Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$,$C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $AJG$ is an equilateral triangle, then the area of $AJG$ is $a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

Peter marks several cells on a $5\times 5$ board. Basil wins if he can cover all marked cells with three-cell corners. The corners must be inside the board and not overlap. What is the least number of cells Peter should mark to prevent Basil from winning? (Cells of the corners must coincide with the cells of the board).

The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? [asy] for (int a = 0; a < 6; ++a) { for (int b = 0; b < 6; ++b) { dot((4*a,3*b)); } } draw((0,0)--(20,0)--(20,15)--(0,15)--cycle); draw((0,0)--(16,12)); draw((0,0)--(16,9)); label(rotate(30)*"Bjorn",(12,6.75),SE); label(rotate(37)*"Alberto",(11,8.25),NW); label("$0$",(0,0),S); label("$1$",(4,0),S); label("$2$",(8,0),S); label("$3$",(12,0),S); label("$4$",(16,0),S); label("$5$",(20,0),S); label("$0$",(0,0),W); label("$15$",(0,3),W); label("$30$",(0,6),W); label("$45$",(0,9),W); label("$60$",(0,12),W); label("$75$",(0,15),W); label("H",(6,-2),S); label("O",(8,-2),S); label("U",(10,-2),S); label("R",(12,-2),S); label("S",(14,-2),S); label("M",(-4,11),N); label("I",(-4,9),N); label("L",(-4,7),N); label("E",(-4,5),N); label("S",(-4,3),N);[/asy] $ \text{(A)}\ 15\qquad\text{(B)}\ 20\qquad\text{(C)}\ 25\qquad\text{(D)}\ 30\qquad\text{(E)}\ 35 $

Davi and George are taking a city tour through Fortaleza, with Davi initially leading. Fortaleza is organized like an $n \times n$ grid. They start in one of the grid's squares and can move from one square to another adjacent square via a street (for each pair of neighboring squares on the grid, there is a street connecting them). Some streets are dangerous. If Davi or George pass through a dangerous street, they get scared and swap who is leading the city tour. Their goal is to pass through every block of Fortaleza exactly once. However, if the city tour ends with George in command, the entire world becomes unemployed and everyone starves to death. Given that there is at least one street that is not dangerous, prove that Davi and George can achieve their goal without everyone dying of hunger.

Let $BE{}$ and $CF$ be heights in the acute-angled triangle $ABC{}$ and let $O{}$ be its circumcenter. The points $M{}$ and $N{}$ are selected on the side $BC{}$ so that $BM=CN.{}$ The line $BE{}$ intersects the circle $(MBF)$ a second time at $P{}$ and the line $CF{}$ intersects the circle $(NCE)$ a second time at $Q.{}$ Prove that the lines $PF, QE$ and $AO{}$ intersect at the same point. [i]Proposed by Luu Dong[/i]

Source: 2017 Canadian Open Math Challenge, Problem A3 ----- Two $1$ × $1$ squares are removed from a $5$ × $5$ grid as shown. [asy] size(3cm); for(int i = 0; i < 6; ++i) { for(int j = 0; j < 6; ++j) { if(j < 5) { draw((i, j)--(i, j + 1)); } } } draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(1,5)); draw((2,5)--(5,5)); draw((0,0)--(2,0)); draw((3,0)--(5,0)); [/asy] Determine the total number of squares of various sizes on the grid.

Let $ABC$ be a triangle with $\angle{A} = 60^\circ$. The midperpendicular of segment $AB$ meets line $AC$ at point $C_1$. The midperpendicular of segment $AC$ meets line $AB$ at point $B_1$. Prove that line $B_1C_1$ touches the incircle of triangle $ABC$.

Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. [i]Proposed by Richard Henner[/i]

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ \[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\] a) Prove that $f(2x)=4f(x)$ for all $x>0$; b) Find all such functions. [i]Nikolai Nikolov, Oleg Mushkarov [/i]

Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ f: (0,\infty)\longrightarrow (0,\infty) $ a non-constant function having the property that $ f\left( x^y\right) = \left( f(x)\right)^{f(y)},\quad\forall x,y>0. $ Show that $ f(xy)=f(x)f(y) $ and $ f(x+y)=f(x)+f(y), $ for all $ x,y>0. $

Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that \[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}. \]

Let $a, b, c$ be positive reals. Show that $\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.$

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have?

Julia and Hansol are having a math-off. Currently, Julia has one more than twice as many points as Hansol. If Hansol scores $6$ more points in a row, he will tie Julia’s score. How many points does Julia have?

How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

Let $ a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n,A,B$ are positive reals such that $ a_i\leq b_i,a_i\leq A$ $(i=1,2,\cdots,n)$ and $\frac{b_1 b_2 \cdots b_n}{a_1 a_2 \cdots a_n}\leq \frac{B}{A}.$ Prove that$$\frac{(b_1+1) (b_2+1) \cdots (b_n+1)}{(a_1+1) (a_2+1) \cdots (a_n+1)}\leq \frac{B+1}{A+1}.$$

Given are the points $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$ in the space. Let $P_{ij} (i,j \in 0,1,2,3)$ be the point determined by the equality: $\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}$. Find the volume of the smallest convex polyhedron which contains all the points $P_{ij}$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 11 \}$ with $A \cup B = X$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum and maximum possible value of $P_A +P_B$ and find all possible equality cases. [i]Proposed by Greece[/i]

Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.

Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?