Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies: (i) $f(1995) = 1996$ (ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$. Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .

Prove or disprove: $\exists n\in N$ , s.t. $324 + 455^n$ is prime.

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

For each integer $n\geq0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2\leq n$. Define a sequence by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k\geq0$. For what positive integers $A$ is this sequence eventually constant?

Find all positive integers $b$ with the following property: there exists positive integers $a,k,l$ such that $a^k + b^l$ and $a^l + b^k$ are divisible by $b^{k+l}$ where $k \neq l$.

The board was placed $$ \begin{array}{rcl}<br /> 1 & = & 1 \\<br /> 2 + 3 + 4 & = & 1 + 8 \\<br /> 5 + 6 + 7 + 8 + 9 & = & 8 + 27\\<br /> 10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64\\<br /> & \ldots &<br /> \end{array}$$ Write such a formula for the $ n $-th row of the array that, with the substitutions $ n = 1, 2, 3, 4 $, would give the above four lines of the array and would be true for every natural $ n $.

A subset of $ n\times n$ table is called even if it contains even elements of each row and each column. Find the minimum $ k$ such that each subset of this table with $ k$ elements contains an even subset

Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's [i]good[/i] if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same; 3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same. Find all $ m,n$ such that there exists a good $ m\times n$ board.

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.

In a unit cube $ABCD - EFGH$, an equilateral triangle $BDG$ cuts out a circle from the circumsphere of the cube. Find the area of the circle.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle . (A. Andjans, Riga)

Let $ \mathbb{K} $ be a field and let be a matrix $ M\in\mathcal{M}_3(\mathbb{K} ) $ having the property that $ \text{tr} (A) =\text{tr} (A^2) =0 . $ Show that there is a $ \mu\in \mathbb{K} $ such that $ A^3=\mu A $ or $ A^3=\mu I. $ [i]Cristinel Mortici[/i]

Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$

Three congruent equilateral triangles $T_1$, $T_2$, and $T_3$ are stacked from left to right inside rectangle $JHMT$ such that the bottom left vertex of $T_1$ is $T$, the bottom side of $T_1$ lies on $\overline{MT}$, the bottom left vertex of $T_2$ is the midpoint of a side of $T_1$, the bottom left vertex of $T_3$ is the midpoint of a side of $T_2$, and the other two vertices of $T_3$ lie on $\overline{JH}$ and $\overline{HM}$, as shown below. Given that rectangle $JHMT$ has area $2022$, find the area of any one of the triangles $T_1$, $T_2$, or $T_3$. [asy] unitsize(0.111111111111111111cm); real s = sqrt(4044/sqrt(75)); real l = 5s/2; real w = s * sqrt(3); pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3; J = (0,w); H = (l,w); M = (l,0); T = (0,0); V1 = (s,0); V2 = (s/2,s * sqrt(3)/2); V3 = (V1+V2)/2; V4 = (3 * s/4+s,s * sqrt(3)/4); V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2); V6 = (V4+V5)/2; V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4); V8 = (l-s/2,w); C1 = (T+V1+V2)/3; C2 = (V3+V4+V5)/3; C3 = (V6+V7+V8)/3; draw(J--H--M--T--cycle); draw(V1--V2--T); draw(V3--V4--V5--cycle); draw(V6--V7--V8--cycle); label("$J$", J, NW); label("$H$", H, NE); label("$M$", M, SE); label("$T$", T, SW); label("$T_1$", C1); label("$T_2$", C2); label("$T_3$", C3); [/asy]

Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$, generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$, denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$. Show that for any $k$, $|S(k)|\leq n-1$.

Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?

Find the integer which is closest to the value of the following expression: $$\left((3 + \sqrt{1})^{2023} - \left(\frac{1}{3 - \sqrt{1}}\right)^{2023} \right) \cdot \left((3 + \sqrt{2})^{2023} - \left(\frac{1}{3 - \sqrt{2}}\right)^{2023} \right) \cdot \ldots \cdot \left((3 + \sqrt{8})^{2023} - \left(\frac{1}{3 - \sqrt{8}}\right)^{2023} \right)$$

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

Let $\omega_1,\omega_2$ be two non-intersecting circles, with circumcenters $O_1,O_2$ respectively, and radii $r_1,r_2$ respectively where $r_1 < r_2$. Let $AB,XY$ be the two internal common tangents of $\omega_1,\omega_2$, where $A,X$ lie on $\omega_1$, $B,Y$ lie on $\omega_2$. The circle with diameter $AB$ meets $\omega_1,\omega_2$ at $P$ and $Q$ respectively. If $$\angle AO_1P+\angle BO_2Q=180^{\circ},$$ find the value of $\frac{PX}{QY}$ (in terms of $r_1,r_2$).

Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?

Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$