Olympiad problems

2006 Oral Moscow Geometry Olympiad, 4

The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal. (A. Zaslavsky)

Brazil L2 Finals (OBM) - geometry, 2002.1

Tags: geometry , area of a triangle , area , symmetric , right triangle

Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$.

2009 Balkan MO Shortlist, G5

Tags: equal area , symmetric , convex quadrilateral , midpoint , geometry , area

Let $ABCD$ be a convex quadrilateral and $S$ an arbitrary point in its interior. Let also $E$ be the symmetric point of $S$ with respect to the midpoint $K$ of the side $AB$ and let $Z$ be the symmetric point of $S$ with respect to the midpoint $L$ of the side $CD$. Prove that $(AECZ) = (EBZD) = (ABCD)$.

2015 Chile National Olympiad, 1

Tags: geometry , construction , ruler , symmetric , parallelogram

On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.

Indonesia Regional MO OSP SMA - geometry, 2017.3

Tags: geometry , symmetric , orthocenter , circumcircle

Given triangle $ABC$, the three altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt point $X$ lies on the circumcircle of triangle $ABC$.

2017 Portugal MO, 2

Tags: geometry , symmetric , isosceles , circumcircle

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

2025 Junior Macedonian Mathematical Olympiad, 4

Tags: inequalities , algebra , symmetric

Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality \[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\] When does the equality hold?

2002 Baltic Way, 1

Tags: equation , symmetric , algebra

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

2003 District Olympiad, 4

Tags: geometry , construction , area of a triangle , symmetric

Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$. a) Prove that the area of triangle $AC'A'$ is twice the area of triangle $ABC$. b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.

2008 District Olympiad, 3

Tags: geometry , 3d geometry , collinear , perpendicular , symmetric

Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.

2018 Czech-Polish-Slovak Junior Match, 5

Tags: geometry , perpendicular , circumcircle , arc midpoint , symmetric

An acute triangle $ABC$ is given in which $AB <AC$. Point $E$ lies on the $AC$ side of the triangle, with $AB = AE$. The segment $AD$ is the diameter of the circumcircle of the triangle $ABC$, and point $S$ is the center of this arc $BC$ of this circle to which point $A$ does not belong. Point $F$ is symmetric of point $D$ wrt $S$. Prove that lines $F E$ and $AC$ are perpendicular.

Kyiv City MO 1984-93 - geometry, 1984.8.1

Tags: area , fixed , symmetric , geometry

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of this quadrilateral does not depend on the choice of the point $M$.

2009 Bundeswettbewerb Mathematik, 3

Tags: midpoint , geometry , symmetric , concurrency

Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.

Ukrainian TYM Qualifying - geometry, II.16

Tags: symmetry , symmetric , geometry

Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?

2005 Miklós Schweitzer, 7

Tags: algebra , symmetric , algebraic number

Let $t\in R$. Prove that $\exists A:R \times R \to R$ such that A is a symmetric, biadditive, nonzero function and $A(tx,x)=0 \,\forall x\in R$ iff t is transcendental or (t is algebraic and t,-t are conjugates over $\mathbb{Q}$).

2018 Portugal MO, 4

Tags: geometry , circumcircle , symmetric , congruent triangles

Let $[ABC]$ be any triangle and let $D, E$ and $F$ be the symmetrics of the circumcenter wrt the three sides. Prove that the triangles $[ABC]$ and $[DEF]$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/6/45bd929dfff87fb8deb09eddb59ef46e0dc0f4.png[/img]

2007 Oral Moscow Geometry Olympiad, 5

Tags: geometry , symmetric , parallel , midpoint

Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel. (A. Zaslavsky)

2015 Saudi Arabia GMO TST, 3

Tags: geometry , right triangle , perpendicular , symmetric

Let $ABC$ be a triangle, with $AB < AC$, $D$ the foot of the altitude from $A, M$ the midpoint of $BC$, and $B'$ the symmetric of $B$ with respect to $D$. The perpendicular line to $BC$ at $B'$ intersects $AC$ at point $P$ . Prove that if $BP$ and $AM$ are perpendicular then triangle $ABC$ is right-angled. Liana Topan

2007 Bulgarian Autumn Math Competition, Problem 9.2

Tags: symmetric , algebra

Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.

2015 Saudi Arabia Pre-TST, 1.1

Tags: geometry , symmetric , cyclic quadrilateral , concyclic

Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic. (Malik Talbi)

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: euler line , excenter , symmetric , parallel , geometry

Let $ABC$ be a triangle, $I_a$ the center of the excircle at side $BC$, and $M$ its reflection across $BC$. Prove that $AM$ is parallel to the Euler line of the triangle $BCI_a$.

V Soros Olympiad 1998 - 99 (Russia), 10.1

Tags: analytic geometry , geometry , symmetry , symmetric

It is known that the graph of the function $y =\frac{a-6x}{2+x}$ is centrally summetric to the graph of the function $y = \frac{1}{x}$ with respect to some point. Find the value of the parameter $a$ and the coordinates of the center of symmetry.

2003 Olympic Revenge, 3

Tags: symmetric , equal segments , geometry

Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$.

Estonia Open Junior - geometry, 2015.1.5

Tags: geometry , circumcircle , arc , symmetric

Let $ABC$ be an acute triangle. The arcs $AB$ and $AC$ of the circumcircle of the triangle are reflected over the lines AB and $AC$, respectively. Prove that the two arcs obtained intersect in another point besides $A$.

Ukrainian TYM Qualifying - geometry, 2014.22

Tags: orthocenter , geometry , incenter , symmetric

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .