Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$. (Nagel Igor)

Let $m, n,$ and $k$ be natural numbers, where $n$ is odd. Prove that $\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}$ is not a natural number.

Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle. by A.Khachaturyan

If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1<a_2<a_3<\cdots<a_n,$ its complex power sum is defined to be $a_1i+a_2i^2+a_3i^3+\cdots+a_ni^n,$ where $i^2=-1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8=-176-64i$ and $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$

Infinitesimal Randall Munroe is glued to the center of a pentagon with side length $1$. At each corner of the pentagon is a confused infinitesimal velociraptor. At any time, each raptor is running at one unit per second directly towards the next raptor in the pentagon (in counterclockwise order). How far does each confused raptor travel before it reaches Randall Munroe?

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

Solve for $x$ if $4x + 1 = 37$. $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Find the smallest $3$-digit number that is the product of two $2$-digit numbers , so that the seven digits of these three numbers are all different.

Determine all real solutions to the system of equations: $$x^2 - y = z^2$$ $$y^2 - z = x^2$$ $$z^2 - x = y^2$$

Let \[ I(R)=\iint\limits_{x^2+y^2 \le R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right) dx dy. \] Find \[ \lim_{R \to \infty}I(R), \] or show that this limit does not exist.

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies \[f(x + y) \leq yf(x) + f(f(x))\] for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$. [i]Proposed by Igor Voronovich, Belarus[/i]

Prove that for any distinct rational numbers $a, b, c$, the number \[\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}+\frac{1}{(a-b)^{2}}\] is the square of some rational number.

In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$? $\textbf{(A) } 1 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 92 \qquad \textbf{(E) } 106$

$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A $ choose in order to secure as large a slice as possible?

[Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$.

Consider the set $S = \{1,2,3, ...,1441\}$. 1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania. 2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$. 3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.

Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$. If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.