A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. [i]AMM Magazine[/i]

What is the value of $$(2^2-2) - (3^2-3) + (4^2-4)?$$ $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 12$

Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)\equal{}f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)\equal{}f(\minus{}\bar z)$

Let $r$ be a natural number. Prove that the quadratic trinomial $x^2 - rx- 1$ does not divide any nonzero polynomial whose coefficients are integers with absolute values less than $r$.

Call a sequence $a_0,a_1,a_2,\dots$ of positive integers defined by $a_k = 25 a_{k - 1} + 96$ for all $k > 0$ a \textit{valid} sequence. Call the \textit{goodness} of a \textit{valid} sequence the maximum value of $\gcd(a_k, a_{k+2024})$ over all $k$. Call a \textit{valid} sequence \textit{best} if it has the maximal \textit{goodness} across all possible \textit{valid} sequences. Find the second largest $a_0$ across all \textit{best} sequences. [i]Individual #13[/i]

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

$27$ students are given a number from $1$ to $27.$ How many ways are there to divide $27$ students into $9$ groups of $3$ with the following condition? (i) The sum of students number in each group is $1\pmod{3}$ (ii) There are no such two students where their numbering differs by $3.$

Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$. (1) Find the 36th term. (2) Find $\int x^2\ln x\ dx$. (3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part? If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$. [i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]

Find the number of elements in the set $$ \left\{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\} $$

Let $V$ be a set of $2019$ points in space where any of the four points are not on the same plane, and $E$ be the set of edges connected between them. Find the smallest positive integer $n$ satisfying the following condition: if $E$ has at least $n$ elements, then there exists $908$ two-element subsets of $E$ such that [list][*]The two edges in each subset share a common vertice, [*]Any of the two subsets do not intersect.[/list]

At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by $0.25$ seconds. If Bonnie’s time was exactly $7.80$ seconds, how long did it take for Wendy to go down the slide? $\textbf{(A)}\ 7.80~ \text{seconds} \qquad \textbf{(B)}\ 8.05~ \text{seconds} \qquad \textbf{(C)}\ 7.55~ \text{seconds} \qquad \textbf{(D)}\ 7.15~ \text{seconds} \qquad $ $\textbf{(E)}\ 7.50~ \text{seconds}$

A variable tetrahedron $ABCD$ has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal $(BC = DA, CA = DB, AB = DC)$; and the vertices $A,B,C$ lie respectively on three fixed spheres with the same center $P$ and radii $3, 4, 12$. What is the maximal length of $PD$?

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$

Let $a,b,c,d$ be real numbers such that $a^2+b^2=1,c^2+d^2=1$ and $ac+bd=0$. Determine all possible values of $ab+cd$.

A palindrome is a positive integer which reads in the same way in both directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ where both numbers are palindromes? [i](3 points)[/i]

Find all (real) solutions of the system \[3x_1-x_2-x_3-x_5 = 0,\]\[-x_1+3x_2-x_4-x_6= 0,\]\[-x_1 + 3x_3 - x_4 - x_7 = 0,\]\[-x_2 - x_3 + 3x_4 - x_8 = 0,\]\[-x_1 + 3x_5 - x_6 - x_7 = 0,\]\[-x_2 - x_5 + 3x_6 - x_8 = 0,\]\[-x_3 - x_5 + 3x_7 - x_8 = 0,\]\[-x_4 - x_6 - x_7 + 3x_8 = 0.\]

Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define \[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \] Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x) = a_{1998}X^{1998} + a_{1997}X^{1997} +...+a_1X + a_0$ be a polynomial with real coefficients such that $P(0) \ne P (-1)$, and let $a, b$ be real numbers. Let $Q(x) = b_{1998}X^{1998} + b_{1997}X^{1997} +...+b_1X + b_0$ be the polynomial with real coefficients obtained by taking $b_k = aa_k + b$ ,$\forall k = 0, 1,2,..., 1998$. Show that if $Q(0) = Q (-1) \ne 0$ , then the polynomial $Q$ has no real roots.

Two different points $X,Y$ are marked on the side $AB$ of a triangle $ABC$ so that $\frac{AX \cdot BX}{CX^2}=\frac{AY \cdot BY}{CY^2}$ . Prove that $\angle ACX=\angle BCY$. I.Zhuk

The number of values of $x$ satisfying the equation \[ \frac{2x^2-10x}{x^2-5x}=x-3 \]is: $\text{(A)} \ \text{zero} \qquad \text{(B)} \ \text{one} \qquad \text{(C)} \ \text{two} \qquad \text{(D)} \ \text{three} \qquad \text{(E)} \ \text{an integer greater than 3}$

$a_1,a_2,\cdots,a_n$ are positive numbers, satisfying that $a_1a_2\cdots a_n=1$. Prove that $(2+a_1)(2+a_2)\cdots(2+a_n)\geq3^n$

Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$