For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$

The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$. Note: The locus is the set of all points of the plane that satisfies the property.

Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs $+$ , $-$ and $\times$. He may also use a special sign $\pm$: computing the values of the resulting expression, he chooses values $+$ or $-$ for every $\pm$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6 \}$, and $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3 \}$. Can Pete construct such an expression: $a)$ If the numbers on the blackboard are $1, 2, 4$; $b)$ For any collection of $100$ distinct real numbers on a blackboard?

A rectangle with perimeter $ 176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? [asy]defaultpen(linewidth(.8pt)); draw(origin--(0,3)--(4,3)--(4,0)--cycle); draw((0,1)--(4,1)); draw((2,0)--midpoint((0,1)--(4,1))); real r = 4/3; draw((r,3)--foot((r,3),(0,1),(4,1))); draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]$ \textbf{(A)}\ 35.2\qquad \textbf{(B)}\ 76\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 84\qquad \textbf{(E)}\ 86$

Let $ABC$ be a triangle with sides $BC = 13$, $CA = 14$ and $AB = 15$. We denote by $I$ the intersection point of the angle bisectors and $M$ to the midpoint of $AB$. The line $IM$ cuts at $P$ at the altitude drawn from $C$. Find the length of $CP$.

Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set \[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}. \]What is the minimum possible value of \[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2 \]$ \textbf{(A)}\ 30\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 50$

Let $P(x) = a_{1998}X^{1998} + a_{1997}X^{1997} +...+a_1X + a_0$ be a polynomial with real coefficients such that $P(0) \ne P (-1)$, and let $a, b$ be real numbers. Let $Q(x) = b_{1998}X^{1998} + b_{1997}X^{1997} +...+b_1X + b_0$ be the polynomial with real coefficients obtained by taking $b_k = aa_k + b$ ,$\forall k = 0, 1,2,..., 1998$. Show that if $Q(0) = Q (-1) \ne 0$ , then the polynomial $Q$ has no real roots.

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $

$2020*N$ is a perfect cube. If $N$ can be expressed as $2^a*5^b*101^c,$ find the least possible value of $a+b+c$ such that $a,b,c$ are all positive integers and not necessarily distinct. [i]Proposed by Ephram Chun[/i]

There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

A game is played between two players on a $10 \times 10$ checkerboard. They move alternately, the first player marking $X$s on vacant cells and the second $O$s. When all $100$ cells have been marked, they calculate two numbers $C$ and $Z$. $C$ is the total number of five consecutive $X$s in a row, a column or a diagonal, so that $6$ consecutive $X$s contribute a count of $2$ to $C$, $7$ consecutive $X$s contribute $3$, and so on. Similarly, $Z$ is the total number of five consecutive Os. The first player wins if $C > Z$, loses if $C < Z$ and draws if $C = Z$. Does the first player have a strategy which guarantees (a) a draw or a win (b) a win regardless of the counter-strategy of the second player? (A Belov)

Determine all sets of real numbers $S$ such that: [list] [*] $1$ is the smallest element of $S$, [*] for all $x,y\in S$ such that $x>y$, $\sqrt{x^2-y^2}\in S$ [/list] [i]Adian Anibal Santos Sepcic[/i]

The circles $(R, r)$ and $(P, \rho)$, where $r > \rho$, touch externally at $A$. Their direct common tangent touches $(R, r)$ at B and $(P, \rho)$ at $C$. The line $RP$ meets the circle $(P, \rho)$ again at $D$ and the line $BC$ at $E$. If $|BC| = 6|DE|$, prove that: [b](a)[/b] the lengths of the sides of the triangle $RBE$ are in an arithmetic progression, and [b](b)[/b] $|AB| = 2|AC|.$

Is the infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$ convergent?

If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is: $\textbf{(A) }\dfrac{x+1}{x-1}\qquad\textbf{(B) }\dfrac{x+2}{x-1}\qquad\textbf{(C) }\dfrac{x}{x-1}\qquad\textbf{(D) }2-x\qquad \textbf{(E) }\dfrac{x-1}{x}$

In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle.  Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively.  Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2\tan^{-1}(1/3)$.  Find $\alpha$.

Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$

For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

A square measuring $15$ by $15$ is partitioned into five rows of five congruent squares as shown below. The small squares are alternately colored black and white as shown. Find the total area of the part colored black. [asy] size(150); defaultpen(linewidth(0.8)); int i,j; for(i=1;i<=5;i=i+1) { for(j=1;j<=5;j=j+1) { if (floor((i+j)/2)==((i+j)/2)) { filldraw(shift((i-1,j-1))*unitsquare,gray); } else { draw(shift((i-1,j-1))*unitsquare); } } } [/asy]