Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.

In a virtually-made world, each citizen, which is assumed to be a point (i.e. without area) and labelled as $1, 2, ...$. To fight against a pandemic, these citizens are required to get vaccinated. After they get vaccinated, they need to be observed for a period of time. Now assume the location that the citizens get observe is a circumference with radius $\frac{1}{4}$ on the plane. For the safety reason, it is required for distance between $m$-th citizen and $n$-th citizen $d_{m, n}$ satisfying the following: $(m+n)d_{m, n}\geq 1$ Here what we consider is the distance on the circumference i.e. the arc length of minor arc formed by two points. Then (a) Choose one of the following which fits the situation in reality. A. The location for observation can mostly have $8$ citizens. B. The location for observation can have the upper limit on the number of citizens which is larger than $8$. C. The location for observation can have any number of citizens. (b) Prove your answer in (a).

$(a)$ Is it possible to partition the segment $[0,1]$ into two sets $A$ and $B$ and to define a continuous function $f$ such that for every $x\in A \ f(x)$ is in $B$, and for every $x\in B \ f(x)$ is in $A$? $(b)$ The same question with $[0,1]$ replaced by $[0,1).$

The polygon $P$, cut out of paper, is bent in a straight line and both halves are glued. Can the perimeter of the polygon $Q$ obtained by gluing be larger than the perimeter of the polygon $P$?

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

Find all positive integers $a,n\ge1$ such that for all primes $p$ dividing $a^n-1$, there exists a positive integer $m<n$ such that $p\mid a^m-1$.

Let $a$, $b$ and $c$ be integers such that $$bc+ad=1$$ $$ac+2bd=1$$ Prove that $a^2+c^2=2b^2+2d^2$

For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that \[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\] holds.

Define a sequence $\{ a_n \}_{n=0}^{\infty}$ by $a_0 = 1,$ $a_1 = 8,$ and $a_n = 2a_{n-1} + a_{n-2}$ for $n \geq 2.$ The infinite sum \[ \sum_{n=1}^{\infty} \int_{0}^{2021\pi/14} \sin(a_{n-1}x)\sin(a_nx)\,dx \] can be expressed as a common fraction $\tfrac{p}{q}.$ Compute $p + q.$

Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.

Let $p_n$ be the $n$-th prime. ($p_1=2$) Define the sequence $(f_j)$ as follows: - $f_1=1, f_2=2$ - $\forall j\ge 2$: if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$ - $\forall j\ge 2$: if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$ (a) Show that all $f_i$ are different (b) from which index onwards are all $f_i$ at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?

A function $ f$ from the integers to the integers is defined as follows: \[ f(n) \equal{} \begin{cases} n \plus{} 3 & \text{if n is odd} \\ n/2 & \text{if n is even} \end{cases} \]Suppose $ k$ is odd and $ f(f(f(k))) \equal{} 27$. What is the sum of the digits of $ k$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$

Let $S$ be an infinite set of integers containing zero, and such that the distances between successive number never exceed a given fixed number. Consider the following procedure: Given a set $X$ of integers we construct a new set consisting of all numbers $x \pm s,$ where $x$ belongs to $X$ and s belongs to $S.$ Starting from $S_0 = \{0\}$ we successively construct sets $S_1, S_2, S_3, \ldots$ using this procedure. Show that after a finite number of steps we do not obtain any new sets, i.e. $S_k = S_{k_0}$ for $k \geq k_0.$

Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that $$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$

The diagram shows two intersecting line segments that form some of the sides of two squares with side lengths $3$ and $6$. Two line segments join vertices of these squares. Find the area of the region enclosed by the squares and segments.

Given a cyclic quadrilateral $ABCD$. Let $X$ be a point on segment $BC$ ($X \not= BC$) such that line $AX$ is perpendicular to the angle bisector of $\angle CBD$, and $Y$ be a point on segment $AD$ ($Y \not= D)$ such that $BY$ is perpendicular to the angle bisector of $\angle CAD$. Prove that $XY$ is parallel to $CD$.

Prove that a prime of the form $2^{2^n}+1$ cannot be the difference of two fifth powers of two positive integers. Pierre.

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

Let be two positive real numbers $ a,b $ whose product is $ 1$ and whose sum is irrational. Prove that for any natural number $ n\ge 2 $ the epression $ \sqrt[n]{a}+\sqrt[n]{b} $ is irrational. [i]Râmbu Gheorghe[/i]

Show that for every prime $p$ and integer $n$, there is an irreducible polynomial of degree $n$ in $\mathbb{Z}_p[x]$ and use that to show there is a field of size $p^n$.

Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions $f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s). Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$

Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode? $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$

Vessel $A$ has liquids $X$ and $Y$ in the ratio $X:Y=8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y=5:9$. What ratio should you mix the liquids in both vessels if you need the mixture to be $X:Y=1:1$? $\textbf{(A)}~4:3$ $\textbf{(B)}~30:7$ $\textbf{(C)}~17:25$ $\textbf{(D)}~7:30$