If Bobby’s age is increased by $6$, it’s a number with an integral (positive) square root. If his age is decreased by $6$, it’s that square root. How old is Bobby?

Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ \[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]

Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$, and whose sum $S(N)$ of the digits divides $N$. (a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit. (b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.

A digital calendar displays the date: day, month, and year, with $2$ digits for the day, $2$ digits for the month, and $2$ digits for the year. For example, $01-01-01$ is January $1$, $2001$ and $05-25-23$ is May $25$, $2023$. In front of the calendar is a mirror. The digits of the calendar are as in the figure [img]https://cdn.artofproblemsolving.com/attachments/c/5/a08a4e34071fff4d33b95b23690254f55b33e1.gif[/img] If $0, 1, 2, 5$, and $8$ are reflected, respectively, in $0, 1, 5, 2$, and $8$, and the other digits lose meaning when reflected, determine how many days of the century, when reflected in the mirror, also correspond to a date.

Show that for any function $ f: (0,\plus{}\infty)\to (0,\plus{}\infty)$ there exist real numbers $ x>0$ and $ y>0$ such that: $ f(x\plus{}y)<yf(f(x)).$ [b]Dan Schwarz[/b]

Milos arranged the numbers $1$ through $49$ into the cells of a $7\times7$ board. Djordje wants to guess the arrangement of the numbers. He can choose a square covering some cells of the board and ask Milos which numbers are found inside that square. At least, how many questions does Djordje need so as to be able to guess the arrangement of the numbers?

In a field, $2023$ friends are standing in such a way that all distances between them are distinct. Each of them fires a water pistol at the friend that stands closest. Prove that at least one person does not get wet.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules: $\cdot$ The frog can jump only in points of $M$ $\cdot$ The frog can't jump more than $1$ time over the same point. $\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$ In how many ways the Frog can make his target?

For any given real $a, b, c$ solve the following system of equations: $$\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right.$$ [i]Proposed by Oleksiy Masalitin, Bogdan Rublov[/i]

Alka finds a number $n$ written on a board that ends in $5.$ She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations: $1.$ Erase the written number $m$ and write it´s cube $m^3$. $2.$ Erase the written number $m$ and write the product $2023m$. Alka performs each operation an even number of times in some order and at least once, she finally obtains the number $r$. If the tens digit of $r$ is an odd number, find all possible values that the tens digit of $n^3$ could have had.

Find all the natural numbers $a,b,c$ such that: 1) $a^2+1$ and $b^2+1$ are primes 2) $(a^2+1)(b^2+1)=(c^2+1)$

Let $\vartriangle ABC$ be an acute triangle. $R$ is a point on arc $BC$. Choose two points $P, Q$ on $AR$ such that $B,P,C,Q$ are concyclic. Let the second intersection of $BP$, $CP$, $BQ$, $CQ$ and the circumcircle of $\vartriangle ABC$ is $P_B$, $P_C$, $Q_B$, $Q_C$, respectively. Let the circumcenter of $\vartriangle P P_BP_C$ and $\vartriangle QQ_BQ_C$ are $O_P$ and $O_Q$, respectively. Prove that $A,O_P,O_Q,R$ are concylic. [i]proposed by andychang[/i]

The [i]traveler ant[/i] is walking over several chess boards. He only walks vertically and horizontally through the squares of the boards and does not pass two or more times over the same square of a board. a) In a $4$x$4$ board, from which squares can he begin his travel so that he can pass through all the squares of the board? b) In a $5$x$5$ board, from which squares can he begin his travel so that he can pass through all the squares of the board? c) In a $n$x$n$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?

Find the value of $x^2+y^2+z^2$ where $x, y, z$ are non-zero and satisfy the following: (1) $x+y+z=3$ (2) $x^2(\frac{1}{y}+\frac{1}{z})+y^2(\frac{1}{z}+\frac{1}{x})+z^2(\frac{1}{x}+\frac{1}{y})=-3$

A [i]figuric [/i] is a convex polyhedron with $26^{5^{2019}}$ faces. On every face of a figuric we write down a number. When we throw two figurics (who don't necessarily have the same set of numbers on their sides) into the air, the figuric which falls on a side with the greater number wins; if this number is equal for both figurics, we repeat this process until we obtain a winner. Assume that a figuric has an equal probability of falling on any face. We say that one figuric rules over another if when throwing these figurics into the air, it has a strictly greater probability to win than the other figuric (it can be possible that given two figurics, no figuric rules over the other). Milisav and Milojka both have a blank figuric. Milisav writes some (not necessarily distinct) positive integers on the faces of his figuric so that they sum up to $27^{5^{2019}}$. After this, Milojka also writes positive integers on the faces of her figuric so that they sum up to $27^{5^{2019}}$. Is it always possible for Milojka to create a figuric that rules over Milisav's? [i]Proposed by Bojan Basic[/i]

Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?

We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

Determine all positive integers n such that $n^2/ (n - 1)!$

Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.

A line in the plane of a triangle $T$ is called an [i]equalizer[/i] if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c,$ with $a$ as small as possible, such that there exists a triangle with side lengths $a,b,c$ that has exactly two distinct equalizers.

For a $ n\times n $ real matrix $ M, $ prove that [b]a)[/b] $ M=0 $ if $ \text{tr} \left(M^tM\right) =0. $ [b]b)[/b] $ ^tM=M $ if $M^tM=M^2. $ [b]c)[/b] $ ^tM=-M $ if $ M^tM=-M^2. $ [b]d)[/b] Give example of a $ 2\times 2 $ real matrix $ A $ satisfying the following: $ \text{(i)} ^tA\cdot A^2=A^3 $ and $ ^tA\neq A $ $ \text{(ii)} ^tA\cdot A^2=-A^3 $ and $ ^tA\neq -A $ [i]Vasile Pop[/i]

In a lake there are several sorts of fish, in the following distribution: $ 18\%$ catfish, $ 2\%$ sturgeon and $ 80\%$ other. Of a catch of ten fishes, let $ x$ denote the number of the catfish and $ y$ that of the sturgeons. Find the expectation of $ \frac {x}{y \plus{} 1}$

Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.