Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity \[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\] Where $\alpha,\beta$ are given real numbers.

In the Republic of Unistan there are $n$ national roads, each of them links two cities exactly. You can travel from one city to another of your choice using a sequence of roads. The President of Unistan ordered to label the national roads with the integers from $1$ to $n$ by an old law: if a city is adjacent to more than one road, the greatest common divisor of the numbers of that roads must be one. Show that you can label the national roads without breaking the law.

The sides of a rhombus have length $a$ and the area is $S$. What is the length of the shorter diagonal?

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 $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

In the set of real numbers solve the system of equations $x^4+y^2+4=5yz$ $y^4+z^2+4=5zx$ $z^4+x^2+4=5xy$

A jobber buys an article at $\$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked? ${{ \textbf{(A)}\ 25.20 \qquad\textbf{(B)}\ 30.00 \qquad\textbf{(C)}\ 33.60 \qquad\textbf{(D)}\ 40.00 }\qquad\textbf{(E)}\ \text{none of these} } $

In the triangle $ABC$, let $AM$ and $CN$ internal bisectors, with $M$ in $BC$ and $N$ in $AB$. Prove that if $$\frac{\angle BNM}{\angle MNC}=\frac{\angle BMN}{\angle NMA}$$ then $ABC$ is isosceles.

Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.

Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Let $ABCD$ be a parallelogram. Construct squares $ABC_1D_1, BCD_2A_2, CDA_3B_3, DAB_4C_4$ on the outer side of the parallelogram. Construct a square having $B_4D_1$ as one of its sides and it is on the outer side of $AB_4D_1$ and call its center $O_A$. Similarly do it for $C_1A_2, D_2B_3, A_3C_4$ to obtain $O_B, O_C, O_D$. Prove that $AO_A = BO_B = CO_C = DO_D$.

Let $ABCD$ be a parallelogram and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$, and $P \in BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.

Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$? Recall that Pascal’s triangle is the pattern of numbers that begins as follows [img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img] where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$. Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.

Prove that there exist infinitely many distinct positive integers, $x$ and $y$, such that $$x^3+y^2|x^2+y^3$$

Let \( I \) and \( M \) be the incenter and the centroid of a scalene triangle \( ABC \), respectively. A line passing through point \( I \) parallel to \( BC \) intersects \( AC \) and \( AB \) at points \( E \) and \( F \), respectively. Reconstruct triangle \( ABC \) given only the marked points \( E, F, I, \) and \( M \). [i]Proposed by Hryhorii Filippovskyi[/i]

Let $\sigma_n$ be a permutation of $\{1,\ldots,n\}$; that is, $\sigma_n(i)$ is a bijective function from $\{1,\ldots,n\}$ to itself. Define $f(\sigma)$ to be the number of times we need to apply $\sigma$ to the identity in order to get the identity back. For example, $f$ of the identity is just $1$, and all other permutations have $f(\sigma)>1$. What is the smallest $n$ such that there exists a $\sigma_n$ with $f(\sigma_n)=k$?

Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$. (i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$. (ii) Show that $ BM\equal{}BC$.

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

If we have a number $x$ at a certain step, then at the next step we have $x+1$ or $-\frac 1x$. If we start with the number $1$, which of the following cannot be got after a finite number of steps? $ \textbf{(A)}\ -2 \qquad\textbf{(B)}\ \dfrac 12 \qquad\textbf{(C)}\ \dfrac 53 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of above} $

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks. [i]Proposed by James Lin[/i]

Let $n \ge 2$ be a positive integer. For every number from $1$ to $n$, there is a card with this number and which is either black or white. A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card. For instance, when $n=5$ and the initial configuration is $(1B, 2B, 3W, 4B,5B)$, the magician can choose $1B, 3W$ on the first move to obtain $(3W, 2B, 3W, 4B, 5B)$ and then $3W, 4B$ on the second move to obtain $(4B, 2B, 3W, 4B, 5B)$. Determine in terms of $n$ all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.

Each morning of her five-day workweek, Jane bought either a $ 50$-cent muffin or a $ 75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$