Found problems: 89
2020 JBMO Shortlist, 1
Let $\triangle ABC$ be an acute triangle. The line through $A$ perpendicular to $BC$ intersects $BC$ at $D$.
Let $E$ be the midpoint of $AD$ and $\omega$ the the circle with center $E$ and radius equal to $AE$. The line
$BE$ intersects $\omega$ at a point $X$ such that $X$ and $B$ are not on the same side of $AD$ and the line $CE$
intersects $\omega$ at a point $Y$ such that $C$ and $Y$ are not on the same side of $AD$. If both of the intersection
points of the circumcircles of $\triangle BDX$ and $\triangle CDY$ lie on the line $AD$, prove that $AB = AC$.
2020 CMIMC Algebra & Number Theory, 8
Let $f:\mathbb N\to (0,\infty)$ satisfy $\prod_{d\mid n} f(d) = 1$ for every $n$ which is not prime. Determine the maximum possible number of $n$ with $1\le n \le 100$ and $f(n)\ne 1$.
MOAA Team Rounds, TO2
The Den has two deals on chicken wings. The first deal is $4$ chicken wings for $3$ dollars, and the second deal is $11$ chicken wings for $ 8$ dollars. If Jeremy has $18$ dollars, what is the largest number of chicken wings he can buy?
MOAA Team Rounds, TO1
The number $2020$ has three different prime factors. What is their sum?
2020 JBMO Shortlist, 2
Viktor and Natalia bought $2020$ buckets of ice-cream and want to organize a degustation schedule with $2020$ rounds such that:
- In every round, both of them try $1$ ice-cream, and those $2$ ice-creams tried in a single round
are different from each other.
- At the end of the $2020$ rounds, both of them have tried each ice-cream exactly once.
We will call a degustation schedule fair if the number of ice-creams that were tried by Viktor before Natalia is equal to the number of ice creams tried by Natalia before Viktor.
Prove that the number of fair schedules is strictly larger than $2020!(2^{1010} + (1010!)^2)$.
[i]Proposed by Viktor Simjanoski, Macedonia
[/i]
2020 CMIMC Geometry, 8
Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$. Compute the eccentricity of $\mathcal E$.
(A [i]parabola[/i] $\mathcal P$ is the set of points which are equidistant from a point, called the [i]focus[/i] of $\mathcal P$, and a line, called the [i]directrix[/i] of $\mathcal P$. An [i]ellipse[/i] $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the [i]foci[/i] of $\mathcal E$. The [i]eccentricity[/i] of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)
2020 CMIMC Team, Estimation
Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.
2020 CMIMC Geometry, 2
Let $ABC$ be a triangle. Points $D$ and $E$ are placed on $\overline{AC}$ in the order $A$, $D$, $E$, and $C$, and point $F$ lies on $\overline{AB}$ with $EF\parallel BC$. Line segments $\overline{BD}$ and $\overline{EF}$ meet at $X$. If $AD = 1$, $DE = 3$, $EC = 5$, and $EF = 4$, compute $FX$.
2020 Iran Team Selection Test, 2
Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?
[i]Proposed by Seyed Reza Hosseini[/i]
2020 CMIMC Geometry, 7
In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $ABC$. The circumcircles of $BFI$ and $CEI$ intersect at $X \neq I$. Find the length of $DX$.
2020 CMIMC Team, 7
Points $P$ and $Q$ lie on a circle $\omega$. The tangents to $\omega$ at $P$ and $Q$ intersect at point $T$, and point $R$ is chosen on $\omega$ so that $T$ and $R$ lie on opposite sides of $PQ$ and $\angle PQR = \angle PTQ$. Let $RT$ meet $\omega$ for the second time at point $S$. Given that $PQ = 12$ and $TR = 28$, determine $PS$.
2020 MOAA, TO4
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
2020 ISI Entrance Examination, 5
Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius $1$ is regular (i.e., has equal sides).
2020 CMIMC Geometry, Estimation
Gunmay picks $6$ points uniformly at random in the unit square. If $p$ is the probability that their convex hull is a hexagon, estimate $p$ in the form $0.abcdef$ where $a,b,c,d,e,f$ are decimal digits. (A [i]convex combination[/i] of points $x_1, x_2, \dots, x_n$ is a point of the form $\alpha_1x_1 + \alpha_2x_2 + \dots + \alpha_nx_n$ with $0 \leq \alpha_i \leq 1$ for all $i$ and $\alpha_1 + \alpha_2 + \dots + \alpha_n = 1$. [i]The convex hull[/i] of a set of points $X$ is the set of all possible convex combinations of all subsets of $X$.)