Found problems: 36
2024 Canadian Open Math Challenge, B3
Let $a,b,c,d$ be four [b]distinct [/b]integers such that:
$$\text{min}(a,b)=2$$
$$\text{min}(b,c)=0$$
$$\text{max}(a,c)=2$$
$$\text{max}(c,d)=4$$
Here $\text{min}(a,b)$ and $\text{max}(a,b)$ denote respectively the minimum and the maximum of two numbers $a$ and $b$.
Determine the fifth smallest possible value for $a+b+c+d$
2017 Canadian Open Math Challenge, C3
Source: 2017 Canadian Open Math Challenge, Problem C3
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Let $XYZ$ be an acute-angled triangle. Let $s$ be the side-length of the square which has two adjacent vertices on side $YZ$, one vertex on side $XY$ and one vertex on side $XZ$. Let $h$ be the distance from $X$ to the side $YZ$ and let $b$ be the distance from $Y$ to $Z$.
[asy]
pair S, D;
D = 1.27;
S = 2.55;
draw((2, 4)--(0, 0)--(7, 0)--cycle);
draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);
label("$X$",(2,4),N);
label("$Y$",(0,0),W);
label("$Z$",(7,0),E);
[/asy]
(a) If the vertices have coordinates $X = (2, 4)$, $Y = (0, 0)$ and $Z = (4, 0)$, find $b$, $h$ and $s$.
(b) Given the height $h = 3$ and $s = 2$, find the base $b$.
(c) If the area of the square is $2017$, determine the minimum area of triangle $XYZ$.
2018 Canadian Open Math Challenge, B1
Source: 2018 Canadian Open Math Challenge Part B Problem 1
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Let $(1+\sqrt2)^5 = a+b\sqrt2$, where $a$ and $b$ are positive integers. Determine the value of $a+b.$
2024 Canadian Open Math Challenge, A3
Colleen has three shirts: red, green, and blue; three skirts: red, green, and grey; three scarves: red, blue, and grey; and three hats: green, blue, and grey.
How many ways are there for her to pick a shirt, a skirt, a scarf, and a hat, so that two of the four clothes are one color and the other two are one other color?
2018 Canadian Open Math Challenge, C1
Source: 2018 Canadian Open Math Challenge Part C Problem 1
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At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center]
$\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$,
[color=transparent](A.)[/color]layer of this pyramid?
$\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers.
[color=transparent](B.)[/color]How many cans are on the bottom layer of the prism?
$\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle.
[color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...)
[color=transparent](C.)[/color]For example, a prism could be composed of the following layers:
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center]
Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.
2024 Canadian Open Math Challenge, A2
Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200 meter side and run in the same direction, Alice running twice as fast as Bob.
After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the building?
2017 Canadian Open Math Challenge, B4
Source: 2017 Canadian Open Math Challenge, Problem B4
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Numbers $a$, $b$ and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.
2018 Canadian Open Math Challenge, C3
Source: 2018 Canadian Open Math Challenge Part C Problem 3
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Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area.
$\text{(a)}$ Prove that $EF$ and $BD$ are parallel.
$\text{(b)}$ Prove that $G$ is the midpoint of $BD$.
$\text{(c)}$ Given that the area of triangle $ABD$ is 4 and the area of triangle $CBD$ is 6,
[color=transparent](C.)[/color]compute the area of triangle $EFG$.
2017 Canadian Open Math Challenge, B2
Source: 2017 Canadian Open Math Challenge, Problem B2
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There are twenty people in a room, with $a$ men and $b$ women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is $106$. Determine the value of $a \cdot b$.
2017 Canadian Open Math Challenge, C4
Source: 2017 Canadian Open Math Challenge, Problem C4
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Let n be a positive integer and $S_n = \{1, 2, . . . , 2n - 1, 2n\}$. A [i]perfect pairing[/i] of $S_n$ is defined to be a partitioning of the $2n$ numbers into $n$ pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if $n = 4$, then a perfect pairing of $S_4$ is $(1, 8),(2, 7),(3, 6),(4, 5)$. It is not necessary for each pair to sum to the same perfect square.
(a) Show that $S_8$ has at least one perfect pairing.
(b) Show that $S_5$ does not have any perfect pairings.
(c) Prove or disprove: there exists a positive integer $n$ for which $S_n$ has at least $2017$ different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)
2017 Canadian Open Math Challenge, A2
Source: 2017 Canadian Open Math Challenge, Problem A2
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An equilateral triangle has sides of length $4$cm. At each vertex, a circle with radius $2$cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$. Determine $a$.
[center][asy]
size(2.5cm);
draw(circle((0,2sqrt(3)/3),1));
draw(circle((1,-sqrt(3)/3),1));
draw(circle((-1,-sqrt(3)/3),1));
draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle);
fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray);
draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle);
fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray);
draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle);
fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray);
[/asy][/center]