Let $f(n)$ count the number of values $0\le k\le n^2$ such that $43\nmid\binom{n^2}{k}$. Find the least positive value of $n$ such that $$43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)$$ [i]Proposed by Adam Bertelli[/i]

What is the difference between the median and the mean of the following data set: $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$?

Find all real $x,y$ such that \[x^2 + 2y^2 + \frac{1}{2} \le x(2y+1) \]

1. In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.(12 points)

Peter is chasing after Rob. Rob is running on the line $y=2x+5$ at a speed of $2$ units a second, starting at the point $(0,5)$. Peter starts running $t$ seconds after Rob, running at $3$ units a second. Peter also starts at $(0,5)$ and catches up to Rob at the point $(17,39)$. What is the value of t?

Let $a, b, c$ be positive numbers such that $a^2+b^2+c^2+abc = 4$. Prove that $$\frac{a + b}{c} +\frac{b + c}{a} +\frac{c + a}{b} \ge a + b + c + \frac{1}{a} + \frac{1}{b} +\frac{1}{c}$$

5. Let $a,b,c,d,e$ be positive rational numbers. Consider integral expressions $f(x)=ax^2+bx+c$ $g(x)=dx+e$ Put $\frac{f(n)}{g(n)}$ an integer for all positive integers $n$. Then, show that $f(x)$ is dividible by $g(x)$.

Find all positive integer $n$ such that for all $i=1,2,\cdots,n$, $\frac{n!}{i!(n-i+1)!}$ is an integer. [i]Proposed by ckliao914[/i]

Let $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and $\deg P(x)=\deg Q(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions, then $ P(x+1)=Q(x-1) $ has a real solution.

Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.

If $x$ is a positive real number, and $n$ is a positive integer, prove that \[[ nx] > \frac{[ x]}1 + \frac{[ 2x]}2 +\frac{[ 3x]}3 + \cdots + \frac{[ nx]}n,\] where $[t]$ denotes the greatest integer less than or equal to $t$. For example, $[ \pi] = 3$ and $\left[\sqrt2\right] = 1$.

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$. (1) Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear. (2) Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.

Let $a,b,$ and $c$ be pairwise distinct complex numbers such that $$a^2=b+6, \quad b^2=c+6, \quad \text{and} \quad c^2=a+6.$$ Compute the two possible values of $a+b+c.$

A frog starts a journey at $(6,9).$ A skip is the act of traveling a positive integer number of units straight south or a positive integer number of units straight west. A jump is the act of traveling one unit straight west. A hop consists of any skip followed by a jump. How many different sequences of hops can the frog take so that the frog's final destination is $(0,0)$? [i]Proposed by Jack Ma[/i]

Let $a_1, a_2, . . . , a_n$ be positive real numbers and $n \ge 1$. Show that $n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})$ When does equality hold?

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that $$n+gcd(n, k)=k.$$

Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\text{Aut } H| > |\text{Aut } G|$? Disprove or provide an example. Here the notation $ |\text{Aut } X|$ for some group $ X$ denotes the number of isomorphisms from $ X$ to itself.

Let $r\geq2$ be a fixed positive integer, and let $F$ be an infinite family of sets, each of size $r$, no two of which are disjoint. Prove that there exists a set of size $r-1$ that meets each set in $F$.

For all positive real numbers $a,b,c$ with $a+b+c=3$, show that $$a^3b+b^3c+c^3a+9\geq 4(ab+bc+ca).$$

Prove the equality $$\prod_{k=1}^{2m}\cos\frac{k\pi}{2m+1}=\frac{(-1)^m}{4m}.$$

Let $x, y, z$ be nonzero real numbers such that $ x + y + z = 0$ and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1 -xyz + \frac{1}{xyz}.$$ Determine the value of the expression ' $$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$

The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$. Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.

Two squares of area $38$ are given. Each of the squares is divided into $38$ connected pieces of unit area by simple curves. Then the two squares are patched together. Show that one can sting the patched squares with $38$ needles so that every piece of each square is stung exactly once.