Compute the unique five-digit positive integer $\underline{abcde}$ such that $a \neq 0, c \neq 0,$ and $$\underline{abcde}=(\underline{ab}+\underline{cde})^2.$$

In the country of Muilejistan, there exists a network of roads connecting all its cities. The network has the particular property that for any two cities, there is a unique path without backtracking (i.e., a path where the traveler never returns along the same road). The longest possible path between two cities is 600 kilometers. For instance, the path from the city of Mlar to the city of Nlar is 600 kilometers. Similarly, the path from the city of Klar to the city of Glar is also 600 kilometers. 1. If Jalim departs from Mlar towards Nlar at noon and Kalim departs from Klar towards Glar also at noon, both traveling at the same speed, prove that they meet at some point on their journey. 2. If the distance in kilometers between any two cities is an integer, prove that the distance from Glar to Mlar is even.

In a cone with height $3$ and base radius $4$, let $X$ be a point on the circumference of the base. Let $Y$ be a point on the surface of the cone such that the distance from $Y$ to the vertex of the cone is $2$, and $Y$ is diametrically opposite $X$ with respect to the base of the cone. The length of the shortest path across the surface of the cone from $X$ to $Y$ can be expressed as $\sqrt{a +\sqrt{b}}$, where a and b are positive integers. Find $a +b$.

Prove that there are exactly $\binom{k}{[k/2]}$ arrays $a_1, a_2, \ldots , a_{k+1}$ of nonnegative integers such that $a_1 = 0$ and $|a_i-a_{i+1}| = 1$ for $i = 1, 2, \ldots , k.$

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

Given a triangle $ABC $, $A {{A} _ {1}} $, $B {{B} _ {1}} $, $C {{C} _ {1}}$ - its chevians intersecting at one point. ${{A} _ {0}}, {{C} _ {0}} $ - the midpoint of the sides $BC $ and $AB$ respectively. Lines ${{B} _ {1}} {{C} _ {1}} $, ${{B} _ {1}} {{A} _ {1}} $and ${ {B} _ {1}} B$ intersect the line ${{A} _ {0}} {{C} _ {0}} $ at points ${{C} _ {2}} $ , ${{A} _ {2}} $ and ${{B} _ {2}} $, respectively. Prove that the point ${{B} _ {2}} $ is the midpoint of the segment ${{A} _ {2}} {{C} _ {2}} $. (Eugene Bilokopitov)

A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it would take for $1$ teaching assistant to grade all the homework assignments.

In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$? j h i e f g a b c d

Denote by $(a, b)$ the greatest common divisor of $a$ and $b$. Let $n$ be a positive integer such that $(n, n + 1) < (n, n + 2) <... < (n,n + 35)$. Prove that $(n, n + 35) < (n,n + 36)$.

Consider the graph $G$ with $100$ vertices, and the minimum odd cycle goes through $13$ vertices. Prove that the vertices of the graph can be colored in $6$ colors in a way that no two adjacent vertices have the same color.

In every acyclic graph with 2022 vertices we can choose $k$ of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of $k$.

Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. $ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 18 $

Prove that there are no functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x,y\in \mathbb{R}:$ $ f(x^2+g(y)) -f(x^2)+g(y)-g(x) \leq 2y$ and $f(x)\geq x^2$. [i]Proposed by Mohammad Ahmadi[/i]

There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?

Gary is baking cakes, one at a time. However, Gary’s not been having much success, and each failed cake will cause him to slowly lose his patience, until eventually he gives up. Initially, a failed cake has a probability of $0$ of making him give up. Each cake has a $1/2$ of turning out well, with each cake independent of every other cake. If two consecutive cakes turn out well, the probability resets to $0$ immediately after the second cake. On the other hand, if the cake fails, assuming that he doesn’t give up at this cake, his probability of breaking on the next failed cake goes from p to $p + 0.5$. If the expected number of successful cakes Gary will bake until he gives up is$ p/q$, for relatively prime $p, q$, find $p + q$.

For positive numbers $a, b, c$ satisfying condition $a+b+c<2$, Prove that $$ \sqrt{a^2 +bc}+\sqrt{b^2 +ca}+\sqrt{c^2 + ab}<3. $$

At the start of the game there are $ r$ red and $ g$ green pieces/stones on the table. Hojoo and Kestutis make moves in turn. Hojoo starts. The person due to make a move, chooses a colour and removes $ k$ pieces of this colour. The number $ k$ has to be a divisor of the current number of stones of the other colour. The person removing the last piece wins. Who can force the victory?

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

Use $[x]$ to represent the greatest integer no more than a real number $x$. Let $$S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,)$$ Prove that there are infinitely many $n$ such that $C_n^{S_n}$ is an even number. [b]PS.[/b] [i]Attached is the original wording which forgets left [/i] [b][ [/b][i]. I hope it is ok where I put it.[/i]

For a permutation $ \sigma\in S_n$ with $ (1,2,\dots,n)\mapsto(i_1,i_2,\dots,i_n)$, define \[ D(\sigma) \equal{} \sum_{k \equal{} 1}^n |i_k \minus{} k| \] Let \[ Q(n,d) \equal{} \left|\left\{\sigma\in S_n : D(\sigma) \equal{} d\right\}\right| \] Show that when $ d \geq 2n$, $ Q(n,d)$ is an even number.

Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.

[asy]draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); draw((0,5)--(-2,7)); label("F",(0,0.5),W); label("D",(0,1.5),W); label("C",(0,2.5),W); label("B",(0,3.5),W); label("A",(0,4.5),W); label("A",(0.5,5),N); label("B",(1.5,5),N); label("C",(2.5,5),N); label("D",(3.5,5),N); label("F",(4.5,5),N); label("0",(0.5,0),N); label("0",(0.5,1),N); label("1",(0.5,2),N); label("1",(0.5,3),N); label("2",(0.5,4),N); label("0",(1.5,0),N); label("0",(1.5,1),N); label("3",(1.5,2),N); label("4",(1.5,3),N); label("2",(1.5,4),N); label("2",(2.5,0),N); label("1",(2.5,1),N); label("5",(2.5,2),N); label("3",(2.5,3),N); label("1",(2.5,4),N); label("1",(3.5,0),N); label("1",(3.5,1),N); label("2",(3.5,2),N); label("0",(3.5,3),N); label("0",(3.5,4),N); label("0",(4.5,0),N); label("1",(4.5,1),N); label("0",(4.5,2),N); label("0",(4.5,3),N); label("0",(4.5,4),N); label("TEST 2",(1,6),N); label("TEST 1",(-2,5),SW);[/asy] The table displays the grade distribution of the $ 30$ students in a mathematics class on the last two tests. For example, exactly one student received a "D" on Test 1 and a "C" on Test 2. What percent of the students received the same grade on both tests? \[ \textbf{(A)}\ 12 \% \qquad \textbf{(B)}\ 25 \% \qquad \textbf{(C)}\ 33 \frac{1}{3} \% \qquad \textbf{(D)}\ 40 \% \qquad \textbf{(E)}\ 50 \% \qquad \]

The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$$ is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$ $\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$

Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. The tangent to $\Gamma_1$ at $B$ intersects $\Gamma_2$ again in $C$, different from $B$; the tangent to $\Gamma_2$ at $B$ intersects $\Gamma_1$ again at $D$, different from $B$. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in $X$ and $Y$, respectively. Let $P$ and $Q$ be the circumcentres of triangles $ACD$ and $XAY$, respectively. Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$. [i]Proposed by Prithwijit De[/i]

Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.