Found problems: 85335
2012 USA TSTST, 2
Let $ABCD$ be a quadrilateral with $AC = BD$. Diagonals $AC$ and $BD$ meet at $P$. Let $\omega_1$ and $O_1$ denote the circumcircle and the circumcenter of triangle $ABP$. Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$. Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$ (other than $B$ and $C$), respectively. Let $M$ and $N$ be the midpoints of minor arcs $\widehat {SP}$ (not including $B$) and $\widehat {TP}$ (not including $C$). Prove that $MN \parallel O_1O_2$.
2019 CHMMC (Fall), 1
Let $ABC$ be an equilateral triangle of side length $6$. Points $D, E$ and $F$ are on sides $AB$, $BC$, and $AC$ respectively such that $AD = BE = CF = 2$. Let circle $O$ be the circumcircle of $DEF$, that is, the circle that passes through points $D, E$, and $F$. What is the area of the region inside triangle $ABC$ but outside circle $O$?
2019 Regional Olympiad of Mexico Southeast, 5
Let $n$ a natural number and $A=\{1, 2, 3, \cdots, 2^{n+1}-1\}$. Prove that if we choose $2n+1$ elements differents of the set $A$, then among them are three distinct number $a,b$ and $c$ such that
$$bc<2a^2<4bc$$
Kvant 2022, M2729
Determine all positive integers $n{}$ and $m{}$ such that $m^n=n^{3m}$.
[i]Proposed by I. Dorofeev[/i]
2008 ITest, 60
Consider the Harmonic Table
\[\begin{array}{c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c}&&&1&&&\\&&\tfrac12&&\tfrac12&&\\&\tfrac13&&\tfrac16&&\tfrac13&\\\tfrac14&&\tfrac1{12}&&\tfrac1{12}&&\tfrac14\\&&&\vdots&&&\end{array}\] where $a_{n,1}=1/n$ and \[a_{n,k+1}=a_{n-1,k}-a_{n,k}.\] Find the remainder when the sum of the reciprocals of the $2007$ terms on the $2007^\text{th}$ row gets divided by $2008$.
2006 MOP Homework, 4
Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.
2021 MIG, 6
Which of the following choices is an even number?
$\textbf{(A) }2 \cdot 0 + 2 - 1\qquad\textbf{(B) }20 + 21\qquad\textbf{(C) }2^0 - 2 + 1\qquad\textbf{(D) }2 - 0 \cdot 2 + 1\qquad\textbf{(E) }2 \cdot 0 + 2 + 1$
2013 IMO, 1
Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\]
[i]Proposed by Japan[/i]
JOM 2015 Shortlist, G7
Let $ABC$ be an acute triangle. Let $H_A,H_B,H_C$ be points on $BC,AC,AB$ respectively such that $AH_A\perp BC, BH_B\perp AC, CH_C\perp AB$. Let the circumcircles $AH_BH_C,BH_AH_C,CH_AH_B$ be $\omega_A,\omega_B,\omega_C$ with circumcenters $O_A,O_B,O_C$ respectively and define $O_AB\cap \omega_B=P_{AB}\neq B$. Define $P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}$ similarly. Define circles $\omega_{AB},\omega_{AC}$ to be $O_AP_{AB}H_C,O_AP_{AC}H_B$ respectively. Define circles $\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}$ similarly.
Prove that there are $6$ pairs of tangent circles in the $6$ circles of the form $\omega_{xy}$.
1986 ITAMO, 3
Two numbers are randomly selected from interval $I = [0, 1]$. Given $\alpha \in I$, what is the probability that the smaller of the two numbers does not exceed $\alpha$?
Is the answer $(100 \alpha)$%, it just seems too easy. :|