Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers $$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$ are greater than $1$.

How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2357$ is one such integer.) $\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }21 \qquad \textbf{(E) }28$

Let $A B C D$ be a unit square. Suppose $M$ and $N$ are points on $B C$ and $C D$ respectively such that the perimeter of triangle $M C N$ is 2 . Let $O$ be the circumcentre of triangle $M A N$, and $P$ be the circumcentre of triangle $M O N$. If $\left(\frac{O P}{O A}\right)^2=\frac{m}{n}$ for some relatively prime positive integers $m$ and $n$, find the value of $m+n$.

Each pair of vertices of a regular 10-sided polygon is connected by a line segment. How many unordered pairs of distinct parallel line segments can be chosen from these segments? [i]Author: Ray Li[/i]

Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$

There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?

Let $n$ be a positive integer, and let $[n]=\{1,2,\dots,n\}$. Find the maximum posible cardinality of a subset $S$ of $[n]$ with the property that there aren't any distinct $a,b,c\in S$ such that $a+b=c$.

Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

$27$ students in a school take French. $32$ students in a school take Spanish. $5$ students take both courses. How many of these students in total take only $1$ language course?

Let $ABCD$ be a tetrahedron and let $H_{a},H_{b},H_{c},H_{d}$ be the orthocenters of triangles $BCD,CDA,DAB,ABC$, respectively. Prove that lines $AH_{a},BH_{b},CH_{c}, DH_{d}$ are concurrent if and only if $AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2$

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

Determine the perfect squares $ \overline{aabcd} $ of five digits such that $ \overline{dcbaa} $ is a perfect square of five digits.

Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.

The number of maps $f$ from $1,2,3$ into the set $1,2,3,4,5$ such that $f(i)\le f(j)$ whenever $i\le j$ is $\textbf{(A)}~60$ $\textbf{(B)}~50$ $\textbf{(C)}~35$ $\textbf{(D)}~30$

Let $S_P$ be the set of all polynomials $P$ with complex coefficients, such that $P(x^2) = P(x)P(x-1)$ for all complex numbers $x$. Suppose $P_0$ is the polynomial in $S_P$ of maximal degree such that $P_0(1) \mid 2016$. Find $P_0(10)$.

Let $ABC$ be a triangle, where $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$. Let $E$ be a point in $AC$ such that $DE \perp AC$. Let $F$ be a point in $AB$ such that $EF \parallel BC$. If $EC = 4$, determine the length of $EF$.

For a positive integer $n$, denote $rad(n)$ as product of prime divisors of $n$. And also $rad(1)=1$. Define the sequence $\{a_i\}_{i=1}^{\infty}$ in this way: $a_1 \in \mathbb N$ and for every $n \in \mathbb N$, $a_{n+1}=a_n+rad(a_n)$. Prove that for every $N \in \mathbb N$, there exist $N$ consecutive terms of this sequence which are in an arithmetic progression.

Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $2^a+3^b+1=6^c$.

Given that $\tan A=\frac{12}{5}$, $\cos B=-\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\cos (A-B)$. $ \textbf{(A) }-\frac{63}{65}\qquad\textbf{(B) }-\frac{64}{65}\qquad\textbf{(C) }\frac{63}{65}\qquad\textbf{(D) }\frac{64}{65}\qquad\textbf{(E) }\frac{65}{63} $

How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the above} $

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram. [asy] import markers; unitsize(1.5 cm); pair A, B, C, U, V, W; A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2; draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, N); dot("$U$", U, NE); dot("$V$", V, NE); dot("$W$", W, NW); [/asy]

A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.

We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country. Prove that one can choose $n$ people into one class such that they come from different groups and different countries.

In the fall of $1996$, a total of $800$ students participated in an annual school clean-up day. The organizers of the event expect that in each of the years $1997$, $1998$, and $1999$, participation will increase by $50 \%$ over the previous year. The number of participants the organizers will expect in the fall of $1999$ is $\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700$