Found problems: 106
2019 CMIMC, 10
Let $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Determine the number of positive integers $2\leq n\leq 50$ such that all coefficients of the polynomial
\[
\left(x^{\varphi(n)} - 1\right) - \prod_{\substack{1\leq k\leq n\\\gcd(k,n) = 1}}(x-k)
\]
are divisible by $n$.
MOAA Team Rounds, 2019.5
Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
MOAA Team Rounds, 2019.10
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.
2019 CMIMC, 5
Let $x_n$ be the smallest positive integer such that $7^n$ divides $x_n^2-2$. Find $x_1+x_2+x_3$.
2019 AMC 8, 8
Gilda has a bag of marbles. She gives $20 \%$ of them to her friend Pedro. The, Gilda gives $10 \%$ of what is left to her other friend, Ebony. Finally, Gilda gives $25 \%$ of what is left in the bag to her brother. What percentage of her original bag does she have left?
$\textbf{(A) } 20 \qquad\textbf{(B) } 33\tfrac{1}{3} \qquad\textbf{(C) } 38 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 54$
2019 ASDAN Math Tournament, 1
What is the greatest positive integer $x$ for which $2^{2^x+1}+2$ is divisible by $17$?
2019 CMIMC, 6
Let $ABC$ be a triangle with $AB=209$, $AC=243$, and $\angle BAC = 60^\circ$, and denote by $N$ the midpoint of the major arc $\widehat{BAC}$ of circle $\odot(ABC)$. Suppose the parallel to $AB$ through $N$ intersects $\overline{BC}$ at a point $X$. Compute the ratio $\tfrac{BX}{XC}$.
2019 CMIMC, 6
There are $100$ lightbulbs $B_1,\ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1,\ldots, S_{100}$ such that for all $1\leq i\leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where here $B_{101} = B_1$). Suppose David chooses whether to flick each switch with probability $\tfrac12$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?
2019 MOAA, 8
Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)
2019 ISI Entrance Examination, 5
A subset $\bf{S}$ of the plane is called [i]convex[/i] if given any two points $x$ and $y$ in $\bf{S}$, the line segment joining $x$ and $y$ is contained in $\bf{S}$. A quadrilateral is called [i]convex[/i] if the region enclosed by the edges of the quadrilateral is a convex set.
Show that given a convex quadrilateral $Q$ of area $1$, there is a rectangle $R$ of area $2$ such that $Q$ can be drawn inside $R$.
2019 CMIMC, 6
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of
$$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$
2019 Macedonia Junior BMO TST, 4
Let the real numbers $a$, $b$, and $c$ satisfy the equations
$(a+b)(b+c)(c+a)=abc$ and $(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9$.
Prove that at least one of $a$, $b$, and $c$ equals $0$.
2019 CMIMC, 10
Suppose $ABC$ is a triangle, and define $B_1$ and $C_1$ such that $\triangle AB_1C$ and $\triangle AC_1B$ are isosceles right triangles on the exterior of $\triangle ABC$ with right angles at $B_1$ and $C_1$, respectively. Let $M$ be the midpoint of $\overline{B_1C_1}$; if $B_1C_1 = 12$, $BM = 7$ and $CM = 11$, what is the area of $\triangle ABC$?
MOAA Team Rounds, 2019.2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
2019 ASDAN Math Tournament, 3
Consider a triangle $\vartriangle ABC$ with $BC = 10$. An excircle is a circle that is tangent to one side of the triangle as well as the extensions of the other two sides; suppose that the excircle opposite vertex $B$ has center $I_2$ and exradius $r_2 = 11$, and suppose that the excircle opposite vertex $C$ has center $I_3$ and exradius $r_3 = 13$. Compute $I_2I_3$.
2019 MOAA, 2
The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?
2019 ASDAN Math Tournament, 2
Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$?
MOAA Team Rounds, 2019.7
Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.
2019 ASDAN Math Tournament, 2
A square and a line intersect at a $45^o$ angle. The line bisects the square into two unequal pieces such that the area of one piece is twice that of the other. If the square has side length $6$, compute the length of the cut due to the line.
[img]https://cdn.artofproblemsolving.com/attachments/6/4/2eb33fb9766497d25d342001cdbae9a7ffd4b4.png[/img]
2019 ASDAN Math Tournament, 3
$5$ monkeys, $5$ snakes, and $5$ tigers are standing in line at the local grocery store, with animals of the same species being indistinguishable. A monkey stands at the front of the line and a tiger stands at the end of the line. Unfortunately, monkeys and tigers are sworn enemies, so monkeys and tigers cannot stand in adjacent places in line. Compute the number of possible arrangements of the line.
2019 CMIMC, 5
In the game of Ric-Rac-Roe, two players take turns coloring squares of a $3 \times 3$ grid in their color; a player wins if they complete a row or column of their color on their turn. If Alice and Bob play this game, picking an uncolored square uniformly at random on their turn, what is the probability that they tie?
2019 CMIMC, 3
How many ordered triples $(a,b,c)$ of integers with $1\le a\le b\le c\le 60$ satisfy $a\cdot b=c$?
2019 ASDAN Math Tournament, 9
Consider triangle $\vartriangle ABC$ with circumradius $R = 10$, inradius $r = 2$ and semi-perimeter $S = 18$. Let $I$ be the incenter, and we extend $AI$, $BI$ and $CI$ to intersect the circumcircle at $D, E$ and $F$ respectively. Compute the area of $\vartriangle DEF$.
2019 CMIMC, 1
The figure below depicts two congruent triangles with angle measures $40^\circ$, $50^\circ$, and $90^\circ$. What is the measure of the obtuse angle $\alpha$ formed by the hypotenuses of these two triangles?
[asy]
import olympiad;
size(80);
defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10));
pair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0));
draw(anglemark((4.25,0),P,(0,4.25),10));
label("$\alpha$",P,2 * NE);
[/asy]
2019 CMIMC, 5
On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles?
[asy]
size(270);
defaultpen(linewidth(0.8));
real r = 0.3, rad = 0.1, shift = 3.7;
pen th = linewidth(5)+gray(0.2);
for(int i=0; i<= 2;i=i+1)
{
for(int j=0; j<= 1;j=j+1)
{
fill(circle((i,j),r),gray(0.8));
fill(circle((i+shift,j),r),gray(0.8));
}
draw((0,1)--(2-rad,1)^^(2,1-rad)--(2,rad)^^(2-rad,0)--(0,0),th);
draw(arc((2-rad,1-rad),rad,0,90)^^arc((2-rad,rad),rad,270,360),th);
draw((shift+1,0)--(shift+1,1-2*rad)^^(shift+1-rad,1-rad)--(shift+rad,1-rad)^^(shift+rad,1+rad)--(shift+2,1+rad),th);
draw(arc((shift+1-rad,1-2*rad),rad,0,90)^^arc((shift+rad,1),rad,90,270),th);
}
[/asy]