Olympiad problems

2007 Postal Coaching, 1

Tags: sum , constant , square , circumcircle , geometry

Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$ is constant with respect to the point $P$.

2000 Greece JBMO TST, 2

Tags: constant , convex quadrilateral , geometry

Let $ABCD$ be a convex quadrilateral with $AB=CD$. From a random point $P$ of it's diagonal $BD$, we draw a line parallel to $AB$ that intersects $AD$ at point $M$ and a line parallel to $CD$ that intersects $BC$ at point $N$. Prove that: a) The sum $PM+PN$ is constant, independent of the position of $P$ on the diagonal $BD$. b) $MN\le BD$. When the equality holds?

1974 Vietnam National Olympiad, 3

Tags: projection , constant , geometry , equal angles , ratio

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$.

2024 Pan-American Girls’ Mathematical Olympiad, 5

Tags: algebra , real , constant , functional equation

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$ for all real numbers $x, y$.

2019 Philippine TST, 3

Tags: absolute value , function , quadratic , functional inequality , constant , inequalities , algebra

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

1980 Poland - Second Round, 6

Tags: geometry , fixed , constant , incircle

Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).

2024 Middle European Mathematical Olympiad, 8

Tags: number theory , sequence , divisibility , constant

Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

2014 Sharygin Geometry Olympiad, 2

Tags: tangent , geometry , constant , segment

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$. (A. Zertsalov, D. Skrobot)

2017 Peru Iberoamerican Team Selection Test, P1

Tags: fixed , geometry , perimeter , ratio , constant , tangent circle

Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.

2009 Belarus Team Selection Test, 2

Tags: algebra , constant , exponential

Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$. D. Bazylev

1949-56 Chisinau City MO, 62

Tags: tetrahedron , 3d geometry , volume , fixed , constant , geometry

On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.

2006 Junior Tuymaada Olympiad, 5

Tags: algebra , quadratic trinomial , trinomial , quadratic , constant , sides of a triangle

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

1911 Eotvos Mathematical Competition, 2

Tags: geometry , fixed , constant , octagon

Let $Q$ be any point on a circle and let $P_1P_2P_3...P_8$ be a regular inscribed octagon. Prove that the sum of the fourth powers of the distances from $Q$ to the diameters $P_1P_5$, $P_2P_6$, $P_3P_7$, $P_4P_8$ is independent of the position of $Q$.

2021 Romania Team Selection Test, 1

Tags: constant , geometry

Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.

2021 Middle European Mathematical Olympiad, 1

Tags: algebra , sequence , constant

Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying \[ x_{n+1}=A-\frac{1}{x_n} \] for every integer $n \ge 1$, has only finitely many negative terms.