Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

A particle moves from the point \(P\) to the point \(Q\) in the Cartesian plane. When it passes through any point \((x,y)\), the particle has an instantaneous speed of \(\sqrt{x + y}\). Compute the minimum time required for the particle to move: (i) from \(P_1=(-1,0)\) to \(Q_1=(1,0)\), and (ii) from \(P_2=(0,1)\) to \(Q_2=(1,1)\). [i]proposed by the ICMC Problem Committee[/i]

Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$.

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Let $p > 2$ be a fixed prime number. Find all functions $f: \mathbb Z \to \mathbb Z_p$, where the $\mathbb Z_p$ denotes the set $\{0, 1, \ldots , p-1\}$, such that $p$ divides $f(f(n))- f(n+1) + 1$ and $f(n+p) = f(n)$ for all integers $n$. [i]Proposed by Grant Yu[/i]

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

There are $100$ lightbulbs $B_1,\ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1,\ldots, S_{100}$ such that for all $1\leq i\leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where here $B_{101} = B_1$). Suppose David chooses whether to flick each switch with probability $\tfrac12$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?

Are there any integers $a,b$ and $c$, not all of them $0$, such that $$a^2=2021b^2+2022c^2~~?$$ [i] (Cosmin Gavrilă)[/i]

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circles, internally tangent at $P$ ($\mathcal{C}_2$ lies inside of $\mathcal{C}_1$). A chord $AB$ of $\mathcal{C}_1$ is tangent to $\mathcal{C}_2$ at $C.$ Let $D$ be the second point of intersection between the line $CP$ and $\mathcal{C}_1.$ A tangent from $D$ to $\mathcal{C}_2$ intersects $\mathcal{C}_1$ for the second time at $E$ and it intersects $\mathcal{C}_2$ at $F.$ Prove that $F$ is the incenter of triangle $ABE.$

Find, with proof, all positive integers $n$ such that neither $n$ nor $n^2$ contain a $1$ when written in base $3$.

Suppse that $n\geq 2$ and $0<x_1<x_2<...<x_n$ are integer numbers. We denote that :\[ S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n. \] (where $A$ is a non-empty subset). Show that if $S_n ,S_{n-1}$ were positive integer numbers , then $\forall k : S_k$ is a positive integer.

Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that \[\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )\] prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that \[\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )\]

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?

The Banana Republic language has more words than letters in its alphabet. Prove that there is a natural number $k$ for which we can choose $k$ different words that use exactly $k$ different letters.

The exchange rate in a Funny-Money machine is $s$ McLoonies for a Loonie or $\frac{1}{s}$ Loonies for a McLoonie, where $s$ is a positive real number. The number of coins returned is rounded off to the nearest integer. If it is exactly in between two integers, then it is rounded up to the greater integer. $(a)$ Is it possible to achieve a one-time gain by changing some Loonies into McLoonies and changing all the McLoonies back to Loonies? $(b)$ Assuming that the answer to $(a)$ is "yes", is it possible to achieve multiple gains by repeating this procedure, changing all the coins in hand and back again each time?

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

In how many ways can a set of $12$ elements be partitioned into six two-element subsets?

Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that \[ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . \] Prove that \[ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . \]

Does the sequence \[11, 111, 1111, 11111, \ldots\] contain any fifth power of a positive integer? Justify your answer.

Is there a quadrilateral, any vertex of which can be moved to another location so that the new quadrilateral is congruent to the original one?

There are $ 5$ dogs, $ 4$ cats, and $ 7$ bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken?

Consider a triangle $ABC$ and points $D,E,F,G,H,I$ in the plane such that $ABED$, $BCGF$ and $ACHI$ are squares exterior to the triangle. Prove that points $D,E,F,G,H,I$ are concyclic if and only if one of the following two statements hold: (i) $ABC$ is an equilateral triangle. (ii) $ABC$ is an isosceles right triangle.

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

Prove that for an integer $ n>\equal{}1$ we have $ n(1\plus{}\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\plus{}\frac{1}{n})\geq (n\plus{}1)(\frac{1}{2}\plus{}\frac{1}{3}\plus{}\dots\frac{1}{n\plus{}1})$

a) The fields of the square table $4\times 4$ are filled with the "+" or "-" signs. You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner square, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only. b) The fields of the square table $8\times 8$ are filled with the "+" or signs except one non-corner field with "-". You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner field, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only.