In a country there are $4^9$ schoolchildren living in four cities. At the end of the school year a state examination was held in 9 subjects. It is known that any two students have different marks at least in one subject. However, every two students from the same city got equal marks at least in one subject. Prove that there is a subject such that every two children living in the same city have equal marks in this subject. [i]Fedor Petrov[/i]

Let $ A_1,B_1,C_1 $ be the feet of the heights of an acute triangle $ ABC. $ On the segments $ B_1C_1,C_1A_1,A_1B_1, $ take the points $ X,Y, $ respectively, $ Z, $ such that $$ \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\ \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . $$ Show that $ AX,BY,CZ, $ are concurrent.

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$ and $\overline{AD}=\overline{BC}, \overline{AB}>\overline{CD}$. Let $E$ be a point such that $\overline{EC}=\overline{AC}$ and $EC \perp BC$, and $\angle ACE<90^{\circ}$. Let $\Gamma$ be a circle with center $D$ and radius $DA$, and $\Omega$ be the circumcircle of triangle $AEB$. Suppose that $\Gamma$ meets $\Omega$ again at $F(\neq A)$, and let $G$ be a point on $\Gamma$ such that $\overline{BF}=\overline{BG}$. Prove that the lines $EG, BD$ meet on $\Omega$.

Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.

The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side.

One morning each member of Angela’s family drank an $ 8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Let real numbers $a,b$ such that $a\ge b\ge 0$. Prove that \[ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .\]

Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.

Assume that the connected graph $G$ has $n$ vertices all with degree at least three. Prove that there exists a spanning tree of $G$ with more than $\frac{2}{9}n$ leaves.

Given two forests $A$ and $B$ with \(V(A) = V(B)\), that is the graphs are over same vertex set. Suppose $A$ has [b]strictly more[/b] edges than $B$. Prove that there exists an edge of $A$ which if included in the edge set of $B$, then $B$ will still remain a forest. Graphs are undirected

There is a queue of $n{}$ girls on one side of a tennis table, and a queue of $n{}$ boys on the other side. Both the girls and the boys are numbered from $1{}$ to $n{}$ in the order they stand. The first game is played by the girl and the boy with the number $1{}$ and then, after each game, the loser goes to the end of their queue, and the winner remains at the table. After a while, it turned out that each girl played exactly one game with each boy. Prove that if $n{}$ is odd, then a girl and a boy with odd numbers played in the last game. [i]Proposed by A. Gribalko[/i]

Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The first term of an arithmetic series of consecutive integers is $ k^2 \plus{} 1$. The sum of $ 2k \plus{} 1$ terms of this series may be expressed as: $ \textbf{(A)}\ k^3 \plus{} (k \plus{} 1)^3\qquad \textbf{(B)}\ (k \minus{} 1)^3 \plus{} k^3\qquad \textbf{(C)}\ (k \plus{} 1)^3\qquad \\ \textbf{(D)}\ (k \plus{} 1)^2\qquad \textbf{(E)}\ (2k \plus{} 1)(k \plus{} 1)^2$

p1. The pattern $ABCCCDDDDABBCCCDDDDABBCCCDDDD...$ repeats to infinity. Which letter ranks in place $2533$ ? p2. Prove that if $a > 2$ and $b > 3$ then $ab + 6 > 3a + 2b$. p3. Given a rectangle $ABCD$ with size $16$ cm $\times 25$ cm, $EBFG$ is kite, and the length of $AE = 5$ cm. Determine the length of $EF$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.png[/img] p4. Consider the following series of statements. It is known that $x = 1$. Since $x = 1$ then $x^2 = 1$. So $x^2 = x$. As a result, $x^2 - 1 = x- 1$ $(x -1) (x + 1) = (x - 1) \cdot 1$ Using the rule out, we get $x + 1 = 1$ $1 + 1 = 1$ $2 = 1$ The question. a. If $2 = 1$, then every natural number must be equal to $ 1$. Prove it. b. The result of $2 = 1$ is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong? p5. To calculate $\sqrt{(1998)(1996)(1994)(1992)+16}$ . someone does it in a simple way as follows: $2000^2-2 \times 5\times 2000 + 5^2 - 5$? Is the way that person can justified? Why? p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than $25,000$ Rp.. Behind every coupon is written one of the following numbers: $9$, $12$, $42$, $57$, $69$, $21$, 15, $75$, $24$ and $81$. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV $21''$. If the restaurant owner provides as much as $10$ $21''$ TV pieces, how many should be handed over to the the customer? p7. Given is the shape of the image below. [img]https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png[/img] The centers of circles $B$, $C$, $D$, and $E$ are placed on the diameter of circle $A$ and the diameter of circle $B$ is the same as the radius of circle $A$. Circles $C$, $D$, and $E$ are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle $A$. What is the ratio of the circumference of the circle $A$ with the sum of the circumferences of circles $B$, $C$, $D$, and $E$? p8. It is known that $a + b + c = 0$. Prove that $a^3 + b^3 + c^3 = 3abc$.

The sequence $ (x_n)$ is defined by $ x_1\equal{}1,x_2\equal{}a$ and $ x_n\equal{}(2n\plus{}1)x_{n\minus{}1}\minus{}(n^2\minus{}1)x_{n\minus{}2}$ $ \forall n \geq 3$, where $ a \in N^*$.For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$.

Find all integers $ n $ $(n\geq2)$ with the property: for every $ n $ distinct disks in a plane with at least a common point one of the disks contains the center of another disk.

Prove that if a function $f:R \to R$ is bounded and its graph is closed as subset of the $R^2$ plane, then the function f is continuous.

For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$ $\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

Several soldiers are standing in a row. After a command each of them turned their head either to the left or to the right. After that every second every soldier performs the following operation simultaneously: 1) if the soldier is facing right and the majority of soldiers to the right of him are facing left, he starts facing left; 2) if the soldier is facing left and the majority of soldiers to the left of him are facing right, he starts facing right; 3) otherwise he does nothing. Prove that at some point the process will stop.

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$. [i]The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.[/i] 1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$. 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$. 3. Prove that $PJ$ is the angle bisector of $\angle APD$.

Find the last digit of $2023^{2023}$. [i]Proposed by Yifan Kang[/i]

In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$. From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C the perpendicular on $OB$ intersects the ray $OD$ at point $H$. Prove that the points $C, D, G$ and $H$ are conlyclic.

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

Dragoons take up $1\times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the y-coordinates of their centers are both at most $6$, regardless of any dragoons in between. For example, a dragoon centered at $(4, 5)$ can re at a dragoon centered at the origin, but a dragoon centered at $(7, 0)$ can not. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?