Found problems: 85335
2002 AMC 12/AHSME, 8
Suppose July of year $ N$ has five Mondays. Which of the following must occur five times in August of year $ N$? (Note: Both months have $ 31$ days.)
$ \textbf{(A)}\ \text{Monday} \qquad
\textbf{(B)}\ \text{Tuesday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Thursday} \qquad
\textbf{(E)}\ \text{Friday}$
2020 BMT Fall, Tie 2
On a certain planet, the alien inhabitants are born without any arms, legs, or noses. Every year, on their birthday, each alien randomly grows either an arm, a leg, or a nose, with equal probability for each. After its sixth birthday, the probability that an alien will have at least $2$ arms, at least $2$ legs, and at least $1$ nose on the day is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2022 Stanford Mathematics Tournament, 2
The incircle of $\triangle ABC$ is centered at $I$ and is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A circle with radius $2$ is centered at each of $D$, $E$, and $F$. Circle $D$ intersects circle $I$ at points $D_1$ and $D_2$. The points $E_1$, $E_2$, $F_1$, and $F_2$ are defined similarly. If the inradius of $\triangle ABC$ is $5$, what is the ratio of the area of the triangle whose sides are formed by extending $D_1D_2$, $E_1E_2$, and $F_1F_2$ to the area of $\triangle ABC$?
2016 Purple Comet Problems, 1
Mike has 12 books, Sean has 9 books, and little Sherry has only 4 books. Find the percentage of these books that Sean has.
2020 Romania EGMO TST, P1
An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.
2012-2013 SDML (Middle School), 12
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
2003 IMO Shortlist, 3
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
2013 Tournament of Towns, 3
Each of $11$ weights is weighing an integer number of grams. No two weights are equal. It is known that if all these weights or any group of them are placed on a balance then the side with a larger number of weights is always heavier. Prove that at least one weight is heavier than $35$ grams.
2023 Austrian MO National Competition, 4
Find all pairs of positive integers $(n, k)$ satisfying the equation $$n!+n=n^k.$$
2016 Federal Competition For Advanced Students, P1, 1
Determine the largest constant $C$ such that
$$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$
holds for all real numbers $x_1, x_2, \cdots , x_6$.
For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds.
(Walther Janous)