \angle FAC$. Determine all possible values of $\angle ABC$.
1997 Canada National Olympiad, 4
The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$. Prove that $\angle OBC=\angle ODC$.
1994 Canada National Olympiad, 5
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
2021 Canadian Junior Mathematical Olympiad, 4
Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$.
Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$
2021 Canadian Mathematical Olympiad Qualification, 5
Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle.
Determine who wins each of the following games:
(a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$.
(b) The starting rectangles are $100 \times 100$ and $100 \times 500$.
2020 Canada National Olympiad, 4
$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, 1
There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.
1996 China National Olympiad, 3
In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.
2021 Canadian Mathematical Olympiad Qualification, 3
$ABCDE$ is a regular pentagon. Two circles $C_1$ and $C_2$ are drawn through $B$ with centers $A$ and $C$ respectively. Let the other intersection of $C_1$ and $C_2$ be $P$. The circle with center $P$ which passes through $E$ and $D$ intersects $C_2$ at $X$ and $AE$ at $Y$. Prove that $AX = AY$.
2020 Canada National Olympiad, 5
Simple graph $G$ has $19998$ vertices. For any subgraph $\bar G$ of $G$ with $9999$ vertices, $\bar G$ has at least $9999$ edges. Find the minimum number of edges in $G$
2021 Canada National Olympiad, 2
Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$.
Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$