Prove that there is no non-constant polynomial $P(x)$ with integer coefficients such that $P(n)$ is a prime number for all positive integers $n$.

In the triangle $ABC$ the median $AM$ is drawn. Is it possible that the radius of the circle inscribed in $\vartriangle ABM$ could be twice as large as the radius of the circle inscribed in $\vartriangle ACM$ ? ( D . Fomin , Leningrad)

In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.

$40$ people, numbered $1$ through $40$ counterclockwise, sit around a circular table. They begin playing a game. Each person is initially considered "alive". Starting with person $1$, the first person eliminates the closest "alive" person to their right (so Person $1$ eliminates Person $2$). Then the next "alive" person, moving counterclockwise, eliminates the closest "alive" person to their right (so since Person $2$ is eliminated, Person $3$ eliminates Person $4$). This process continues until there is only $1$ "alive" person remaining. What is the number of the last "alive" person? [asy] usepackage("cancel", "makeroom, thicklines"); usepackage("bm"); size(15cm); picture p; draw(p, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(p, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(p, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(p, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(p, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); add(shift(-20, 0) * p); draw((-11, 0)--(-8,0), EndArrow); picture q; draw(q, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(q, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i != 1) label(q, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(q, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(q, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 1; i+=2) { //label(q, "\bm\xcancel{~}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(q, "\xcancel{2}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(q); draw((9,0)--(12,0), EndArrow); picture r; draw(r, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { if(i % 2 == 1) label(r, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i % 2 != 1) label(r, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(r, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(r, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 4; i+=2) { label(r, "\xcancel{" + string(40 - i) +"}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(r, "\xcancel{" + string(i + 1) + "}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(shift(20, 0) * r); [/asy] [center]In the last step here, Person $39$ eliminates Person $40$. Next turn, Person $1$ eliminates the closest person to his right, Person $3$.[/center]

Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$

Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings. [i](5 points)[/i]

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

Let $f(x)=2x^3-2x$. For what positive values of $a$ do there exist distinct $b,c,d$ such that $(a,f(a)),(b,f(b)),(c,f(c)),(d,f(d))$ is a rectangle?

Let $R$ be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of $R$ ? $\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}7 \qquad \textbf{(E)}\hspace{.05in}8 $

Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid. [center]<see attached>[/center] Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

In elections to the City Duma, each voter, if he goes to the polls, casts a vote for himself (if he is a candidate) and for those candidates who are his friends. The forecast of the sociological service of the mayor's office is considered good if it correctly predicts the number of votes cast for at least one of the candidates, and bad otherwise. Prove that for any forecast, voters can turn out to vote in such a way that this forecast turns out to be bad.

Let $ABC$ be a right angled triangle with $\angle A = 90^o$ and $AD$ its altitude. We draw parallel lines from $D$ to the vertical sides of the triangle and we call $E, Z$ their points of intersection with $AB$ and $AC$ respectively. The parallel line from $C$ to $EZ$ intersects the line $AB$ at the point $N$. Let $A' $ be the symmetric of $A$ with respect to the line $EZ$ and $I, K$ the projections of $A'$ onto $AB$ and $AC$ respectively. If $T$ is the point of intersection of the lines $IK$ and $DE$, prove that $\angle NA'T = \angle ADT$.

On the sides $AB, \, \, BC, \, \, CA$ of the triangle $ABC$ the points ${{C} _ {1}}, \, \, {{A} _ { 1}},\, \, {{B} _ {1}}$ are selected respectively, that are different from the vertices. It turned out that $\Delta {{A} _ {1}} {{B} _ {1}} {{C} _ {1}}$ is equilateral, $\angle B{{C}_{1}}{{A}_{1}}=\angle {{C}_{1}}{{B}_{1}}A$ and $\angle B{{A}_{1}}{{C}_{1}}=\angle {{A}_{1}}{{B}_{1}}C$ . Is $ \Delta ABC$ equilateral?

A rectangle $ABCD$ has side lengths $AB=6 \text{ miles}$ and $BC=9\text{ miles}.$ A pigeon hovers at point $P$, which is 5 miles above some randomly chosen point inside $ABCD$. Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as $\tfrac{a}{b}$, what is $ab$? [i]2022 CCA Math Bonanza Lightning Round 2.2[/i]

Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

In the following sum: $1 + 2 + 3 + 4 + 5 + 6$, if we remove the first two “+” signs, we obtain the new sum $123 + 4 + 5 + 6 = 138$. By removing three “$+$” signs, we can obtain $1 + 23 + 456 = 480$. Let us now consider the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$, in which some “$+$” signs are to be removed. What are the three smallest multiples of $100$ that we can get in this way?

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$).

Two distinct positive even integers sum to $8.$ Determine the larger of the $2$ integers.

Let $ f ( x , y ) = ( x + y ) / 2 , g ( x , y ) = \sqrt { x y } , h ( x , y ) = 2 x y / ( x + y ) $, and let $$ S = \{ ( a , b ) \in \mathrm { N } \times \mathrm { N } | a \neq b \text { and } f( a , b ) , g ( a , b ) , h ( a , b ) \in \mathrm { N } \} $$ where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$.

We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.

Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE\cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC\cap BD \equal{} E'$. Suppose also that $ eABD'\cap eAC'E \equal{} A''$, $ eBCE'\cap eBD'A \equal{} B''$, $ eCDA'\cap eCE'B \equal{} C''$, $ eDEB'\cap eDA'C \equal{} D''$, $ eEAC'\cap eEB'D \equal{} E''$. Prove that $ AA'', BB'', CC'', DD'', EE''$ are concurrent. (Here $ l_1\cap l_2 \equal{} P$ means that $ P$ is the intersection of lines $ l_1$ and $ l_2$. Also $ eA_1A_2A_3\cap eB_1B_2B_3 \equal{} Q$ means that $ Q$ is the intersection of the circumcircles of $ \Delta A_1A_2A_3$ and $ \Delta B_1B_2B_3$.)

A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.