Navvye/Olympiad-Problems-CSV huggingface dataset:
1990 Vietnam National Olympiad
A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.

(a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron?

(b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.
1990 Vietnam National Olympiad
A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' = MB \cdot MB' = MC \cdot MC'$.
1990 Vietnam National Olympiad
Suppose $ f(x)=a_0x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) = f(2x^3 + x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.
1990 Vietnam National Olympiad
The children sitting around a circle are playing the game as follows. At first the teacher gives each child an even number of candies (bigger than $ 0$, may be equal, maybe different). A certain child gives half of his candies to his neighbor on the right. Then the child who has just received candies does the same if he has an even number of candies, otherwise he gets one candy from the teacher and then does the job; and so on. Prove that after several steps there will be a child who will be able, giving the teacher half of his candies, to make the numbers of candies of all the children equal.
1989 Vietnam National Olympiad
Let $ n$ and $ N$ be natural number. Prove that for any $ \alpha$, $ 0\le\alpha\le N$, and any real $ x$, it holds that $${ |\sum_{k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}$$
1989 Vietnam National Olympiad
The Fibonacci sequence is defined by $ F_1 = F_2 = 1$ and $ F_{n+1} = F_n +F_{n-1}$ for $ n > 1$. Let $ f(x) = 1985x^2 + 1956x + 1960$. Prove that there exist infinitely many natural numbers $ n$ for which $ f(F_n)$ is divisible by $ 1989$. Does there exist $ n$ for which $ f(F_n) + 2$ is divisible by $ 1989$?
1989 Vietnam National Olympiad
A square $ ABCD$ of side length $ 2$ is given on a plane. The segment $ AB$ is moved continuously towards $ CD$ until $ A$ and $ C$ coincide with $ C$ and $ D$, respectively. Let $ S$ be the area of the region formed by the segment $ AB$ while moving. Prove that $ AB$ can be moved in such a way that $ S <\frac{5\pi}{6}$.
1989 Vietnam National Olympiad
Are there integers $ x$, $ y$, not both divisible by $ 5$, such that $ x^2 + 19y^2 = 198\cdot 10^{1989}$?
1989 Vietnam National Olympiad
The sequence of polynomials $ \left\{P_n(x)\right\}_{n=0}^{+\infty}$ is defined inductively by $ P_0(x) = 0$ and $ P_{n+1}(x) = P_n(x)+\frac{x - P_n^2(x)}{2}$. Prove that for any $ x \in [0, 1]$ and any natural number $ n$ it holds that $ 0\le\sqrt x- P_n(x)\le\frac{2}{n + 1}$.
1989 Vietnam National Olympiad
Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.
1988 Vietnam National Olympiad
There are $ 1988$ birds in $ 994$ cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so?
1988 Vietnam National Olympiad
Suppose $ P(x) = a_nx^n+\cdots+a_1x+a_0$ be a real polynomial of degree $ n > 2$ with $ a_n = 1$, $ a_{n-1} = -n$, $ a_{n-2} =\frac{n^2 - n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.
1988 Vietnam National Olympiad
The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?
1988 Vietnam National Olympiad
A bounded sequence $ (x_n)_{n\ge 1}$ of real numbers satisfies $ x_n + x_{n + 1} \ge 2x_{n + 2}$ for all $ n \ge 1$. Prove that this sequence has a finite limit.
1988 Vietnam National Olympiad
Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 + px^2 + qx + p = 0$, where $ q\neq 1$. Show that $ p \le - 3\sqrt 3$ and $ q > 1$.
1988 Vietnam National Olympiad
Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.
1987 Vietnam National Olympiad
Let $ u_1$, $ u_2$, $ \ldots$, $ u_{1987}$ be an arithmetic progression with $ u_1 = \frac {\pi}{1987}$ and the common difference $ \frac {\pi}{3974}$. Evaluate

$$ S = \sum_{\epsilon_i\in\left\{ - 1, 1\right\}}\cos\left(\epsilon_1 u_1 + \epsilon_2 u_2 + \cdots + \epsilon_{1987} u_{1987}\right)

$$
1987 Vietnam National Olympiad
Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 = 365$, $ x_{n+1} = x_n\left(x^{1986} + 1\right) + 1622$, and $ y_0 = 16$, $ y_{n+1} = y_n\left(y^3 + 1\right) - 1952$, for all $ n \ge 0$. Prove that $ \left|x_n- y_k\right|\neq 0$ for any positive integers $ n$, $ k$.
1987 Vietnam National Olympiad
Let be given $ n \ge 2$ lines on a plane, not all concurrent and no two parallel. Prove that there is a point which belongs to exactly two of the given lines.
1987 Vietnam National Olympiad
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be positive real numbers ($ n \ge 2$) whose sum is $ S$. Show

that $$ \sum_{i=1}^n\frac{a_i^{2^{k}}}{\left(S-a_i\right)^{2^t-1}}\ge\frac{S^{1+2^k-2^t}}{\left(n-1\right)^{2^t-1}n^{2^k-2^t}}$$ for any nonnegative integers $ k$, $ t$ with $ k \ge t$. When does equality occur?
1987 Vietnam National Olympiad
Let $ f : [0, +\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to+\infty}f(x)$.
1987 Vietnam National Olympiad
Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.
1986 Vietnam National Olympiad
Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2- 4a_ix_i + \left(a_i - 1\right)^2 \le 0$. Prove that $$ \sqrt{\sum_{i=1}^n\frac{x_i^2}{n}}\le\sum_{i=1}^n\frac{x_i}{n}+1$$
1986 Vietnam National Olympiad
Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that $$ \frac{R}{r}\ge 1+\frac{1}{\cos\frac{\pi}{1986}}$$ and find the total area of the surface of the pyramid when the equality occurs.
1986 Vietnam National Olympiad
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) = 2^y$ for $ y = 1, 2, \ldots, n + 1$. Compute $ M(n + 2)$.
1986 Vietnam National Olympiad
Let $ ABCD$ be a square of side $ 2a$. An equilateral triangle $ AMB$ is constructed in the plane through $ AB$ perpendicular to the plane of the square. A point $ S$ moves on $ AB$ such that $ SB=x$. Let $ P$ be the projection of $ M$ on $ SC$ and $ E$, $ O$ be the midpoints of $ AB$ and $ CM$ respectively.

(a) Find the locus of $ P$ as $ S$ moves on $ AB$.

(b) Find the maximum and minimum lengths of $ SO$.
1986 Vietnam National Olympiad
Find all $ n > 1$ such that the inequality $$ \sum_{i=1}^nx_i^2\ge x_n\sum_{i=1}^{n-1}x_i$$ holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$.
1986 Vietnam National Olympiad
A sequence of positive integers is constructed as follows: the first term is $ 1$, the following two terms are $ 2$, $ 4$, the following three terms are $ 5$, $ 7$, $ 9$, the following four terms are $ 10$, $ 12$, $ 14$, $ 16$, etc. Find the $ n$-th term of the sequence.
1985 Vietnam National Olympiad
Find all pairs $ (x, y)$ of integers such that $ x^3 - y^3 = 2xy + 8$.
1985 Vietnam National Olympiad
Find all functions $ f \colon \mathbb{Z} \mapsto \mathbb{R}$ which satisfy:

i) $ f(x)f(y) = f(x + y) + f(x - y)$ for all integers $ x$, $ y$

ii) $ f(0) \neq 0$

iii) $ f(1) = \frac {5}{2}$
1985 Vietnam National Olympiad
A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.
1985 Vietnam National Olympiad
Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n - 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) = \gcd (b, m)$.
1985 Vietnam National Olympiad
Find all real values of parameter $ a$ for which the equation in $ x$

$$ 16x^4 - ax^3 + (2a + 17)x^2 - ax + 16 = 0

$$

has four solutions which form an arithmetic progression.
1985 Vietnam National Olympiad
A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB +OC}{2}$, $ \frac{OC + OA}{2}$ and $ \frac{OA + OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.
1984 Vietnam National Olympiad
$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root.

$(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.
1984 Vietnam National Olympiad
The sequence $(u_n)$ is defined by $u_1 = 1, u_2 = 2$ and $u_{n+1} = 3u_n - u_{n-1}$ for $n \ge 2$. Set $v_n =\sum_{k=1}^n \text{arccot }u_k$. Compute $\lim_{n\to\infty} v_n$.
1984 Vietnam National Olympiad
A square $ABCD$ of side length $2a$ is given on a plane $\Pi$. Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a.$

$(a)$ Let $M \in BC$ and $N \in CD$ be two variable points.

$i$. Find the positions of $M,N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular and $BM \cdot DN$ is minimum.

$ii$. Find $M$ and $N$ such that $\angle MAN = 45^{\circ}$ and the volume of $SAMN$ attains an extremum value. Find these values.

$(b)$ Let $Q$ be a point such that $\angle AQB = \angle AQD = 90^{\circ}$. The line $DQ$ intersects the plane $\pi$ through $AB$ perpendicular to $\Pi$ at $Q'$.

$i$. Find the locus of $Q'$.

$ii$. Let $K$ be the locus of points $Q$ and let $CQ$ meet $K$ again at $R$. Let $DR$ meets $\Pi$ at $R'$. Prove that $sin^2 \angle Q'DB + sin^2 \angle R'DB$ is independent of $Q$.
1984 Vietnam National Olympiad
$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$.

$(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.
1984 Vietnam National Olympiad
Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy

$$xP(x - a) = (x - b)P(x).$$
1984 Vietnam National Olympiad
Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha, \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively.

$(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$

$(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$

$(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.
1983 Vietnam National Olympiad
Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?
1983 Vietnam National Olympiad
$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$

$(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$.

.
1983 Vietnam National Olympiad
A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.
1983 Vietnam National Olympiad
Show that it is possible to express $1$ as a sum of $6$, and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.
1983 Vietnam National Olympiad
Decide whether $S_n$ or $T_n$ is larger, where

$$S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}$$
1983 Vietnam National Olympiad
Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.
1982 Vietnam National Olympiad
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$
1982 Vietnam National Olympiad
For a given parameter $m$, solve the equation

$$x(x + 1)(x + 2)(x + 3) + 1 - m = 0.$$
1982 Vietnam National Olympiad
Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that

$$S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.$$
1982 Vietnam National Olympiad
Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.
1982 Vietnam National Olympiad
Let $p$ be a positive integer and $q, z$ be real numbers with $0\le q\le 1$ and $q^{p+1}\le z\le 1$. Prove that

$$\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.$$
1982 Vietnam National Olympiad
Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.
1981 Vietnam National Olympiad
Prove that a triangle $ABC$ is right-angled if and only if

$$\sin A + \sin B + \sin C = \cos A + \cos B + \cos C  + 1$$
1981 Vietnam National Olympiad
Consider the polynomials

$$f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;$$

$$g(p) = p^3 + 2p + m.$$

Find all integral values of $m$ for which $f$ is divisible by $g$.
1981 Vietnam National Olympiad
A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.
1981 Vietnam National Olympiad
Solve the system of equations

$$x^2 + y^2 + z^2 + t^2 = 50;$$

$$x^2 - y^2 + z^2 - t^2 = -24;$$

$$xy = zt;$$

$$x - y + z - t = 0.$$
1981 Vietnam National Olympiad
Let $p, q$ be real numbers with $0 < p < q$ and let $t_1, t_2, \cdots, t_n$ be real numbers in the interval $[p, q]$. Denote by $A$ and $B$ the arithmetic means of $t_1, t_2, \cdots, t_n$ and of $t_1^2, t_2^2,\cdots , t_n^2$, respectively. Prove that

$$\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.$$
1981 Vietnam National Olympiad
Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\Delta ABC$ is tangent to $O_1O_2$.
1980 Vietnam National Olympiad
Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{i})$ is an odd integer. Prove that

$$\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1$$
1980 Vietnam National Olympiad
Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that

$$\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2$$
1980 Vietnam National Olympiad
Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.
1980 Vietnam National Olympiad
Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$
1980 Vietnam National Olympiad
Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?
1980 Vietnam National Olympiad
Let be given an integer $n\ge 2$ and a positive real number $p$. Find the maximum of

$$\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},$$

where $x_i$ are non-negative real numbers with sum $p$.
1979 Vietnam National Olympiad
Show that for all $x > 1$ there is a triangle with sides, $x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.$
1979 Vietnam National Olympiad
Find all real numbers $a, b, c$ such that $x^3 + ax^2 + bx + c$ has three real roots $\alpha, \beta,\gamma$ (not necessarily all distinct) and the equation $x^3 + \alpha^3 x^2 + \beta^3 x + \gamma^3$ has roots $\alpha^3, \beta^3,\gamma^3$ .
1979 Vietnam National Olympiad
$ABC$ is a triangle. Find a point $X$ on $BC$ such that :

area $ABX$ / area $ACX$ = perimeter  $ABX$ / perimeter $ACX$.
1979 Vietnam National Olympiad
For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.
1979 Vietnam National Olympiad
Find all real numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.
1979 Vietnam National Olympiad
$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.
1978 Vietnam National Olympiad
Find all three digit numbers $\overline{abc}$ such that $2 \cdot  \overline{abc} = \overline{bca} + \overline{cab}$.
1978 Vietnam National Olympiad
Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.
1978 Vietnam National Olympiad
The triangle $ABC$ has angle $A = 30^o$ and $AB = \frac{3}{4}  AC$. Find the point $P$ inside the triangle which minimizes $5 PA + 4 PB + 3 PC$.
1978 Vietnam National Olympiad
Find three rational numbers $\frac{a}{d}, \frac{b}{d}, \frac{c}{d}$ in their lowest terms such that they form an arithmetic progression and $\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}$.
1978 Vietnam National Olympiad
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?
1978 Vietnam National Olympiad
Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.
1977 Vietnam National Olympiad
Find all real $x$ such that $ \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}> \frac{x - 1}{x}$
1977 Vietnam National Olympiad
Show that there are $1977$ non-similar triangles such that the angles $A, B, C$

satisfy $\frac{sin A + sin B + sin C}{cos A + cos B + cos C} = \frac{12}{7}$  and $sin A sin B sin C = \frac{12}{25}$.
1977 Vietnam National Olympiad
Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?
1977 Vietnam National Olympiad
$p(x) $ is a real polynomial of degree $3$.

Find necessary and sufficient conditions on its coefficients in order that $p(n)$ is integral for every integer $n$.
1977 Vietnam National Olympiad
The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le  1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le  \frac{ k(n + 1 - k)}{2}$ for all $k$.
1977 Vietnam National Olympiad
The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.
1976 Vietnam National Olympiad
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.
1976 Vietnam National Olympiad
Find all triangles $ABC$ such that $\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}$, where, as usual, $a, b, c$ are the lengths of sides $BC, CA, AB$ and $R$ is the circumradius.
1976 Vietnam National Olympiad
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
1976 Vietnam National Olympiad
Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$
1976 Vietnam National Olympiad
$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$  at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$  is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .
1976 Vietnam National Olympiad
Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i $ with sum $1$.
1975 Vietnam National Olympiad
The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$. Find $a^8 + b^8 + c^8$.
1975 Vietnam National Olympiad
Solve this equation

$\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0$
1975 Vietnam National Olympiad
Let $ABCD$ be a tetrahedron with $BA \perp  AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot  BD = AB^2$.
1975 Vietnam National Olympiad
Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$  whose only digit is $5$.
1975 Vietnam National Olympiad
Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.
1974 Vietnam National Olympiad
Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.
1974 Vietnam National Olympiad
i) How many integers $n$ are there such that $n$ is divisible by $9$ and $n+1$ is divisible by $25$?

ii) How many integers $n$ are there such that $n$ is divisible by $21$ and $n+1$ is divisible by $165$?

iii) How many integers $n$ are there such that $n$ is divisible by $9, n + 1$ is divisible by $25$, and $n + 2$ is divisible by $4$?
1974 Vietnam National Olympiad
Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$. The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively.

i) Prove that  circumcircle of $ARS$ always passes the fixed point $H$.

ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$. Prove that the ratio $RR_1/SS_1$ is constant.

iii) The point $K$ is symmetric to $H$ with respect to $M$. The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$. Prove that$ \angle BHR = \angle DHR, \angle DHS = \angle CHS$.
1974 Vietnam National Olympiad
$C$ is a cube side $1$. The $12$ lines containing the sides of the cube meet at plane $p$ in $12$ points. What can you say about the $12$ points?