Let $ABC$ be a scalene triangle with incenter $I$ and incircle $\omega$. Let the tangency points of $\omega$ to $BC,AC\text{ and } AB$ be $D,E,F$ respectively. Let the line $EF$ intersect the circumcircle of $ABC$ at the points $G, H$. Assume that $E$ lies between the points $F$ and $G$. Let $\Gamma$ be a circle that passes through $G$ and $H$ and that is tangent to $\omega$ at the point $M$ which lies on different semi-planes with $D$ with respect to the line $EF$. Let $\Gamma$ intersect $BC$ at points $K$ and $L$ and let the second intersection point of the circumcircle of $ABC$ and the circumcircle of $AKL$ be $N$. Prove that the intersection point of $NM$ and $AI$ lies on the circumcircle of $ABC$ if and only if the intersection point of $HB$ and $GC$ lies on $\Gamma$.

In an ${8}$×${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart . Then find the mininum value of $n$ .

A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$. $(a)$ Prove that $R\geq r_1+r_2+r_3$ $(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Prove that \[ \sqrt{(1^k+2^k)(1^k+2^k+3^k)\ldots (1^k+2^k+\ldots +n^k)}\] \[ \ge 1^k+2^k+\ldots +n^k-\frac{2^{k-1}+2\cdot 3^{k-1}+\ldots + (n-1)\cdot n^{k-1}}{n}\] for all integers $n,k \ge 2$.

Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

The following apply: $a,b,c,d \ge 0$ and $abcd=1$ Prove that $$ a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd \ge 10$$

Let $n\ge 4$ be a positive integer. Consider any set $A$ formed by $n$ distinct real numbers such that the following condition holds: for every $a\in A$, there exist distinct elements $x, y, z \in A$ such that $\left| x-a \right|, \left| y-a \right|, \left| z-a \right| \ge 1$. For each $n$, find the greatest real number $M$ such that $\sum_{a\in A}^{}\left| a \right|\ge M$.

Let $ABCD$ be a convex quadrilateral with $\angle A=60^o$. Let $E$ and $Z$ be the symmetric points of $A$ wrt $BC$ and $CD$ respectively. If the points $B,D,E$ and $Z$ are collinear, then calculate the angle $\angle BCD$.

A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?

How many positive integer factors does the following expression have? \[ \sum_{n=1}^{999} \log_{10} \left(\frac{n+1}{n} \right) \] [i]2022 CCA Math Bonanza Tiebreaker Round #1[/i]

Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female? $\textbf{(A) }\dfrac13\qquad\textbf{(B) }\dfrac4{11}\qquad\textbf{(C) }\dfrac25\qquad\textbf{(D) }\dfrac49\qquad\textbf{(E) }\dfrac12$

Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

a) Prove that, for any real $x>0$, it is true that $x^3-3x\ge -2$ . b) Prove that, for any real $x,y,z>0$, it is true that $$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}+2\left(\frac{y}{xz}+\frac{z}{xy}+\frac{x}{yz} \right)\ge 9$$ . When we have equality ?

The number $ 121_b$, written in the integral base $ b$, is the square of an integer, for $ \textbf{(A)}\ b \equal{} 10,\text{ only} \qquad \textbf{(B)}\ b \equal{} 10 \text{ and } b \equal{} 5, \text{ only} \qquad \textbf{(C)}\ 2 \leq b \leq 10 \qquad \textbf{(D)}\ b > 2 \qquad \textbf{(E)}\ \text{no value of }b$

Prove that if a natural number $ N $ can be represented in the form the sum of three squares of integers divisible by $3$, then it is also is represented as the sum of three squares of integers that are not divisible by $3$.

Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose?

Let $I$ be the incenter and let $\Gamma$ be the incircle of $\vartriangle ABC$, and let $P = \Gamma \cap BC$. Let $Q$ denote the intersection of $\Gamma$ and the line passing through $P$ parallel to $AI$. Let $\ell$ be the tangent line to $\Gamma$ at $Q$ and let $\ell \cap AB = S$, $\ell \cap AC = R$. If $AB = 7$, $BC = 6$, $AC = 5$, what is $RS$?

In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.

Find all primes $p,q$ such that $p ^3-q^7=p-q$.

Find the largest $2$-digit number $N$ which is divisible by $4$, such that all integral powers of $N$ end with $N$.

There are $n$ guests at a gathering. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. Determine all possible values of $n$.

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

Let $ABCD$ be a convex quadrilateral with $\angle{BAD} = 2\angle{BCD}$ and $AB = AD$. Let $P$ be a point such that $ABCP$ is a parallelogram. Prove that $CP = DP$.