Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.

Estimate the smallest value of $r$ such that a circle of radius $r$ can contain $19$ non-overlapping circles of radius $1$. Express your answer to the nearest hundredth. For example, $11.00$, $5.60$, and $1.34$ are valid responses, but $11$ and $1.342$ are not. An invalid response will receive a score of zero.

Let $ABC$ be a triangle such that $AB = 5$, $AC = 8$, and $\angle BAC = 60^{\circ}$. Let $P$ be a point inside the triangle such that $\angle APB = \angle BPC = \angle CPA$. Lines $BP$ and $AC$ intersect at $E$, and lines $CP$ and $AB$ intersect at $F$. The circumcircles of triangles $BPF$ and $CPE$ intersect at points $P$ and $Q \neq P$. Then $QE + QF=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$. [i]Proposed by Ankan Bhattacharya[/i]

Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$ Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$ Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct? [i]Proposed by Ivan Chan Kai Chin[/i]

On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let $ M$ be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when $ M$ is divided by 10.

Let $ABC$ be a triangle satisfying $AB<AC$. Let $M$ be the midpoint of $BC$. A point $P$ lies on the segment $AB$ with $AP>PB$. A point $Q$ lies on the segment $AC$ with $\angle MPA = \angle AQM$. The perpendicular bisectors of $BC$ and $PQ$ intersect at $S$. Prove that $\angle BAC + \angle QSP = \angle QMP$.

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Tim is thinking of a positive integer between $2$ and $15,$ inclusive, and Ted is trying to guess the integer. Tim tells Ted how many factors his integer has, and Ted is then able to be certain of what Tim's integer is. What is Tim's integer?

Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.

A little boy wrote the numbers $1,2,\cdots,2011$ on a blackboard. He picks any two numbers $x,y$, erases them with a sponge and writes the number $|x-y|$. This process continues until only one number is left. Prove that the number left is even.

Prove the inequality \[ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. \]

In $\bigtriangleup$$ABC$ $BL$ is bisector. Arbitrary point $M$ on segment $CL$ is chosen. Tangent to $\odot$$(ABC)$ at $B$ intersects $CA$ at $P$. Tangents to $\odot$$BLM$ at $B$ and $M$ intersect at point $Q$. Prove that $PQ$$\parallel$$BL$.

Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.

Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be? [i]I. Bogdanov, D. Fon-Der-Flaass[/i]

Daniel chooses some distinct subsets of $\{1, \dots, 2019\}$ such that any two distinct subsets chosen are disjoint. Compute the maximum possible number of subsets he can choose. [i]Proposed by Ankan Bhattacharya[/i]

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

Find all real numbers $x,y,z$ satisfying the following system of equations: \begin{align*} xy+z&=-30\\ yz+x &= 30\\ zx+y &=-18 \end{align*}

Compute the number of positive integers $n$ less than $100$ for which $1+2+\dots+n$ is not divisible by $n.$

Suppose that $n$ triangles are given in the plane such that any three of them have a common vertex, but no four of them do. Find the greatest possible $n$.

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$. Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$. Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .

Prove that: for all positive integers $n\ge 2,$ the polynomial$$(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1$$ is irreducible in $\mathbb{Q}[x].$

Let $G$ be a planar graph, which is also bipartite. Is it always possible to assign a vertex to each face of the graph such that no two faces have the same vertex assigned to them? [i]Submitted by Dávid Matolcsi, Budapest[/i]