For a sequence of $10$ coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails-Heads”, and “Tails-Tails” sequences is recorded. These four numbers are then multiplied to get the [i]Tiger number[/i] of the sequence of flips. How many such sequences have a [i]Tiger number [/i] of $24$?

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$

We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$. [i]Proposed by Ariel García[/i]

In the triangle $ABC$ angle $C$ is a right angle. On the side $AC$ point $D$ has been found, and on the segment $BD$ point K has been found such that $\angle ABC = \angle KAD = \angle AKD$. Prove that $BK = 2DC$.

Let $ABCD$ be a quadrilateral inscribed a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E, AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN,Q$ be the intersection of $MN$ and $GH$. a) Prove that $IK$ and $OE$ are parallel. b) Prove that $PK$ is perpendicular to $IQ$.

Five times $A$'s money added to $B$'s money is more than $\$51.00$. Three times $A$'s money minus $B$'s money is $\$21.00$. If $a$ represents $A$'s money in dollars and $b$represents $B$'s money in dollars, then: $ \textbf{(A)}\ a>9, b>6 \qquad\textbf{(B)}\ a>9, b<6 \qquad\textbf{(C)}\ a>9, b=6\qquad$ $\textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad\textbf{(E)}\ 2a=3b $

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

All sides of a convex pentagon are equal, and all angles are different. Prove that the maximum and minimum angles are adjacent to the same side of the pentagon.

An inscribed $n$-gon ($n > 3$), is divided into $n-2$ triangles by diagonals which meet only in vertices. What is the maximum possible number of congruent triangles obtained? (An inscribed $n$-gon is an $n$-gon where all its vertices lie on a circle) [i]Proposed by Boris Frenkin - Russia[/i]

Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number? $ \textbf{(A)}\ \frac {47}{72}\qquad \textbf{(B)}\ \frac {37}{56}\qquad \textbf{(C)}\ \frac {2}{3}\qquad \textbf{(D)}\ \frac {49}{72}\qquad \textbf{(E)}\ \frac {39}{56}$

Find the maximum of the expression $$||...||x_1 - x_2|- x_3| -... | - x_{2023}|,$$ where $x_1,x_2,..., x_{2023}$ are distinct natural numbers between $1$ and $2023$.

Let $p$ be a prime number and let $X$ be a finite set containing at least $p$ elements. A collection of pairwise mutually disjoint $p$-element subsets of $X$ is called a $p$-family. (In particular, the empty collection is a $p$-family.) Let $A$(respectively, $B$) denote the number of $p$-families having an even (respectively, odd) number of $p$-element subsets of $X$. Prove that $A$ and $B$ differ by a multiple of $p$.

solve this diophantine equation y^2 = x^3 - 16

Let $A_1A_2A_3\ldots A_n$ be a regular $n$-gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$. Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$. Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$, and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$. Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$. [i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]

Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm, and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepiped $m$ parallel to line $A_1C_1$ is drawn on the plane $\gamma$. a) Draw a section of a parallelepiped with plane $\gamma$. b) Justify what geometric figure is this section, and find its area. (Alexander Shkolny)

[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$. [b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? [b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ . [b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day? [b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$. [b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces? [b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$. [b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? [img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img] [b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$? [b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$? [b]p12.[/b] What is the largest prime number that is a factor of $160,401$? [b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$? [b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$. [b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$. [b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ . [b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold: $\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$, $\bullet$ $k_1 + k_2 + ...+ k_n = 0$. Determine the number of spicy integers less than $10^6$. [b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$ $$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system. [b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not. [b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.

For an integer $n$, denote $A =\sqrt{n^{2}+24}$ and $B =\sqrt{n^{2}-9}$. Find all values of $n$ for which $A-B$ is an integer.

The number $64$ has the property that it is divisible by its units digit. How many whole numbers between $10$ and $50$ have this property? $\text{(A)}\ 15 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 20$

There are prime numbers $a$, $b$, and $c$ such that the system of equations $$a \cdot x - 3 \cdot y + 6 \cdot z = 8$$ $$b \cdot x + 3\frac12 \cdot y + 2\frac13 \cdot z = -28$$ $$c \cdot x - 5\frac12 \cdot y + 18\frac13 \cdot z = 0$$ has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.

Show that there are infinitely many positive odd integers $n$ such that $n^5+2n+1$ is expressible as a sum of squares of two coprime integers.

Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy \[a | b+c+bc, b | c+a+ca, c | a+b+ab.\] Prove that at least one of the numbers $a, b, c$ is not prime.

The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people? $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 29 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 36$

For all real $x$, the real-valued function $y=f(x)$ satisfies \[ y''-2y'+y=2e^x. \] (a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real $x$? Explain. (b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real $x$? Explain.