Found problems: 12
2017 Canadian Open Math Challenge, A4
Source: 2017 Canadian Open Math Challenge, Problem A4
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Three positive integers $a$, $b$, $c$ satisfy
$$4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.$$
Determine the sum of $a + b + c$.
2017 Canadian Open Math Challenge, A3
Source: 2017 Canadian Open Math Challenge, Problem A3
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Two $1$ × $1$ squares are removed from a $5$ × $5$ grid as shown.
[asy]
size(3cm);
for(int i = 0; i < 6; ++i) {
for(int j = 0; j < 6; ++j) {
if(j < 5) {
draw((i, j)--(i, j + 1));
}
}
}
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
draw((0,5)--(1,5));
draw((2,5)--(5,5));
draw((0,0)--(2,0));
draw((3,0)--(5,0));
[/asy]
Determine the total number of squares of various sizes on the grid.
2017 Canadian Open Math Challenge, A1
Source: 2017 Canadian Open Math Challenge, Problem A1
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The average of the numbers $2$, $5$, $x$, $14$, $15$ is $x$. Determine the value of $x$ .
2017 Canadian Open Math Challenge, B3
Source: 2017 Canadian Open Math Challenge, Problem B3
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Regular decagon (10-sided polygon) $ABCDEFGHIJ$ has area $2017$ square units. Determine
the area (in square units) of the rectangle $CDHI$.
[asy]
pair A,B,C,D,E,F,G,H,I,J;
A = (0.809016994375, 0.587785252292);
B = (0.309016994375, 0.951056516295);
C = (-0.309016994375, 0.951056516295);
D = (-0.809016994375, 0.587785252292);
E = (-1, 0);
F = (-0.809016994375, -0.587785252292);
G = (-0.309016994375, -0.951056516295);
H = (0.309016994375, -0.951056516295);
I = (0.809016994375, -0.587785252292);
J = (1, 0);
label("$A$",A,NE);
label("$B$",B,NE);
label("$C$",C,NW);
label("$D$",D,NW);
label("$E$",E,E);
label("$F$",F,E);
label("$G$",G,SW);
label("$H$",H,S);
label("$I$",I,SE);
label("$J$",J,2*dir(0));
fill(C--D--H--I--cycle,mediumgrey);
draw(polygon(10));
[/asy]
2017 Canadian Open Math Challenge, C1
Source: 2017 Canadian Open Math Challenge, Problem C1
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For a positive integer $n$, we define function $P(n)$ to be the sum of the digits of $n$ plus the number of digits of $n$. For example, $P(45) = 4 + 5 + 2 = 11$. (Note that the first digit of $n$ reading from left to right, cannot be $0$).
$\qquad$(a) Determine $P(2017)$.
$\qquad$(b) Determine all numbers $n$ such that $P(n) = 4$.
$\qquad$(c) Determine with an explanation whether there exists a number $n$ for which $P(n) - P(n + 1) > 50$.
2017 Canadian Open Math Challenge, B1
Source: 2017 Canadian Open Math Challenge, Problem B1
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Andrew and Beatrice practice their free throws in basketball. One day, they attempted a total of $105$ free throws between them, with each person taking at least one free throw. If Andrew made exactly $1/3$ of his free throw attempts and Beatrice made exactly $3/5$ of her free throw attempts, what is the highest number of successful free throws they could have made between them?
2017 Canadian Open Math Challenge, C2
Source: 2017 Canadian Open Math Challenge, Problem C2
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A function $f(x)$ is periodic with period $T > 0$ if $f(x + T) = f(x)$ for all $x$. The smallest such number $T$ is called the least period. For example, the functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$.
$\qquad$(a) Let a function $g(x)$ be periodic with the least period $T = \pi$. Determine the least period of $g(x/3)$.
$\qquad$(b) Determine the least period of $H(x) = sin(8x) + cos(4x)$
$\qquad$(c) Determine the least periods of each of $G(x) = sin(cos(x))$ and $F(x) = cos(sin(x))$.
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 $XYZ$.
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 $a$.
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
-----
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]