Found problems: 36
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 $a + b + c$.
2018 Canadian Open Math Challenge, B4
Source: 2018 Canadian Open Math Challenge Part B Problem 4
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Determine the number of $5$-tuples of integers $(x_1,x_2,x_3,x_4,x_5)$ such that
$\text{(a)}$ $x_i\ge i$ for $1\le i \le 5$;
$\text{(b)}$ $\sum_{i=1}^5 x_i = 25$.
2024 Canadian Open Math Challenge, B1
For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$.
Let $s(n)$ denote the sum of the first $n$ factorials, i.e.
$$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$
Find the remainder when $s(2024)$ is divided by $8$
2018 Canadian Open Math Challenge, B3
Source: 2018 Canadian Open Math Challenge Part B Problem 3
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The [i]doubling sum[/i] function is defined by
\[D(a,n)=\overbrace{a+2a+4a+8a+...}^{\text{n terms}}.\]
For example, we have
\[D(5,3)=5+10+20=35\]
and
\[D(11,5)=11+22+44+88+176=341.\]
Determine the smallest positive integer $n$ such that for every integer $i$ between $1$ and $6$, inclusive, there exists a positive integer $a_i$ such that $D(a_i,i)=n.$
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.
2018 Canadian Open Math Challenge, A2
Source: 2018 Canadian Open Math Challenge Part A Problem 2
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Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v\times w\times x\times y\times z.$ What is the sum of the integers?
2024 Canadian Open Math Challenge, C3
Let $ABC$ be a triangle for which the tangent from $A$ to the circumcircle intersects line $BC$ at $D$, and let $O$ be the circumcenter. Construct the line $l$ that passes through $A$ and is perpendicular to $OD$. $l$ intersects $OD$ at $E$ and $BC$ at $F$. Let the circle passing through $ADO$ intersect $BC$ again at $H$. It is given that $AD=AO=1$.
a) Find $OE$
b) Suppose for this part only that $FH=\frac{1}{\sqrt{12}}$: determine the area of triangle $OEF$.
c) Suppose for this part only that $BC=\sqrt3$: determine the area of triangle $OEF$.
d) Suppose that $B$ lies on the angle bisector of $DEF$. Find the area of the triangle $OEF$.
2024 Canadian Open Math Challenge, A4
Consider the sequence of consecutive even numbers starting from 0, arranged in a staggered format, where each row contains one more number than the previous row.
The beginning of this arrangement is shown below:
$0$
$2\; 4$
$6\;\underline{8}\;10$
$12\: 14\: 16\: 18$
$20\: 22 \: 24 \: 26\: 28 $
The number in the middle of the third row is 8. What is the number in the middle of the 101-st row?
2024 Canadian Open Math Challenge, A1
Two locations A and B are connected by a 5-mile trail which features a lookout C. A group of 15 hikers started at A and walked along the trail to C. Another group of 10 hikers started at B and walked along the trail to C. The total distance travelled to C by all hikers from the group that started in A was equal to the total distance travelled to C by all hikers from the group that started in B.
Find the distance (in miles) from A to C along the trail.
2018 Canadian Open Math Challenge, A4
Source: 2018 Canadian Open Math Challenge Part A Problem 4
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In the sequence of positive integers, starting with $2018, 121, 16, ...$ each term is the square of the sum of digits of the previous term. What is the $2018^{\text{th}}$ term of the sequence?
2018 Canadian Open Math Challenge, A1
Source: 2018 Canadian Open Math Challenge Part A Problem 1
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Suppose $x$ is a real number such that $x(x+3)=154.$ Determine the value of $(x+1)(x+2)$.
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 $x$ .
2018 Canadian Open Math Challenge, B2
Source: 2018 Canadian Open Math Challenge Part B Problem 2
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Let ABCD be a square with side length 1. Points $X$ and $Y$ are on sides $BC$ and $CD$ respectively such that the areas of triangels $ABX$, $XCY$, and $YDA$ are equal. Find the ratio of the area of $\triangle AXY$ to the area of $\triangle XCY$.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZi9lLzAzZjhhYzU0N2U0MGY2NGZlODM4MWI4Njg2MmEyMjhlY2M3ZjgzLnBuZw==&rn=YjIuUE5H[/img][/center]
2024 Canadian Open Math Challenge, C4
Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$.
a) Compute the number of excellent polynomials with degree at most 3.
b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$.
c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.
2024 Canadian Open Math Challenge, C1
Let the function $f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}$ for all real values $x,y$ and $t\not=\pm1$
a) Evaluate $f(2,0,3)$ and $f(0,2,3)$.
b) Show that $f(x,y,0)=f(y,x,0)$ for any values of $(x,y)$.
c) Show that $f(x,y,t)=f(y,x,t)$ for any values of $(x,y)$ and $t\not=\pm1$.
d) Given
$$g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}$$
for all real values $x,y$ and $s\not=\frac{\pi}{2}+2\pi k$, where $k$ is an integer number, show that $g(x,y,s)=g(y,x,s)$ for any values of $(x,y)$ and $s$ in the domain of $g(x,y,s)$.
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]
2024 Canadian Open Math Challenge, B4
Initially, the integer $80$ is written on a blackboard. At each step, the integer $x$ on the blackboard is replaced with an integer chosen uniformly at random among [0,x−1], unless $x=0$ , in which case it is replaced by an integer chosen uniformly at random among [0,2024]. Let $P(a,b)$ be the probability that after $a$ steps, the integer on the board is $b$. Determine
$$\lim_{x\to\infty}\frac{P(a,80)}{P(a,2024)}$$
(that is, the value that the function $\frac{P(a,80)}{P(a,2024)}$ approaches as $a$ goes to infinity).
2018 Canadian Open Math Challenge, A3
Source: 2018 Canadian Open Math Challenge Part A Problem 3
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Points $(0,0)$ and $(3\sqrt7,7\sqrt3)$ are the endpoints of a diameter of circle $\Gamma.$ Determine the other $x$ intercept of $\Gamma.$
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?
2018 Canadian Open Math Challenge, C4
Source: 2018 Canadian Open Math Challenge Part C Problem 4
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Given a positive integer $N$, Matt writes $N$ in decimal on a blackboard, without writing any of the leading 0s. Every minute he takes two consicutive digits, erases them, and replaces them with the last digit of their product. Any leading zeroes created this way are also erased. He repeats this process for as long as he likes. We call the positive integer $M$ [i]obtainable[/i] from $N$ if starting from $N$, there is a finite sequence of moves that Matt can make to produce the number $M$. For example, 10 is obtainible from 251023 via
\[2510\underline{23}\rightarrow\underline{25} 106\rightarrow 1\underline{06}\rightarrow 10\]
$\text{(a)}$ Show that 2018 is obtainablefrom 2567777899.
$\text{(b)}$ Find two positive integers $A$ and $B$ for which there is no positive integer $C$
[color=transparent](B.)[/color] such that both $A$ and $B$ are obtainablefrom $C$
$\text{(c)}$ Let $S$ be any finite set of positive integers, none of which contains the digit 5
[color=transparent](C.)[/color] in its decimal representation. Prove that there exists a positive integer $N$
[color=transparent](C.)[/color] for which all elements of $S$ are obtainable from $N$.
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 $F(x) = cos(sin(x))$.
2018 Canadian Open Math Challenge, C2
Source: 2018 Canadian Open Math Challenge Part C Problem 2
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Alice has two boxes $A$ and $B$. Initially box $a$ contains $n$ coins and box $B$ is empty. On each turn, she may either move a coin from box $a$ to box $B$, or remove $k$ coins from box $A$, where $k$ is the current number of coins in box $B$. She wins when box $A$ is empty.
$\text{(a)}$ If initially box $A$ contains 6 coins, show that Alice can win in 4 turns.
$\text{(b)}$ If initially box $A$ contains 31 coins, show that Alice cannot win in 10 turns.
$\text{(c)}$ What is the minimum number of turns needed for Alice to win if box $A$ initially contains 2018 coins?
2024 Canadian Open Math Challenge, C2
a) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $15$?
b) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $13$?
c) How many ways are there to pair up the elements of $\{1,2,\dots,2024\}$ into $1012$ pairs so that each pair has sum at least $2001$?
2024 Canadian Open Math Challenge, B2
David wanted to calculate the volume of a prism with an equilateral triangular base. He was given the height of the prism $H=15$ and the height of the base $h=6$. He accidentally swapped the values of $H$ and $h$ in his calculations. By what number should he multiply his result to get the correct volume?