Let $p$ be a prime. Let $f(x)$ be the number of ordered pairs $(a, b)$ of positive integers less than $p$, such that $a^b \equiv x \pmod p$. Suppose that there do not exist positive integers $x$ and $y$, both less than $p$, such that $f(x) = 2f(y)$, and that the maximum value of $f$ is greater than $2018$. Find the smallest possible value of $p$.

For a positive integer $n$, there is a school with $n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is $k$, show that the maximum number of well-formed sets is not greater than $3^{(n+k)/3}$. Here, an empty set and a set with one student is regarded as well-formed as well.

The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .

A sequence \( a_1, a_2, \ldots \) of real numbers satisfies the following condition: for every positive integer \( k \geq 2 \), there exists a positive integer \( i < k \) such that \( a_i + a_k = k \). It is known that for some \( j \), the fractional parts of the numbers \( a_j \) and \( a_{j+1} \) are equal. Prove that for some positive integers \( x \neq y \), the equality \[ a_x - a_y = x - y \] holds. [i]The fractional part of a real number \( a \) is defined as the number \( \{a\} \in [0, 1) \), which satisfies the condition \( a = n + \{a\} \), where \( n \) is an integer. For example, \( \{-3\} = 0 \), \( \{3.14\} = 0.14 \), and \( \{-3.14\} = 0.86 \).[/i] [i]Proposed by Mykhailo Shtandenko[/i]

Arthur has a regular 11-gon. He labels the vertices with the letters in $CORONAVIRUS$ in consecutive order. Every non-ordered set of 3 letters that forms an isosceles triangle is a member of a set $S$, i.e. $\{C, O, R\}$ is in $S$. How many elements are in $S$? [i]Proposed by Sammy Chareny[/i]

Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$. Find the area of $S$.

A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display? $ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$

The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$. Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.

The $12$ islands of the Bonanza archipelago are labeled $A,B,C,\dots,K,L$. Some of the islands are connected by bridges, as indicated in the diagram below. Tristan wants be able to walk from island to island crossing each bridge exactly once (he doesn't care if he visits a given island more than once, or whether he starts and ends on the same island). Submit a pair of unconnected islands such that if they are connected by a bridge, Tristan can accomplish his goal. [img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC80L2M0MTU1ZDVmZTRlNjQ5MmQ5ZTNhN2U3NTQwZDRhMzRmNjk1YTk4LnBuZw==&rn=bWJncmFwaHB1enpsZS5wbmc=[/img] [i]2017 CCA Math Bonanza Team Round #4[/i]

Let $A_1A_2A_3A_4$ be a quadrilateral inscribed in a circle $C$. Show that there is a point $M$ on $C$ such that $MA_1 -MA_2 +MA_3 -MA_4 = 0.$

Positive real sequences $\{ a_n \}$ and $\{ b_n \}$ satisfy the following conditions for all positive integers $n$. [list] [*] $a_{n+1}b_{n+1}= a_n^2 + b_n^2$ [*] $a_{n+1}+b_{n+1}=a_nb_n$ [*] $a_n \geq b_n$ [/list] Prove that there exists positive integer $n$ such that $\frac{a_n}{b_n}>2023^{2023}.$

Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

In a three-digit positive integer $M$, the number of hundreds is less than the number of tenths and the number of tenths is less than the number of ones. The arithmetic mean of the integer three-digit numbers obtained by arranging the number $M$ and its numbers ends with the number $5$. Find all such three-digit numbers $M$.

$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\ If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\ [i](Vlad Robu)[/i]

There are $14$ students who have particiated to a $3$ hour test consisting on $15$ short problems. Each student has solved a different number of problems and each problem has been solved by a different number of students. Prove that there exists a student who has solved exactly $5$ problems.

Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.

Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $ 7$ songs, which was more than any other girl, and Mary sang $ 4$ songs, which was fewer than any other girl. How many songs did these trios sing? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$

Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$. If $AB=11$ and $BC=8$, find the area of the quadrilateral $AEFB$.

The jury of an Olympiad has $30$ members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most $15$ sessions there will be no more exclusions. (Note that nobody votes about his own competence.)

[b]Problem 2[/b] Determine all pairs $(n, m)$ of positive integers satisfying the equation $$5^n = 6m^2 + 1\ . $$

In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied. One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors. The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore. [i]Note: The order of the numbers of the list is not important.[/i]

[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars? [b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second triangle, similar to the first one, has one side of length $12$. Find the sum of all possible lengths of the second triangle's longest side. [b]p3.[/b] Hen Hao runs two laps around a track. Her overall average speed for the two laps was $20\%$ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap? [b]p4.[/b] Square $ABCD$ has side length $2$. Circle $\omega$ is centered at $A$ with radius $2$, and intersects line $AD$ at distinct points $D$ and $E$. Let $X$ be the intersection of segments $EC$ and $AB$, and let $Y$ be the intersection of the minor arc $DB$ with segment $EC$. Compute the length of $XY$ . [b]p5.[/b] Hen Hao rolls $4$ tetrahedral dice with faces labeled $1$, $2$, $3$, and $4$, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square. [b]p6.[/b] Let $N \ge 11$ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $1$ to $N$, inclusive. Later, during the live drawing, another group of six numbers from $1$ to $N$ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $N$? [b]p7.[/b] An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$? [b]p8.[/b] Let $a_1, a_2, a_3, ...$ be a geometric sequence of positive reals such that $a_1 < 1$ and $(a_{20})^{20} = (a_{18})^{18}$. What is the smallest positive integer n such that the product $a_1a_2a_3...a_n$ is greater than $1$? [b]p9.[/b] In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$. [b]p10.[/b] Farmer James places the numbers $1, 2,..., 9$ in a $3\times 3$ grid such that each number appears exactly once in the grid. Let $x_i$ be the product of the numbers in row $i$, and $y_i$ be the product of the numbers in column $i$. Given that the unordered sets $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$? [i]Proposed by Ray Li[/i]

Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.