Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

[b]10.[/b] Given a triangle $ABC$, construct outwards over the sides $AB, BC, CA$ similiar isosceles triangles $ABC_{1}, BCA_{1}$ and $CAB_{1}$. Prove that the straight lines $AA_{1}. BB_{1}$ and $CC_{1}$ are concurrent. Is this statemente true in elliptic and hyperbolic geometry, too? [b](G. 19)[/b]

Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

The natural number $A$ has three digits added to its right. The resulting number turned out to be equal to the sum of all natural numbers from $1$ to $A$. Find $A$.

A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth? [asy] path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(3,0)*p); draw(shift(3,1)*p); draw(shift(4,0)*p); draw(shift(4,1)*p); draw(shift(7,0)*p); draw(shift(7,1)*p); draw(shift(7,2)*p); draw(shift(8,0)*p); draw(shift(8,1)*p); draw(shift(8,2)*p); draw(shift(9,0)*p); draw(shift(9,1)*p); draw(shift(9,2)*p); [/asy] $ \text{(A)}\ 11\qquad\text{(B)}\ 12\qquad\text{(C)}\ 13\qquad\text{(D)}\ 14\qquad\text{(E)}\ 15 $

$N$ unfair coins (with heads and tails on the sides) are thrown, with the $k^{th}$ coin has got a chance of $\frac{1}{2k + 1}$ to land on tails.How high is the probability that an odd number of coins will show tails?

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows: $\circ$ The first member $s_1$ is 2. $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation. $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation. $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation. The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$.\\ Does this sequence contain a) 2001, b) 2006?

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

We print the terms of the sequence $ (n_1, n_2, \ldots, n_k) $, where $ n_1 = 1000 $, and $ n_j $ for $ j > 1 $ is an integer selected randomly from the range $ [0, n_{j-1 } - 1] $ (each number in this range is equally likely to be selected). We stop printing when the selected number is zero, i.e. $ n_{k-1} $, $ n_k = 0 $, The length $ k $ of the sequence $ (n_1, n_2, \ldots, n_k) $ is a random variable. Prove that the expected value of this random variable is greater than 7.

Given an acute triangle $ABC$ with $AB < AC$.Let $\Omega $ be the circumcircle of $ ABC$ and $M$ be centeriod of triangle $ABC$.$AH$ is altitude of $ABC$.$MH$ intersect with $\Omega $ at $A'$.prove that circumcircle of triangle $A'HB$ is tangent to $AB$. A.I.Golovanov, A. Yakubov

Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$ [i]Proposed by Ray Li[/i]

Decompose the polynomial $$x^8 + x^4 +1$$ to factors of at most second degree.

On the plane there is a finite set of points $Z$ with the property that no two distances of the points of the set $Z$ are equal. We connect the points $ A, B $ belonging to $ Z $ if and only if $ A $ is the point closest to $ B $ or $ B $ is the point closest to $ A $. Prove that no point in the set $Z$ will be connected to more than five others.

The incircle $(I)$ of a triangle $ABC$ touches $BC,CA,AB$ at $D,E, F$. Let $ID, IE, IF$ intersect $EF, FD,DE$ at $X,Y,Z$, respectively. The lines $\ell_a, \ell_b, \ell_c$ through $A,B,C$ respectively and are perpendicular to $YZ,ZX,XY$ . Prove that $\ell_a, \ell_b, \ell_c$ are concurrent at a point that is on the line segment joining $I$ and the centroid of triangle $ABC$ . Nguyễn Minh Hà

Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? [asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.5,2); B = (1.5,2); C = (1.5,1); D = (0.5,1); E = (0,1); F = (0,0); G = (1,0); H = (1,1); I = (2,1); J = (2,0); draw(A--B); draw(C--B); draw(D--A); draw(F--E); draw(I--J); draw(J--F); draw(G--H); draw(A--J); filldraw(A--B--C--I--J--cycle,grey); draw(E--I); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, NW); dot("$F$", F, SW); dot("$G$", G, S); dot("$H$", H, N); dot("$I$", I, NE); dot("$J$", J, SE);[/asy] $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac7{24} \qquad \textbf{(C)}\ \frac13 \qquad \textbf{(D)}\ \frac38 \qquad \textbf{(E)}\ \frac5{12}$

Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

Triangle $\triangle ABC$ has $AB= 3$, $BC = 4$, and $AC = 5$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. If line $AN$ intersects the circumcircle of triangle $\triangle BMC$ at points $X$ and $Y$, then $XY^2 = \frac{m}{n}$ for some relatively prime positive integers $m,n$. Find $m+n$. [i]Proposed by [b]Th3Numb3rThr33[/b][/i]

(V.Yasinsky, Ukraine) Suppose $ X$ and $ Y$ are the common points of two circles $ \omega_1$ and $ \omega_2$. The third circle $ \omega$ is internally tangent to $ \omega_1$ and $ \omega_2$ in $ P$ and $ Q$ respectively. Segment $ XY$ intersects $ \omega$ in points $ M$ and $ N$. Rays $ PM$ and $ PN$ intersect $ \omega_1$ in points $ A$ and $ D$; rays $ QM$ and $ QN$ intersect $ \omega_2$ in points $ B$ and $ C$ respectively. Prove that $ AB \equal{} CD$.

Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]

Determine the natural numbers that cannot be written as $\lfloor n + \sqrt{n} + \frac{1}{2} \rfloor$ for any $n \in \mathbb{N}$.