Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime. [i]Proposed by Jack Gurev[/i]

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.

Let $A \in \mathcal{M}_n(\mathbb{C})$ be a matrix such that $(AA^*)^2=A^*A$, where $A^*=(\bar{A})^t$ denotes the Hermitian transpose (i.e., the conjugate transpose) of $A$. (a) Prove that $AA^*=A^*A$. (b) Show that the non-zero eigenvalues of $A$ have modulus one.

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ PQ $ be a chord cut by the line $ AB $ on circle $ S_3 $, and $ K $ be the midpoint of $ CD $. Prove that $ \angle PKC = \angle QKC $.

Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]

In a scalene triangle $ABC$, the incircle touches $BC, AC$ and $AB$ at $D, E, F$ respectively. Let $K$ be the foot of the perpendicular from $A$ onto $BC$, and $M$ the midpoint of $BC$. Let $AD$ intersect the incircle again at $X$, and $BE$ at $Y$. Given that $E,F,K,M$ are concyclic, prove that $AX=XY=YD$.

Suppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$. Compute the number of possible integer values of $n$.

Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.

A sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ is called [i]good[/i], if $a_i$ are non-negative integers, and $a_{i+1}-a_{i}$ is either $0$ or $1$ for all $1\le i\le m-1$. Fix a positive integer $n$, and Ivan has a whiteboard with some ones written on it. In each step, he may erase any good sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ that appears on the whiteboard, and then he writes the number $2^k$ such that $$2^{k-1}<2^{a_1}+2^{a_2}+\cdots+2^{a_m}\le 2^{k}$$ Suppose Ivan starts with the least possible number of ones to obtain $2^n$ after some steps, determine the minimum number of steps he will need in order to do so. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively: \[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\] Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$

If a, b, c are positive real numbers with $ a+b+c=1 $. Find the minimum value of $ \sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{1}{\sqrt{abc}} $

Given positive integers $a, b,$ and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}.$

Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence $$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$ is convergent and calculate its limit.

A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID? [i]Proposed by Richard Chen[/i]

Amy painted a dart board over a square clock face using the "hour positions" as boundaries. [See figure.] If $t$ is the area of one of the eight triangular regions such as that between $12$ o'clock and $1$ o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between $1$ o'clock and $2$ o'clock, then $\frac{q}{t}=$ [asy] size((80)); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--(.9,0)--(3.1,4)--(.9,4)--(3.1,0)--(2,0)--(2,4)); draw((0,3.1)--(4,.9)--(4,3.1)--(0,.9)--(0,2)--(4,2)); [/asy] $ \textbf{(A)}\ 2\sqrt{3}-2 \qquad\textbf{(B)}\ \frac{3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(D)}\ \sqrt{3} \qquad\textbf{(E)}\ 2 $

[b]p1.[/b] Prove that for every point inside a regular polygon, the average of the distances to the sides equals the radius of the inscribed circle. The distance to a side means the shortest distance from the point to the line obtained by extending the side. [b]p2.[/b] Suppose we are given positive (not necessarily distinct) integers $a_1, a_2,..., a_{1997}$ . Show that it is possible to choose some numbers from this list such that their sum is a multiple of $1997$. [b]p3.[/b] You have Blue blocks, Green blocks and Red blocks. Blue blocks and green blocks are $2$ inches thick. Red blocks are $1$ inch thick. In how many ways can you stack the blocks into a vertical column that is exactly $12$ inches high? (For example, for height $3$ there are $5$ ways: RRR, RG, GR, RB, BR.) [b]p4.[/b] There are $1997$ nonzero real numbers written on the blackboard. An operation consists of choosing any two of these numbers, $a$ and $b$, erasing them, and writing $a+b/2$ and $b-a/2$ instead of them. Prove that if a sequence of such operations is performed, one can never end up with the initial collection of numbers. [b]p5.[/b] An $m\times n$ checkerboard (m and n are positive integers) is covered by nonoverlapping tiles of sizes $2\times 2$ and $1\times 4$. One $2\times 2$ tile is removed and replaced by a $1\times 4$ tile. Is it possible to rearrange the tiles so that they cover the checkerboard? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.

There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.

Find the remainder when $7^{7^7}$ is divided by $1000$.

In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$. Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.

Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play? $\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 11$

In the diagram below, $A$ and $B$ trisect $DE$, $C$ and $A$ trisect $F G$, and $B$ and $C$ trisect $HI$. Given that $DI = 5$, $EF = 6$, $GH = 7$, find the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/d/5/90334e1bf62c99433be41f3b5e03c47c4d4916.png[/img]

A circle touches the sides of an angle with vertex $A$ at points $B$ and $C.$ A line passing through $A$ intersects this circle in points $D$ and $E.$ A chord $BX$ is parallel to $DE.$ Prove that $XC$ passes through the midpoint of the segment $DE.$