Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?

Let $\Gamma$ is a circle with diameter $AB$. Let $\ell$ be the tangent of $\Gamma$ at $A$, and $m$ be the tangent of $\Gamma$ through $B$. Let $C$ be a point on $\ell$, $C \ne A$, and let $q_1$ and $q_2$ be two lines that passes through $C$. If $q_i$ cuts $\Gamma$ at $D_i$ and $E_i$ ($D_i$ is located between $C$ and $E_i$) for $i = 1, 2$. The lines $AD_1, AD_2, AE_1, AE_2$ intersects $m$ at $M_1, M_2, N_1, N_2$ respectively. Prove that $M_1M_2 = N_1N_2$.

Let $I$ be the point of intersection of the angle bisectors of the $\vartriangle ABC$, $W_1,W_2,W_3$ be point of intersection of lines $AI, BI, CI$ with the circle circumscribed around the triangle, $r$ and $R$ be the radii of inscribed and circumscribed circles respectively. Prove the inequality $$IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}$$

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ ABC$ be a triangle. Circle $ \omega$­ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$­ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of ­$ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.

Consider 6 points on a plane such that 8 of the distances between them are equal to 1. Prove that there are at least 3 points that form an equilateral triangle.

A subset $S$ of the natural numbers is called [i]dense [/i] for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.

$PABCDE$ is a pyramid with $ABCDE$ a convex pentagon. A plane meets the edges $PA, PB, PC, PD, PE$ in points $A', B', C', D', E'$ distinct from $A, B, C, D, E$ and $P$. For each of the quadrilaterals $ABB'A', BCC'B, CDD'C', DEE'D', EAA'E'$ take the intersection of the diagonals. Show that the five intersections are coplanar.

In the country of [i]Sugarland[/i], there are $13$ students in the IMO team selection camp. $6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation. Is it possible that all $13$ students have a chance of being a team member? [i]Proposed by Morteza Saghafian[/i]

(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 (and before 10000)?

You have a 1 by 2024 grid of squares in a column, vertices labelled with coordinates $(0,0)$ to $(1,2024)$. Place a weed at $(0,0)$. When a weed is attempting to be placed at coordinates $(x,y)$, it will be placed with a $50\%$ probability if and only if exactly one of the vertices $(x-1, y)$ or $(x, y-1)$ has a weed on it, otherwise the attempt will fail with probability $1$. The placement attempts are made in the following order: For each vertex with $x$ coordinate $0$, attempt a placement for each vertex starting from $y$ coordinate $0$, incrementing by $1$ until $2024$. Then, attempts will be made on the vertices with $x$ coordinate $1$ in the same fashion. Each placement attempt is made exactly once. The probability that a weed appears on $(1,2024)$ after placing the weed at $(0,0)$ and attempting to place weeds on every vertex is $p$. Estimate $9p\cdot2^{2025}$ to the nearest integer. \\\\ Your score will be calculated by the function $\max(0, \lfloor\frac{2000\log_{10}A}{(A - S)^2+100\log_{10}A}\rfloor)$, where $S$ is your submission and $A$ is the true answer. [i]Lightning 5.2[/i]

a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle. b) Prove the same statement for the arbitrary tangential polygon. c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.

Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.

Let $C_1$ and $C_2$ be two different circles , and let their radii be $r_1$ and $r_2$ , the two circles are passing through the two points $A$ and $B$ (i)Let $P_1$ on $C_1$ and $P_2$ on $C_2$ such that the line $P_1P_2$ passes through $A$. Prove that $P_1B \cdot r_2 = P_2B \cdot r_1$ (ii)Let $DEF$ be a triangle that it's inscribed in $C_1$ , and let $D'E'F'$ be a triangle that is inscribed in $C_2$ . The lines $EE'$,$DD'$ and $FF'$ all pass through $A$ . Prove that the triangles $DEF$ and $D'E'F'$ are similar (iii)The circle $C_3$ also passes through $A$ and $B$ . Let $l$ be a line that passes through $A$ and cuts circles $C_i$ in $M_i$ with $i = 1,2,3$ . Prove that the value of$$\frac{M_1M_2}{M_1M_3}$$is constant regardless of the position of $l$ Provided that $l$ is different from $AB$

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Show that it is possible to color the set of integers \[M=\{ 1, 2, 3, \cdots, 1987 \},\] using four colors, so that no arithmetic progression with $10$ terms has all its members the same color.

Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ [i]nice[/i] if for every $k = 2 ,3, \cdots , n$, we have that $a_1^2 + a_2^2 + \cdots + a_k^2$ is [b]not[/b] divisible by $k$. Determine which positive integers $n>1$ have a [i]nice[/i] arrangement.

Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.

Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive numbers such that \[a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.\] Show that the sequence contains an irrational number.

A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, the sum of the unit outgoing vectors along the edges of the graph is zero. Prove that the total length of the web is equal to the number of its vertices on the circle.

Find the real roots of the equation $x^2 + 2ax + \frac{1}{16} = -a +\sqrt{ a^2 + x - \frac{1}{16} }$ , $\left(0 < a < \frac14 \right)$ .

An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$

You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?