Given that the expression $\frac{20^{21}}{20^{20}} +\frac{20^{20}}{20^{21}}$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Ada Tsui[/i]

In each square of a $101\times 101$ board, except the central one, is placed either a sign " turn" or a sign " straight". The chess piece " car" can enter any square on the boundary of the board from outside (perpendicularly to the boundary). If the car enters a square with the sign " straight" then it moves to the next square in the same direction, otherwise (in case it enters a square with the sign " turn") it turns either to the right or to the left ( its choice). Can one place the signs in such a way that the car never enter the central square?

Let $n$ be an odd integer and $m=\phi(n)$ be the Euler's totient function. Call a set of residues $T=\{a_1, \cdots, a_k\} \pmod n$ to be [i]good[/i] if $\gcd(a_i, n) > 1$ $\forall i$, and $\gcd(a_i, a_j) = 1, \forall i \neq j$. Define the set $S_n$ consisting of the residues $$\sum_{i=1}^k a_i ^m\pmod{n}$$ over all possible residue sets $T=\{a_1,\cdots,a_k\}$ that is good. Determine $|S_n|$. [i]Proposed by Anzo Teh Zhao Yang[/i]

Find all triples $(a,b,k)$, $k \geq 2$, of positive integers such that $(a^k+b)(b^k+a)$ is a power of two.

Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.

In the cells of a $5 \times 5$ grid there are some lamps. If a lamp is touched, it is turned on and it lights up all of its neighbouring cells, including its own cell. If a cell is lit up and there is a lamp in it, the lamp is also turned on and lights up its neighbouring cells, including its own. What is the smallest number of lamps needed to light up all of the cells with just one touch? [i](Note: Two cells are neighbours if they have a common side or vertex.)[/i]

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

Let $a$ and $b$ be positive real numbers satisfying $$a -12b = 11 -\frac{100}{a} \,\,\,\,and \,\,\,\, a -\frac{12}{b}= 4 -\frac{100}{a}.$$ Then $a + b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

Suppose that $ a,b$ are positive integers for which $ A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a}$ is an integer.Prove that $ A\equal{}3$.

Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.

(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win? (b) What is the answer to this question if the second player is permitted to colour two squares at once? (DG Azov) PS. (a) for Juniors, (a),(b) for Seniors

An ant sits at vertex $A$ of unit square $ABCD$. He needs to get to point $C$, where the entrance to the anthill is located. Points $A$ and $C$ are separated by a vertical wall in the form of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path that an ant must overcome in order to get into the anthill.

Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*} S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\ S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PC<PB \\ S_3=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PA<PC \\ S_4=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PC<PA \\ S_5=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PA<PB \\ S_6=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PB<PA\end{align*} Suppose that the ratio of the area of the largest region to the area of the smallest non-empty region is $49 : 1$. Determine the ratio $AC : AB$.

A $2024$-[i]table [/i]is a table with two rows and $2024$ columns containg all the numbers $1,2,\dots,4048$. Such a table is [i]evenly coloured[/i] if exactly half of the numbers in each row, and one number in each column, is coloured red. The [i]red sum[/i] in an evenly coloured $2024$-table is the sum of all the red numbers in the table. Let $N$ be the largest number such that every $2024$-table has an even colouring with red sum $\ge N$. Determine $N$, and find the number of $2024$-tables such that every even colouring of the table has red sum $\le N$.

Let $ABTCD$ be a convex pentagon with area $22$ such that $AB = CD$ and the circumcircles of triangles $TAB$ and $TCD$ are internally tangent. Given that $\angle{ATD} = 90^{\circ}, \angle{BTC} = 120^{\circ}, BT = 4,$ and $CT = 5$, compute the area of triangle $TAD$.

Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.

Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.

A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?

When the expression $$(xy-5x+3y-15)^n$$ for some positive integer $n$ is expanded and like terms are combined, the expansion contains at least 2021 distinct terms. Compute the minimum possible value of $n$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 4)[/i]

Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK $.

In triangle $ ABC$, $ P$ is the midpoint of side $ BC$. Let $ M\in(AB)$, $ N\in(AC)$ be such that $ MN\parallel BC$ and $ \{Q\}$ be the common point of $ MP$ and $ BN$. The perpendicular from $ Q$ on $ AC$ intersects $ AC$ in $ R$ and the parallel from $ B$ to $ AC$ in $ T$. Prove that: (a) $ TP\parallel MR$; (b) $ \angle MRQ\equal{}\angle PRQ$. [i]Mircea Fianu[/i]

Given is an acute angled triangle $ABC$ with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$ and $P$ be the point of intersection of $\omega$ and $k$ other than $A$. Assume that $BP$ and $CP$ intersect $\omega$ for the second time at points $Q$ and $R$, respectively. If $D$ is the foot of the altitude from $A$ to $BC$ and $S$ is the point of the intersection of $\omega$ and $QD$, prove that $HR = HS$.

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$