Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?

Oh no! While playing Mario Party, Theo has landed inside the Bowser Zone. If his next roll is between $1$ and $5$ inclusive, Bowser will shoot his ``Zero Flame" that sets a player's coin and star counts to zero. Fortunately, Theo has a double dice block, which lets him roll two fair $10$-sided dice labeled $1$-$10$ and take the sum of the rolls as his "roll". If he uses his double dice block, what is the probability he escapes the Bowser zone without losing his coins and stars? [i]Proposed by Connor Gordon[/i]

How many ways are there to cut a $1$ by $1$ square into $8$ congruent polygonal pieces such that all of the interior angles for each piece are either $45$ or $90$ degrees? Two ways are considered distinct if they require cutting the square in different locations. In particular, rotations and reflections are considered distinct.

Ephram Chun is a senior and math captain at Lexington High School. He is well-loved by the freshmen, who seem to only listen to him. Other than being the father figure that the freshmen never had, Ephramis also part of the Science Bowl and Science Olympiad teams along with being part of the highest orchestra LHS has to offer. His many hobbies include playing soccer, volleyball, and the many forms of chess. We hope that he likes the questions that we’ve dedicated to him! [b]p1.[/b] Ephram is scared of freshmen boys. How many ways can Ephram and $4$ distinguishable freshmen boys sit together in a row of $5$ chairs if Ephram does not want to sit between $2$ freshmen boys? [b]p2.[/b] Ephram, who is a chess enthusiast, is trading chess pieces on the black market. Pawns are worth $\$100$, knights are worth $\$515$, and bishops are worth $\$396$. Thirty-four minutes ago, Ephrammade a fair trade: $5$ knights, $3$ bishops, and $9$ rooks for $8$ pawns, $2$ rooks, and $11$ bishops. Find the value of a rook, in dollars. [b]p3.[/b] Ephramis kicking a volleyball. The height of Ephram’s kick, in feet, is determined by $$h(t) = - \frac{p}{12}t^2 +\frac{p}{3}t ,$$ where $p$ is his kicking power and $t$ is the time in seconds. In order to reach the height of $8$ feet between $1$ and $2$ seconds, Ephram’s kicking power must be between reals $a$ and $b$. Find is $100a +b$. [b]p4.[/b] Disclaimer: No freshmen were harmed in the writing of this problem. Ephram has superhuman hearing: He can hear sounds up to $8$ miles away. Ephramstands in the middle of a $8$ mile by $24$ mile rectangular grass field. A freshman falls from the sky above a point chosen uniformly and randomly on the grass field. The probability Ephram hears the freshman bounce off the ground is $P\%$. Find $P$ rounded to the nearest integer. [img]https://cdn.artofproblemsolving.com/attachments/4/4/29f7a5a709523cd563f48176483536a2ae6562.png[/img] [b]p5.[/b] Ephram and Brandon are playing a version of chess, sitting on opposite sides of a $6\times 6$ board. Ephram has $6$ white pawns on the row closest to himself, and Brandon has $6$ black pawns on the row closest to himself. During each player’s turn, their only legal move is to move one pawn one square forward towards the opposing player. Pawns cannot move onto a space occupied by another pawn. Players alternate turns, and Ephram goes first (of course). Players take turns until there are no more legal moves for the active player, at which point the game ends. Find the number of possible positions the game can end in. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$. The sum of their ages was $\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$

Let $ k$ and $ m$ be positive integers such that $ \displaystyle\frac12\left(\sqrt{k\plus{}4\sqrt{m}}\minus{}\sqrt{k}\right)$ is an integer. (a) Prove that $ \sqrt{k}$ is rational. (b) Prove that $ \sqrt{k}$ is a positive integer.

It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?

Written in mm/dd format, a date is called [i]cute[/i] if the month is divisible by the day. For example, the date [i]cute[/i] is a [i]cute[/i] date because $8$ is divisible by $2$. Find the number of [i]cute[/i] dates in a year. [i]Proposed by Andy Xu[/i]

Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

Let $P(n)=(n-1^3)(n-2^3)\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \mid P(n)$ for any $n>2023$. If $d$ is product of $m$ not necessarily distinct primes, find $m$.

Sketch the graph of $u(x, 1)$, if $0 \le x\le1$, \[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true. [i](Swiss Mathematical Olympiad 2011, Final round, problem 4)[/i]

A circle has circumference $8\pi.$ Determine its radius.

Prove that the equation $$ (x - a) (x - c) + 2 (x - b) (x - d) = 0,$$ in which $ a < b < c < d $, has two real roots.

[b]1.[/b] For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$. For example, $\frac{1}{4} = 0.25000\dots$, $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$. a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$, it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$, we have $a_i = p_1$ or $a_1 = q_i$. b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$, the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$.

Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.

Let $f$ be a linear function. Compute the slope of $f$ if $$\int_3^5f(x)dx=0\text{ and }\int_5^7f(x)dx=12.$$

Let $f:[0,1]\to [0,1]$ be a continuous function. We say $f\equiv 0$ if $f(x)=0$ for all $x\in [0,1]$ and similarly $f\not\equiv 0$ if there exists at least one $x\in [0,1]$ such that $f(x)\neq 0$. Suppose $f\not\equiv 0$, $f \circ f \not\equiv 0$ but $f \circ f \circ f \equiv 0$. Do there exists such an $f$? If yes construct such an function, if no prove it

If $ x,y$ are positive real numbers with sum $ 2a$, prove that : $ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$ When does equality hold ? Babis

A frog jumps around on the integers on the number line. If it lands on an even number $n$, it jumps to the number $\frac{n}{2}$ . If it lands on an odd number $n$, it jumps to the number $n + 5$. At some point it lands on the number $25$. At which numbers may it have been three jumps ago?

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

Let $P$ be a polynomial with integer coefficients of degree $d$. For the set $A = \{ a_1, a_2, ..., a_k\}$ of positive integers we denote $S (A) = P (a_1) + P (a_2) + ... + P (a_k )$. The natural numbers $m, n$ are such that $m ^{d+ 1} | n$. Prove that the set $\{1, 2, ..., n\}$ can be subdivided into $m$ disjoint subsets $A_1, A_2, ..., A_m$ with the same number of elements such that $S (A_1) = S(A_2) = ... = S (A_m )$.

A sequence of numbers $ a_1, a_2, a_3, ...$ satisfies (i) $ a_1 \equal{} \frac{1}{2}$ (ii) $ a_1\plus{}a_2 \plus{} \cdots \plus{} a_n \equal{} n^2 a_n \ (n \geq 1)$ Determine the value of $ a_n \ (n \geq 1)$.

Prove that every finite triangle-free graph can be embedded as an induced subgraph in a finite triangle-free graph of diameter 2.

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.