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Neka je $n \geqslant 4$ prirodni broj i neka su $x_1, x_2, \ldots, x_n$ realni brojevi takvi da je

$$x_1 + x_2 + \cdots + x_n \geqslant n \quad \text{i} \quad x_1^2 + x_2^2 + \cdots + x_n^2 \geqslant n^2.$$

Dokaži da postoji $i \in \{1, 2, \ldots, n\}$ takav da je $x_i \geqslant 2$.

Odredi najmanji realni broj $D$ takav da nejednakost

$$\frac{a + b}{a + 2b} + \frac{b + c}{b + 2c} + \frac{c + a}{c + 2a} < D$$

vrijedi za sve pozitivne realne brojeve $a$, $b$ i $c$.

Neka su $a$, $b$ i $c$ pozitivni realni brojevi takvi da je $a + b + c = 3$. Dokaži da je

$$\frac{a^4}{b^2 + c} + \frac{b^4}{c^2 + a} + \frac{c^4}{a^2 + b} \geqslant \frac{3}{2}.$$

Dokaži da za pozitivne realne brojeve $a$, $b$ i $c$ za koje je $a + b + c = 3$ vrijedi $$\frac{a^2}{a + b^2} + \frac{b^2}{b + c^2} + \frac{c^2}{c + a^2} \geq \frac{3}{2}.$$

Unutar šiljastokutnog trokuta $ABC$ dana je točka $S$ takva da je $\measuredangle SAB = \measuredangle SBC = \measuredangle SCA$. Pravci $AS$, $BS$, $CS$ sijeku redom kružnice opisane trokutima $SBC$, $SCA$, $SAB$ u točkama $A_1$, $B_1$, $C_1$. Dokaži nejednakost $$P(A_1CB) + P(B_1AC) + P(C_1BA) \geq 3P(ABC).$$

Dani su pozitivni realni brojevi $x$, $y$ i $z$ takvi da je $x + y + z = 18xyz$. Dokaži nejednakost

$$\frac{x}{\sqrt{x^2 + 2yz + 1}} + \frac{y}{\sqrt{y^2 + 2xz + 1}} + \frac{z}{\sqrt{z^2 + 2xy + 1}} \geqslant 1.$$

Za prirodni broj $n \geqslant 2$, neka su $x_1, x_2, \ldots, x_n$ realni brojevi različiti od nule takvi da je $x_1 + x_2 + \cdots + x_n = 0$. Dokaži da postoje različiti prirodni brojevi $i$ i $j$ ($i, j \leqslant n$) takvi da je

$$\frac{1}{2} \leqslant \left| \frac{x_i}{x_j} \right| \leqslant 2.$$

Odredi sve funkcije $f: \mathbb{R} \to \mathbb{R}$ takve da je $f(1) \geqslant 0$ i da za sve realne brojeve $x$ i $y$ vrijedi

$$f(x) - f(y) \geqslant (x - y)f(x - y).$$

U ovisnosti o prirodnom broju $k$, odredi najmanji realni broj $D_k$ takav da je

$$(abc)^2 + (bcd)^2 + (cda)^2 + (dab)^2 \leqslant D_k$$

za sve nenegativne realne brojeve $a, b, c, d$ za koje je $a^k + b^k + c^k + d^k = 4$.

Neka su $a_1, a_2, \ldots, a_n$ pozitivni realni brojevi takvi da je $a_1 + a_2 + \cdots + a_n = 1$.

Dokaži nejednakost:

$$\frac{a_1^3}{a_1^2 + a_2 a_3} + \frac{a_2^3}{a_2^2 + a_3 a_4} + \cdots + \frac{a_{n-1}^3}{a_{n-1}^2 + a_n a_1} + \frac{a_n^3}{a_n^2 + a_1 a_2} \geqslant \frac{1}{2}.$$

Dan je realni broj $\alpha \geqslant \frac{1}{2}$. Dokaži da za pozitivne realne brojeve $x, y, z$ vrijedi nejednakost: $$x(x - y)(\alpha x - y) + y(y - z)(\alpha y - z) + z(z - x)(\alpha z - x) \geqslant 0.$$

Neka $m$ prirodni broj. Dano je $2^{m}$ papira i na svakom od njih napisan je broj $1$. U svakom potezu dozvoljeno je izabrati dva različita papira, pobrisati brojeve $a$ i $b$ koji pišu na tim papirima te na oba papira napisati broj $a + b$.

Dokaži da nakon $2^{m-1}m$ poteza zbroj brojeva na svim papirima iznosi najmanje $4^{m}$.

Dokaži da za sve pozitivne realne brojeve $x$, $y$, $z$ vrijedi nejednakost $$\frac{x^2}{xy + z} + \frac{y^2}{yz + x} + \frac{z^2}{zx + y} \geqslant \frac{(x + y + z)^3}{3[x^2(y + 1) + y^2(z + 1) + z^2(x + 1)]}.$$

Niz $a_1, a_2, \ldots$ pozitivnih realnih brojeva zadovoljava uvjet

$$a_{k+1} \geq \frac{ka_k}{a_k^2 + k - 1}$$

za svaki prirodni broj $k$. Dokaži da je

$$a_1 + a_2 + \cdots + a_n \geq n$$

za svaki $n \geq 2$.

Dan je prirodni broj $n$. Dokaži da za sve realne brojeve $x_1, x_2, \ldots, x_n \geq 0$ vrijedi nejednakost

$$\left(x_1 + \frac{x_2}{2} + \cdots + \frac{x_n}{n}\right) \cdot \left(x_1 + 2x_2 + \cdots + nx_n\right) \leq \frac{(n+1)^2}{4n} \left(x_1 + x_2 + \cdots + x_n\right)^2.$$

Odredi najmanji realni broj $C$ takav da je za sve pozitivne realne brojeve $a_1, a_2, a_3, a_4$ i $a_5$ moguće odabrati međusobno različite indekse $i, j, k, l$ tako da vrijedi

$$\left| \frac{a_i}{a_j} - \frac{a_k}{a_l} \right| \leqslant C.$$

Točka $M$ se nalazi u unutrašnjosti trokuta $ABC$. Pravac $AM$ siječe kružnicu opisanu trokutu $MBC$ još jednom u točki $D$, pravac $BM$ kružnicu opisanu trokutu $MCA$ još jednom u točki $E$, a pravac $CM$ kružnicu opisanu trokutu $MAB$ još jednom u točki $F$. Dokaži da vrijedi

$$\frac{|AD|}{|MD|} + \frac{|BE|}{|ME|} + \frac{|CF|}{|MF|} \geqslant \frac{9}{2}.$$

Dokaži da za sve pozitivne realne brojeve $a$, $b$ i $c$ vrijedi

$$\frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} + \sqrt{\frac{ab + bc + ca}{a^2 + b^2 + c^2}} \geqslant \frac{5}{2}.$$

Neka su $a$, $b$ i $c$ pozitivni realni brojevi. Dokaži da vrijedi

$$\frac{3}{2} < \frac{4a + b}{a + 4b} + \frac{4b + c}{b + 4c} + \frac{4c + a}{c + 4a} < 9.$$

Neka su $a$, $b$ i $c$ pozitivni realni brojevi takvi da je $a + b + c = 2$. Dokaži da vrijedi

$$\frac{(a - 1)^2}{b} + \frac{(b - 1)^2}{c} + \frac{(c - 1)^2}{a} \geqslant \frac{1}{4} \left( \frac{a^2 + b^2}{a + b} + \frac{b^2 + c^2}{b + c} + \frac{c^2 + a^2}{c + a} \right).$$

Neka su $a$, $b$ i $c$ pozitivni realni brojevi. Dokaži da vrijedi $$\frac{\sqrt{a + b + c} + \sqrt{a}}{b + c} + \frac{\sqrt{a + b + c} + \sqrt{b}}{c + a} + \frac{\sqrt{a + b + c} + \sqrt{c}}{a + b} \geq \frac{9 + 3\sqrt{3}}{2\sqrt{a + b + c}}.$$

Neka je $n$ prirodni broj i neka su $x_1, x_2, \ldots, x_n$ realni brojevi takvi da je $$x_1 + x_2 + \cdots + x_n = 0 \quad \text{i} \quad x_1^2 + x_2^2 + \cdots + x_n^2 = 1.$$

Ako je $a$ najmanji, a $b$ najveći broj među brojevima $x_1, x_2, \ldots, x_n$, dokaži da je $ab \leq -\dfrac{1}{n}$.

Neka su $a$, $b$ i $c$ pozitivni realni brojevi za koje vrijedi $ab + bc + ca = 1$.

Dokaži da je $$\frac{4}{a + b + c} \leq (a + b)(c\sqrt{3} + 1).$$

Dokaži da za sve realne brojeve $x_1, x_2, \ldots, x_{100}$ vrijedi nejednakost

$$\sum_{1 \leqslant i < j \leqslant 100} \frac{(x_j - x_i)^2}{j^2 - i^2} \geqslant \frac{1}{101} \sum_{1 \leqslant i \leqslant 50} (x_{101-i} - x_i)^2.$$

Neka je $a_1, a_2, a_3, \ldots$ niz pozitivnih realnih brojeva takav da za svaki prirodan broj $n \geqslant 2$ vrijedi

$$a_n - a_{n+2} \leqslant (a_{n-1} - a_{n+1}) \cdot \frac{a_{n+1} + a_{n+2}}{a_{n-1} + a_n}.$$

Dokaži da je $a_{100} \geqslant a_{102}$.

Neka su $x, y, z$ pozitivni realni brojevi takvi da je $xy + yz + zx = 3$. Dokaži da vrijedi

$$\frac{x + 3}{y + z} + \frac{y + 3}{x + z} + \frac{z + 3}{x + y} + 3 \geqslant 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$

Odredi sve funkcije $f: \mathbb{N} \to \mathbb{N}$ takve da za sve $m,n \in \mathbb{N}$ vrijedi nejednakost

$$m^{f(n)} - n^{f(m)} \leq mn.$$

Za realan broj kažemo da je velik ako mu je apsolutna vrijednost veća ili jednaka $1$. Za svaki prirodan broj $m$, odredi najveći realan broj $C_m$ takav da za bilo kojih $m$ velikih brojeva $a_1, a_2, \ldots, a_m$ vrijedi

$$a_1^2 + (a_1 + a_2)^2 + \ldots + (a_1 + a_2 + \ldots + a_m)^2 \geq C_m.$$

For what values of the variable $x$ does the following inequality hold: $$\frac{4x^{2}}{(1-\sqrt{1+2x})^{2}}<2x+9?$$

Let $a, b, c$ be the sides of a triangle, and $T$ its area. Prove: $a^2 + b^2 + c^2 \geq 4\sqrt{3}T$. In what case does equality hold?

Consider triangle $P_1P_2P_3$ and a point $P$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers $$\frac{P_1P}{PQ_1}, \frac{P_2P}{PQ_2}, \frac{P_3P}{PQ_3}$$ at least one is $\leq 2$ and at least one is $\geq 2$.

Determine all real numbers $x$ which satisfy the inequality: $$\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}.$$

Suppose $a, b, c$ are the sides of a triangle. Prove that $$a^2(b + c - a) + b^2(c + a - b) + c^2(a + b - c) \leq 3abc.$$

Determine all values $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality $$2 \cos x \leq \left| \sqrt{1 + \sin 2x} - \sqrt{1 - \sin 2x} \right| \leq \sqrt{2}.$$

Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius 1 with centers $A, B, C, D$ cover the parallelogram if and only if $$a \leq \cos \alpha + \sqrt{3} \sin \alpha.$$

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is $\leq 1/8$.

Prove that for all real numbers $x_1, x_2, y_1, y_2, z_1, z_2$, with $x_1 > 0$, $x_2 > 0$, $x_1y_1 - z_1^2 > 0$, $x_2y_2 - z_2^2 > 0$, the inequality

$$\frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}$$

is satisfied. Give necessary and sufficient conditions for equality.

Let $a, b$ and $n$ be integers greater than 1, and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_n$ are numbers in the system with base $a$, and $B_{n-1}$ and $B_n$ are numbers in the system with base $b$; these are related as follows: $$A_n = x_n x_{n-1} \cdots x_0, \quad A_{n-1} = x_{n-1} x_{n-2} \cdots x_0,$$ $$B_n = x_n x_{n-1} \cdots x_0, \quad B_{n-1} = x_{n-1} x_{n-2} \cdots x_0,$$ $$x_n \neq 0, \quad x_{n-1} \neq 0.$$

Prove: $$\frac{A_{n-1}}{A_n} < \frac{B_{n-1}}{B_n} \text{ if and only if } a > b.$$

The real numbers $a_0, a_1, \ldots, a_n, \ldots$ satisfy the condition: $$1 = a_0 \leq a_1 \leq a_2 \leq \cdots \leq a_n \leq \cdots.$$

The numbers $b_1, b_2, \ldots, b_n, \ldots$ are defined by $$b_n = \sum_{k=1}^{n} \left(1 - \frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}}.$$

(a) Prove that $0 \leq b_n < 2$ for all $n$.

(b) Given $c$ with $0 \leq c < 2$, prove that there exist numbers $a_0, a_1, \ldots$ with the above properties such that $b_n > c$ for large enough $n$.

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $\triangle ABC$. Prove that $$(AB + BC + CA)^2 \leq 6(AD^2 + BD^2 + CD^2).$$

For what tetrahedra does equality hold?

Prove that the following assertion is true for $n = 3$ and $n = 5$, and that it is false for every other natural number $n > 2$: If $a_1, a_2, \ldots, a_n$ are arbitrary real numbers, then $$(a_1 - a_2)(a_1 - a_3) \cdots (a_1 - a_n) + (a_2 - a_1)(a_2 - a_3) \cdots (a_2 - a_n)$$ $$+ \cdots + (a_n - a_1)(a_n - a_2) \cdots (a_n - a_{n-1}) \geq 0$$

Let $A = (a_{ij})$ $(i, j = 1, 2, \ldots, n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij} = 0$, the sum of the elements in the $i$th row and the $j$th column is $\geq n$. Prove that the sum of all the elements of the matrix is $\geq n^2/2$.

Find all solutions $(x_1, x_2, x_3, x_4, x_5)$ of the system of inequalities $$(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0$$ $$(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0$$ $$(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0$$ $$(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0$$ $$(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0$$ where $x_1, x_2, x_3, x_4, x_5$ are positive real numbers.

Point $O$ lies on line $g$; $\overrightarrow{OP_1}, \overrightarrow{OP_2}, \ldots, \overrightarrow{OP_n}$ are unit vectors such that points $P_1, P_2, \ldots, P_n$ all lie in a plane containing $g$ and on one side of $g$. Prove that if $n$ is odd, $$\left|\overrightarrow{OP_1} + \overrightarrow{OP_2} + \cdots + \overrightarrow{OP_n}\right| \geq 1$$

Here $\left|\overrightarrow{OM}\right|$ denotes the length of vector $\overrightarrow{OM}$.

Let $a_1, a_2, \ldots, a_n$ be $n$ positive numbers, and let $q$ be a given real number such that $0 < q < 1$. Find $n$ numbers $b_1, b_2, \ldots, b_n$ for which

(a) $a_k < b_k$ for $k = 1, 2, \ldots, n$,

(b) $q < \frac{b_{k+1}}{b_k} < \frac{1}{q}$ for $k = 1, 2, \ldots, n - 1$,

(c) $b_1 + b_2 + \cdots + b_n < \frac{1+q}{1-q}(a_1 + a_2 + \cdots + a_n)$.

In the triangle $ABC$, prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $$\sin A \sin B \leq \sin^2 \frac{C}{2}.$$

Determine all possible values of $$S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}$$ where $a$, $b$, $c$, $d$ are arbitrary positive numbers.

Let $x_i, y_i$ $(i = 1, 2, \ldots, n)$ be real numbers such that

$$x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n.$$

Prove that, if $z_1, z_2, \cdots, z_n$ is any permutation of $y_1, y_2, \cdots, y_n$, then

$$\sum_{i=1}^{n}(x_i - y_i)^2 \leq \sum_{i=1}^{n}(x_i - z_i)^2.$$

Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.

Four real constants $a$, $b$, $A$, $B$ are given, and $$f(\theta) = 1 - a\cos\theta - b\sin\theta - A\cos 2\theta - B\sin 2\theta.$$ Prove that if $f(\theta) \geq 0$ for all real $\theta$, then $$a^2 + b^2 \leq 2 \text{ and } A^2 + B^2 \leq 1.$$