In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"? $ \textbf{(A) } \frac{1}{22,050} \qquad \textbf{(B) } \frac{1}{21,000}\qquad \textbf{(C) } \frac{1}{10,500}\qquad \textbf{(D) } \frac{1}{2,100} \qquad \textbf{(E) } \frac{1}{1,050} $

On every side of a square $ABCD$, we consider three points different (to each other). a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square. b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and yellow when in the sun. Initially the ratio of green to yellow frogs was 3:1. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is 4:1. What is the difference between the number of green frogs and yellow frogs now? $\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24$

$ABCD$ is a regular tetrahedron of side length $4.$ Four congruent spheres are inside $ABCD$ such that each sphere is tangent to exactly three of the faces, the spheres have distinct centers, and the four spheres are concurrent at one point. Let $v$ be the volume of one of the spheres. If $v^2$ can be written as $\tfrac{a}{b}\pi^2,$ where $a$ and $b$ are relatively prime positive integers, find $a+b.$

Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.

Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$ [i]Proposed by Wong Jer Ren[/i]

Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by $\$2$ from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then: $ \textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}$ $\qquad\textbf{(B)}\ \text{The middle book sells for \$36 }$ $\qquad\textbf{(C)}\ \text{The cheapest book sells for \$4 }$ $\qquad\textbf{(D)}\ \text{The most expensive book sells for \$64 }$ $\qquad\textbf{(E)}\ \text{None of these is correct } $

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

Let $$f(x) = \left|\left|\cdots\left|\left|\left|\left|x\right|-1\right|-2\right|-3\right|-\cdots \right|-10\right|.$$ Compute $f(1)+f(2)+\cdots+f(54)+f(55).$

A pack of MIT students are holding an escape room, where students may compete in teams of 4, 5, or 6. There is \$60 dollars worth of prize money in Amazon gift cards for the winning team. If each gift card can contain any whole number of dollars, what is the minimum number of gift cards required so that the prize money can be distributed evenly among any team? [i]Lightning 4.1[/i]

Let $ABC$ be a triangle, having no right angles, and let $D$ be a point on the side $BC$. Let $E$ and $F$ be the feet of the perpendiculars drawn from the point $D$ to the lines $AB$ and $AC$ respectively. Let $P$ be the point of intersection of the lines $BF$ and $CE$. Prove that the line $AP$ is the altitude of the triangle $ABC$ from the vertex $A$ if and only if the line $AD$ is the angle bisector of the angle $CAB$.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Which of the following is the same as \[ \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}? \]$ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ \minus{}\frac23 \qquad \textbf{(C)}\ \frac23 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac{14}{3}$

Prove that there is a positive integer $M$ for which the following statement holds: For all prime numbers $p$, there is an integer $n$ for which $\sqrt{p} \le n \le M\sqrt{p}$ and $p \mod n \le \frac{n}{2020}$ . Note: Here, $p \mod n$ denotes the unique integer $r \in {0, 1, ..., n - 1}$ for which $n|p -r$. In other words, $p \mod n$ is the residue of $p$ upon division by $n$.

Prove that the product of three consecutive positive integers is never a perfect square.

[b]p1.[/b] Solve the equation $3^x + 9^x = 27^x$. [b]p2.[/b] An equilateral triangle in inscribed in a circle of area $1$ m$^2$. Then the second circle is inscribed in the triangle. Find the radius of the second circle. [b]p3.[/b] Solve the inequality: $2\sqrt{x^2 - 5x + 4} + 3\sqrt{x^2 + 2x - 3} \le 5\sqrt{6 - x - x^2}$ [b]p4.[/b] Peter and John played a game. Peter wrote on a blackboard all integers from $1$ to $18$ and offered John to choose $8$ different integers from this list. To win the game John had to choose 8 integers such that among them the difference between any two is either less than $7$ or greater than $11$. Can John win the game? Justify your answer. [b]p5.[/b] Prove that given $100$ different positive integers such that none of them is a multiple of $100$, it is always possible to choose several of them such that the last two digits of their sum are zeros. [b]p6.[/b] Given $100$ different squares such that the sum of their areas equals $1/2$ m$^2$ , is it possible to place them on a square board with area $1$ m$^2$ without overlays? Justify your answer. PS. You should use hide for answers.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Let $ A,B\in\mathcal{M} \left( \mathbb{R} \right) $ that satisfy $ AB=O_3. $ Prove that: [b]a)[/b] The function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ defined as $ f(x)=\det \left( A^2+B^2+xBA \right) $ is a polynomial one, of degree at most $ 2. $ [b]b)[/b] $ \det\left( A^2+B^2 \right)\ge 0. $

Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence: $\bullet$ $n_{i+1}\geq n_i$ $\bullet$ There is at least one number $i$, such that $n_i=2022$ $\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real $x, y$ holds equality $$f(xf(y)) + f(xy) = 2f(x)y$$ [i]Proposed by Arseniy Nikolaev[/i]

I found this inequality in "Topics in Inequalities" (I 85) For all positive reals $x,y,z$ with $xyz=1$ prove: \[ \frac{x^9+y^9}{x^6+x^3y^3+y^6}+\frac{y^9+z^9}{y^6+y^3z^3+z^6}+\frac{z^9+x^9}{z^6+z^3x^3+x^6}\geq 2 \]

[u]Set 4[/u] [b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$. [b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$. [b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$. [u]Set 5[/u] [b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$. [b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$. [b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 6[/u] [b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$. [b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$? [b]G18.[/b] Deja vu? Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$. What integer was removed ?