Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.

Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $

Let $ n$ be an integer, $ n\geq 2$, and the integers $ a_1,a_2,\ldots,a_n$, such that $ 0 < a_k\leq k$, for all $ k \equal{} 1,2,\ldots,n$. Knowing that the number $ a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n$ is even, prove that there exists a choosing of the signs $ \plus{}$, respectively $ \minus{}$, such that \[ a_1 \pm a_2 \pm \cdots \pm a_n\equal{} 0. \]

Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pile actually has. What is the minimum number of steps we have to take?

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.

Let $BF$ and $CN$ be the altitudes of the acute triangle $ABC$. Bisectors the angles $ACN$ and $ABF$ intersect at the point $T$. Find the radius of the circle circumscribed around the triangle $FTN$, if it is known that $BC = a$. (Grigory Filippovsky)

Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and \[f(n)=n - f(f(n-1)), \quad \forall n \geq 2.\] Prove that $f(n+f(n))=n $ for each positive integer $n.$

Given is a triangle $\triangle{ABC}$,with $AB<AC<BC$,inscribed in circle $c(O,R)$.Let $D,E,Z$ be the midpoints of $BC,CA,AB$ respectively,and $K$ the foot of the altitude from $A$.At the exterior of $\triangle{ABC}$ and with the sides $AB,AC$ as diameters,we construct the semicircles $c_1,c_2$ respectively.Suppose that $P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1$ and $R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2$.Finally,let $M$ be the intersection of the lines $PS,RT$. [b]i.[/b] Prove that the lines $PR,ST$ intersect at $A$. [b]ii.[/b] Prove that the lines $PR\cap MD$ intersect on $c$. [asy]import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); draw((0.,0.)--(5.,0.), uququq); draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); draw((0.6699432366054657,0.)--(5.,3.257603675597893)); draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); /* dots and labels */ dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); label("$A$", (0.7472169504504719,2.65), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.2,-0.4), NE * labelscalefactor); dot((5.,0.),linewidth(3.pt) + dotstyle); label("$C$", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); dot((2.5,0.),linewidth(3.pt) + dotstyle); label("$D$", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$E$", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); label("$Z$", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); label("$K$", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); label("$P$", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); label("$S$", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); label("$T$", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); label("$R$", (3.445433497126431,4.375025554412117), NE * labelscalefactor); dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); label("$M$", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Let $[a,b]$ be a given interval of real numbers not containing integers. Prove that there exists $N>0$ such that $[Na,Nb]$ does not contain integer numbers and the length of the interval $[Na,Nb]$ exceedes $\dfrac 16$.

A rectangular box has a total surface area of $94$ square inches. The sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals? ${ \textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2} $

Equal line segments are marked in triangle $ABC$. Find its angles. [img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]

A [i]mirrored polynomial[/i] is a polynomial $f$ of degree $100$ with real coefficients such that the $x^{50}$ coefficient of $f$ is $1$, and $f(x) = x^{100} f(1/x)$ holds for all real nonzero $x$. Find the smallest real constant $C$ such that any mirrored polynomial $f$ satisfying $f(1) \ge C$ has a complex root $z$ obeying $|z| = 1$.

The tangents at points $B$ and $C$ on a given circle meet at point $A$. Let $Q$ be a point on segment $AC$ and let $BQ$ meet the circle again at $P$. The line through $Q $ parallel to $AB$ intersects $BC$ at $J$. Prove that $PJ$ is parallel to $AC$ if and only if $BC^2 = AC\cdot QC$.

[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b] A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness [b]One[/b] said exactly one of the four witnesses is a liar. Witness [b]Two[/b] said exactly two of the four witnesses is a liar. Witness [b]Three[/b] said exactly three of the four witnesses is a liar. Witness [b]Four[/b] said exactly four of the four witnesses is a liar. Assume that each of the statements is either true or false.How many of the winesses are liars?

Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote \[ S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|. \] a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series \[ \sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c) \] are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent? b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series \[ S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d) \] are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?