Poland Junior 2006-21 113p

geometry problems from Polish Junior Mathematical Olympiads, Round 1, 2 & finals 

with aops links in the names

collected inside aops

 rounds 1+2 , finals

2006- 21 Round 1

2006 Polish Junior round 1 p2

A convex quadrilateral with the following properties is given:

$\bullet$ a circle can be entered in a quadrilateral,

$\bullet$ the diagonals of the quadrilateral are perpendicular.

Prove that one of the diagonals of the quadrilateral divides the other into two halves.

2006 Polish Junior round 1 p3

$99$ points were selected in a circle with a radius of $10$. . Prove that inside circle there is a point distant from each of the selected points by more than $1$.

2006 Polish Junior round 1 p7

On the plane, points $A, B, C, D$ are given. Point $B$ is the midpoint of segment$AC$, and $AB = BC = BD = 17$ and $AD = 16$. Calculate the length of segment $CD$.

2007 Polish Junior round 1 p2

A triangle $ABC$ is given with $\angle ACB = 90^o$ and $AC \ne BC$. Points $P$ and $Q$ are such that the quadrilateral $APBQ$ is a square. Prove that lines $CP$ and $CQ$ are perpendicular.

2007 Polish Junior round 1 p4

In the triangle $ABC$, point $M$ is the midpoint of side AB and $\angle ACB = 120^o$. Prove that

$$CM \ge \frac{\sqrt3}{6}AB.$$

2007 Polish Junior round 1 p6

Is there a tetrahedron in which at least one face is an obtuse triangle and the center of the sphere circumscribed around this tetrahedron lies in its inside? Justify your answer.

2007 Polish Junior round 1 p7

Ten were chosen from all the vertices of the regular $17$-gon. Show that among the selected points there are four that are vertices of the trapezoid.

2008 Polish Junior round 1 p2

A convex quadrilateral $ABCD$ of area $1$ is given. Point $K$ is symmetrical to point $B$ with respect to point $A$, point $L$ is symmetrical to point $C$ with respect to point $B$, point $M$ is symmetrical to point $D$ with respect to point $C$, point $N$ is symmetrical to point $A$ about point $D$. Find the area of the quadrilateral $KLMN$.

2008 Polish Junior round 1 p5

A circle with radius $1$ is inscribed in a convex quadrilateral $ABCD$. This circle is tangent to the sides $AB, BC, CD, DA$ at points $K, L, M, N$ respectively. It is known that $\angle KLM = 4 \angle AKN$ and $\angle KNM = 4 \angle BKL$. Calculate the length of the segment $LN$.

2008 Polish Junior round 1 p7

Is there such a quadrilateral pyramid and such a plane crossing all its side edges that the area of the obtained cross-section is greater than that of the base of the pyramid? Justify your answer.

2009 Polish Junior round 1 p2

A cuboid with a square base is given. The diagonal of this cuboid has length $d$ and its surface area is $b$. Calculate the sum of the lengths of all edges of the cuboid.

2009 Polish Junior round 1 p3

Given is the square $ABCD$ of side $1$ and a straight line $\ell$passing through his center. Let $a, b, c, d$ denote the distances of points $A, B, C, D$, respectively from line $\ell$. Prove that $a^2+ b^2+ c^2+ d^2=1$.

2009 Polish Junior round 1 p5

In triangle $ABC$, the angle bisector of $ACB$ intersects side $AB$ at point $D$. The lengths of sides $BC$ and $AC$ are equal to $a$ and $b$, respectively, and the length of segment $CD$ is equal to $d$. Prove that $d <\frac{2ab}{a + b}.$

2009 Polish Junior round 1 p7

Is there a polyhedron whose orthogonal projections on some three planes are quadrilateral, hexagon and octagon respectively? Justify your answer.

2010 Polish Junior round 1 p2

A trapezoid $ABCD$ with bases $AB$ and $CD$ is given. Find all the points $P$ lying inside the trapezoid and satisfying the equality $[PAB] + [PCD] = [PBC] + [PDA]$, where $[XY Z]$ is the area of the triangle $XYZ$.

2010 Polish Junior round 1 p4

The regular $18$-gon $A_1A_2...A_{18}$ is given. Prove that the quadrilateral bounded by lines $A_2A_7$, $A_3A_{15}$, $A_6A_{12}$ and $A_{10}A_{17}$ is a rectangle. Is this rectangle a square?

2010 Polish Junior round 1 p7

A regular tetrahedron with edge $1$ was cut by a plane so that a quadrilateral was obtained in the cross-section. What is the smallest possible perimeter of this quadrilateral? Justify your answer.

2011 Polish Junior round 1 p2

In some tetrahedron, each vertex is connected by a segment with the center of the circle inscribed on the opposite face. It turned out that the segments obtained are the heights of the tetrahedron. Show that this tetrahedron is regular

2011 Polish Junior round 1 p4

A convex hexagon $ABCDEF$ is given. Point $X$ is inside this hexagon. The points $K, L, M, N, P, Q$ are the midpoints of the sides $AB, BC, CD, DE, EF, FA$ respectively, . Prove that the sum of the areas of the quadrilaterals $QAKX$, $LCMX$, $NEPX$ does not depend on the choice point $X$.

2011 Polish Junior round 1 p6

A convex quadrilateral $ABCD$ is inscribed in a circle. Its diagonals intersect at point $E$ and the angle $BEC$ is obtuse. A straight line passing through point $C$ and perpendicular to the straight line $AC$ intersects the line through point $B$ and perpendicular to line $BD$ at point $F$. Show that the lines $EF$ and $AD$ are perpendicular.

2012 Polish Junior round 1 p2

A triangle $ABC$ is given with $AC = BC$. Point $D$ lies on the side $AB$, where $BD = 2AD$ and the angle $BCD$ is right. Find the measure of the angle $BAC$.

2012 Polish Junior round 1 p3

There are two rectangles with equal areas and equal perimeters. Prove length og the diagonals of both rectangles are also equal.

2012 Polish Junior round 1 p5

In a convex pentagon $ABCDE$, the angles at the vertices $B$ and $D$ are right. Prove that the perimeter of triangle $ACE$ is not less than $2BD$.

2012 Polish Junior round 1 p7

Let $ABCDA'B'C'D'$ be a cube as shown in the picture. Points $K, L, M, N$ are the midpoints of the edges $AD$, $BC$, $A'B'$, $C'D'$ respectively. Points $P$ and $Q$ are on the segments $KM$ and $LN$, respectively . The edge of the cube is $2$. Prove that $PQ\ge \sqrt2$.

2013 Polish Junior round 1 p2

A convex quadrilateral $ABCD$ is given with $AD + BC = CD$. Bisectors of angles $BCD$ and $CDA$ meet at point $S$. Prove that $AS = BS$.

2013 Polish Junior round 1 p5

Inside the pyramid $SABCD$ , with base the convex quadrilateral $ABCD$, you can inscribe a sphere. Prove that

$\angle ASB + \angle CSD =\angle BSC + \angle DSA$.

2013 Polish Junior round 1 p6

Segments $AD$ and $BE$ are the altitudes of the acute triangle $ABC$. On the outside of the triangle $ABC$, were built the square $ABKL$ and thre rectangles $BDMN$ , $AEPQ$, with $BN = BC$ and $AQ = AC$. Prove that the sum of the areas of the rectangles $BDMN$ and $AEPQ$ is equal to the area of the square $ABKL$.

2014 Polish Junior round 1 p3

Points $E$ and $F$ lie on sides $BC$ and $CD$, respectively, of rectangle A$BCD$, where triangle $AEF$ is equilateral. Point $M$ is the midpoint of segment $AF$. Prove that the triangle $BCM$ is equilateral.

2014 Polish Junior round 1 p5

A convex quadrilateral $ABCD$ is given. The points $K$ and $L$ are by the midpoints of the sides $AB$ and $CD$, respectively. Show that if the areas of the quadrilaterals $BCLK$ and $DAKL$ are equal, then the quadrilateral $ABCD$ is a trapezoid.

2014 Polish Junior round 1 p6

Point $P$ lies on the sphere circumscribed around a cube. Prove that sum of squares of distances of point $P$ from the vertices of the cube does not depend on the choice of point $P$.

2015 Polish Junior round 1 p2

An equilateral triangle $ABC$ is inscribed in the circle $o$. Point $D$ lies on the shorter arc $BC$ of circle $o$. Point $E$ is symmetrical to point $B$ wrt line $CD$. Prove that points $A, D, E$ lie on one straight line.

2015 Polish Junior round 1 p5

On both sides of the river with parallel banks, there are two houses $A$ and $B$, with the line $AB$ not perpendicular to the banks of the river (see figure). Where should the bridge be built, perpendicular to the river banks, that the roads from both houses to the bridge, running in a straight line, were of equal length? Give the appropriate construction with a compass and a ruler and justify its correctness.

2015 Polish Junior round 1 p7

A tetrahedron $ABCD$ is given where $\angle ACB =\angle ADB = 90^o$ and $AC = CD = DB$. Prove that $AB <2 CD$.

2016 Polish Junior round 1 p2

Inside the square $ABCD$, point $P$ is selected such that $AP = AB$ and $\angle CPD = 90^o$. Prove that $DP = 2CP$.

2016 Polish Junior round 1 p4

A triangle $ABC$ is given where $\angle ACB = 60^o$ . The circle $o$ is cirxumscribed around this triangle. Point $X$ is the midpoint of this arc $BC$ of circle $o$ that does not include point $A$, and point $Y$ is the midpoint of this arc $CA$ of circle $o$, that does not contain point $B$. Prove that the line $XY$ is tangent to the inscribed circle into triangle $ABC$.

2016 Polish Junior round 1 p7

Is there such a pyramid $ABCDS$, the base of which is the rectangle $ABCD$, such that every two side edges have different lengths, and moreover, the equality $AS + CS = BS + DS$ is satisfied ? Justify your answer.

2017 Polish Junior round 1 p2

Given an acute triangle $ABC$ with $\angle ACB = 45^o$. Let $BCED$ and $ACFG$ be squares outside the triangle $ABC$. Prove that the midpoint of segment $DG$ coincides with the center of the circumscribed circle around the triangle $ABC$.

2017 Polish Junior round 1 p4

The quadrilateral $ABCD$ is inscribed in a circle where $\angle ABC = 60^o$ and $BC = CD$. Prove $AB = AD + DC$

2017 Polish Junior round 1 p6

The base of the pyramid $ABCD$ is an equilateral triangle $ABC$ with side $1$. Further $\angle ADB = \angle BDC = \angle CDA = 90^o$. Calculate the volume of the pyramid $ABCD$.

2018 Polish Junior round 1 p2

Inside parallelogram $ABCD$ is point $P$, such that $PC = BC$. Show that line $BP$ is perpendicular to line which connects middles of sides of line segments $AP$ and $CD$.

2018 Polish Junior round 1 p4

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Bisectors of $AD$ and $BC$ intersect line segments $BC$ and $AD$ respectively in points $P$ and $Q$. Show that $\angle APD = \angle BQC$.

2018 Polish Junior round 1 p7

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

2019 Polish Junior round 1 p2

A convex quadrilateral $ABCD$ is given in which $\angle DAB = \angle ABC = 45^o$ and $DA = 3$, $AB = 7\sqrt2$, $BC = 4$. Calculate the length of side $CD$.

2019 Polish Junior round 1 p5

A parallelogram $ABCD$ is given. On the diagonal BD, a point $P$ is selected such that $AP = BD$ is satisfied. Point $Q$ is the midpoint of segment $CP$. Prove that $\angle BQD = 90^o$.

2019 Polish Junior round 1 p7

A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.

2020 Polish Junior round 1 p2

Points $P$ and $Q$ lie on the sides $AB$, $BC$ of the triangle $ABC$, such that $AC=CP =PQ=QB$ and $A \neq P$ and $C \neq Q$. If $\sphericalangle ACB = 104^{\circ}$, determine the measures of all angles of the triangle $ABC$.

2020 Polish Junior round 1 p4

Let $ABCD$ be the rectangle. Points $E$, $F$ lies on the sides $BC$ and $CD$ respectively, such that $\sphericalangle EAF = 45^{\circ}$ and $BE = DF$. Prove that area of the triangle $AEF$ is equal to the sum of the areas of the triangles $ABE$ and $ADF$.

2020 Polish Junior round 1 p7

Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.

2021 Polish Junior round 1 p2

A triangle $ABC$ is given with $AC = BC = 5$. The altitude of this triangle drawn from vertex $A$ has length $4$. Calculate the length of the altitude of $ABC$ drawn from vertex $C$.

2021 Polish Junior round 1 p4

A convex quadrilateral $ABCD$ is given where $\angle DAB =\angle ABC = 120^o$ and $CD = 3$,$BC = 2$, $AB = 1$. Calculate the length of segment $AD$.

2021 Polish Junior round 1 p7

The figure below, composed of four regular pentagons with a side length of $1$, was glued in space as follows. First, it was folded along the broken sections, by combining the bold sections, and then formed in such a way that colored sections formed a square. Find the length of the segment $AB$ created in this way.

2006- 21 Round 2

2006 Polish Junior round 2 p1

A certain prism has twice as many vertices as a certain pyramid. Which of these polyhedra has more faces and how many more?

2006 Polish Junior round 2 p3

An acute triangle $ABC$ is given with $BAC = 45^o$. Altitudes of this triangle intersect at point $H$. Prove that $| AH | = | BC |$.

2006 Polish Junior round 2 p5

Given is a convex hexagon $ABCDEF$ with angles at the vertices of $A,B, C, D$ equal to $90^o, 128^o, 142^o, 90^o$ respectively. Prove that the area of this hexagon is less than $\frac12 | AD |^2$.

2007 Polish Junior round 2 p2

Each angle of the hexagon $ABCDEF$ has a measure of $120^o$. Prove that the perpendicular bisectors of $AB$, $CD$, and $EF$ intersect at one point.

2007 Polish Junior round 2 p5

The $ABC$ triangle is the base of the $ABCS$ pyramid in which $\angle ASB = \angle BSC = \angle CSA = 20^o$. Show that the perimeter of the triangle $ABC$ is not less than its length each of the edges $AS, BS$ and $CS$ .

2008 Polish Junior round 2 p2

The point $S$ lies inside the regular hexagon $ABCDEF$. Prove that the sum of the areas of the triangles $ABS, CDS, EFS$ is equal to half the the area of the hexagon $ABCDEF$.

2008 Polish Junior round 2 p5

Is it possible to cut a cube with a flat cut in two solids with equal volumes to obtain a pentagon in cross section? Justify your answer.

2009 Polish Junior round 2 p3

Given is a parallelogram $ABCD$ and a point $E$ belonging to side of $BC$. Through point $D$ we draw a line $k$ parallel to line $AE$. On the line $k$ we choose points $K, L$ such that the quadrilateral $AEKL$ is a parallelogram. Prove that the parallelograms $ABCD$ and $AEKL$ have equal areas.

2009 Polish Junior round 2 p5

A regular hexagonal pyramid was cut by a plane, which cuts through all its side edges. In section a convex hexagon $ABCDEF$ was obtained. Prove it's diagonals $AD, BE$ and $CF$ intersect at one point.

2010 Polish Junior round 2 p2

There is an trapezoid $ABCD$ with bases $AB$ and $CD$ , in which $\angle BAD =\angle ABC = 60^o$ and $CD <BC$. On the side $BC$ of this trapezoid, point $E$ is selected such that $EB = CD$. Prove that $BD = AE$.

2010 Polish Junior round 2 p5

Is there a pyramid with base quadrilateral , and with each edge of the side perpendicular to some edge of the base? Justify your answer.

Caution: Perpendicular lines in space do not have to intersect.

2011 Polish Junior round 2 p1

A convex pentagon $ABCDE$ is given, in which the areas of the triangles $ABD$, $BCE$, $CDA$, $DEB$ and $EAC$ are equal. Show that each diagonal of this pentagon is parallel to one of it's sides.

2011 Polish Junior round 2 p5

Given a regular tetrahedron with a radius of $1$. Prove that there can be $6$ spheres in this tetrahedron with a radius of $\frac12$ , in such a way that every two balls have at most one common point.

2012 Polish Junior round 2 p3

Is there such a triangle with sides of length $a, b, c$, of which the area is $\frac14 (ab + bc)$? Justify your answer.

2012 Polish Junior round 2 p5

A convex quadrilateral $ABCD$ is given in which $\angle DAB +\angle BCD = \angle ABC$. The point $O$ is the center of the circle around the triangle $ABC$. Show that the point $O$ is equidistant from the lines $AD$ and $CD$.

2013 Polish Junior round 2 p2

Is there an acute triangle in which the lengths of all sides and all altitudes are integers? Justify your answer.

2014 Polish Junior round 2 p2

In the trapezoid $ABCD$, the points $M$ and $N$ are the midpoints of the bases $AB$ and $CD$, respectively. Point $P$ lies on the segment $MN$. Prove that the triangles $ADP$ and $BCP$ have equal areas.

2014 Polish Junior round 2 p5

In the triangle $ABC$, point $D$ is the midpoint of side $AB$ and point $E$ is the midpoints of the segment $CD$. Prove that if $\angle CAE = \angle BCD$, is $AC = CD$.

2015 Polish Junior round 2 p2

A triangle $ABC$ is given where $AC <BC$. Points $D$ and $E$ lie on the sides $BC$ and $AC$ of this triangle, respectively, with $AE = BD$. Prove that the perpendicular bisectors of $AB$ and $DE$ intersect at a point on the circle circumscribed around the triangle $ABC$.

2015 Polish Junior round 2 p5

An equilateral triangle $ABC$ is given. Let $P$ be the point inside this triangle. The lines $AP, BP, CP$ intersect the segments $BC, CA, AB$ at points $D, E, F$, respectively. Is it possible to select the point $P$ in such a way that exactly four among the triangles $AEP, AFP, BFP, BDP, CDP, CEP$ have equal areas? Justify your answer.

2016 Polish Junior round 2 p2

A parallelogram $ABCD$ is given. On the sides $AB$ and $AD$ lie points $X$ and $Y$ different from $A$, respectively, such that $AD = DX$ and $AB = BY$. Prove $CX = CY$.

2016 Polish Junior round 2 p4

Point $M$ is the midpoint of side $AB$ of $ABC$ with $\angle BAC + \angle MCB = 90^o$ Show that the triangle $ABC$ is isosceles or right.

2017 Polish Junior round 2 p2

Show that if the diagonals of a certain trapezoid are perpendicular, then the sum of the lengths of the bases of this trapezoid is not greater than the sum the length of the legs of this trapezoid.

2017 Polish Junior round 2 p5

Is there a convex polyhedron that every internal angle of each face is right or obtuse and that has exactly $100$ edges? Justify your answer.

2018 Polish Junior round 2 p2

Let $ABC$ be an acute traingle with $AC \neq BC$. Point $K$ is a foot of altitude through vertex $C$. Point $O$ is a circumcenter of $ABC$. Prove that areas of quadrilaterals $AKOC$ and $BKOC$ are equal.

2018 Polish Junior round 2 p4

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Points $P$ and $Q$ lie on diagonals $AC$ and $BD$, respectively and $\angle APD = \angle BQC$. Prove that $\angle AQD = \angle BPC$.

2019 Polish Junior round 2 p2

Let $ABCD$ be the trapezium with bases $AB$ and $CD$, such that $\sphericalangle ABC = 90^{\circ}$. The bisector of angle $BAD$ intersects the segment $BC$ in the point $P$. Show that if $\sphericalangle APD = 45^{\circ}$, then area of quadrilateral $APCD$ is equal to the area of the triangle $ABP$.

2019 Polish Junior round 2 p4

Let $ABC$ be such a triangle, that $AB = 3\cdot BC$. Points $P$ and $Q$ lies on the side $AB$ and $AP = PQ = QB$. A point $M$ is the midpoint of the side $AC$. Prove that $\sphericalangle PMQ = 90^{\circ}$.

2020 Polish Junior round 2 p2

Let $ABCD$ be the parallelogram, such that angle at vertex $A$ is acute. Perpendicular bisector of the segment $AB$ intersects the segment $CD$ in the point $X$. Let $E$ be the intersection point of the diagonals of the parallelogram $ABCD$. Prove that $XE = \frac{1}{2}AD$.

2020 Polish Junior round 2 p4

Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $\sphericalangle ACB > 90^{\circ}.$

Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$

2021 Polish Junior round 2 p2

Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.

2021 Polish Junior round 2 p4

Points $K$ and $L$ are on the sides $BC$ and CD, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.

2006 - 2021 finals

(didn't take place in 2020) 

2006 Polish Junor finals p2

A parallelogram $ABCD$ is given. Point $E$ belongs to the side $AB$ and point $F$ to side $AD$. The straight line $EF$ crosses the line $CB$ at $P$ and the line $CD$ at $Q$. Prove that the area of a triangle $CEF$ is equal to the area of the triangle $APQ$.

2006 Polish Junor finals p4

A tetrahedron is given such that each dihedral angle defined by its adjacent faces is acute or right . The vertices of this tetrahedron lie on the sphere with the center $S$. Can point $S$ can it lie outside this polyhedron? Justify your answer.

2007 Polish Junor finals p3

In an acute triangle $ABC$, points $M$ and $N$ are respectively the midpoints of the sides $AC$ and $BC$. The altitude of triangle $ABC$ from vertex $C$ intersects the segment $MN$ at point $D$. The perpendicular bisector of $AB$ intersects the segment $MN$ at point $E$. Show, that $MD = NE$.

2007 Polish Junor finals p5

Is there a pyramid with base square where each side is a right triangle? Justify your answer.

2008 Polish Junor finals p3

A triangle $ABC$ is given with $AC> BC$. Point $P$ is the orthogonal projection of point $B$ on the bisector of angle $ACB$. Point $M$ is the midpoint of $AB$. Knowing that $BC = a$, $CA = b$, $AB = c$, calculate the length of the segment $PM$.

2008 Polish Junor finals p5

A square pyramid is given, each of which the edge has a length of $1$. This pyramid is cut by a plane crossing all its side edges , such that a convex quadrilateral $ABCD$ not being a trapezoid in the cross-section was obtained. Lines $AB$ and $CD$ intersect at point $P$. Find all the values that the distance of point $P$ from the plane can take the base of the pyramid.

2009 Polish Junor finals p3

Given is a circle with center $S$ and a point $D$ lying on the circle. The chord $AB$ intersects the segment $SD$ at point $C$, different from point $S$. Prove that $AB> 2CD$.

2009 Polish Junor finals p5

Is there a convex polyhedron with an odd number of edges and each face with an even number of sides? Justify your answer.

2010 Polish Junor finals p3

Point $P$ is inside the triangle $ABC$. Points $D, E, F$ are points symmetric to the point $P$ with respect to the lines $BC, CA, AB$, respectively. Show that if the triangle $DEF$ is equilateral then the lines $AD, BE$ and $CF$ intersect at one point.

2010 Polish Junor finals p5

Is there a convex polyhedron with exactly $100$ faces, at least one of which is $99$-gon and from every vertex there are exactly 3 egdes? Justify your answer.

2011 Polish Junor finals p3

Point $I$ is the center of the circle inside the triangle $ABC$. The circle tangent to the line $AI$ at point $I$ and passing through point $B$ intersects the side of $BC$ at a point $P$ (not $B$). Lines $IP$ and $AC$ intersect at point $Q$. Prove that point $I$ is the midpoint of segment $PQ$.

2011 Polish Junor finals p5

Inside the circle of radius $1$ there are points $A_1, A_2, A_3, ..., A_{100}$. Prove that there is one point $P$ at the edge of the circle for which $P A_1 + P A_2 + ... + P A_{100} \ge 100$.

2012 Polish Junor finals p2

A convex quadrilateral $ABCD$ is given. Points $K$ and $L$ are the midpoints of the sides $BC$ and $AD$, respectively. Perpendicular bisectors of segments $AB$ and $CD$ intersect the segment $KL$ at points $P$ and $Q$, respectively. Show that if $KP = LQ$, then the lines $AB$ and $CD$ are parallel.

2012 Polish Junor finals p5

Is it possible to identify four points on the surface of each tetrahedron, which are the vertices of a square and no two of which lie on one face of the tetrahedron? Justify your answer.

2013 Polish Junor finals p3

A triangle $ABC$ is given with $\angle ACB = 120^o$ . Point $M$ is the midpoint of the side $AB$. On the segments $AC$ and $BC$, points $P$ and $Q$, respectively, were selected such that $AP=PQ=QB$. Prove that $\angle PMQ = 90^o$.

2013 Polish Junor finals p5

Is there a convex polyhedron which has an odd number of faces and each vertex has an even number of edges? Justify your answer.

2014 Polish Junor finals p2

A triangle $ABC$ is given with $AC = 8$ and $BC = 10$. Point $M$ is the midpoint of side $AB$. Circle with center at point $M$ has a radius of length $1$. Show that exists on this circle exactly one such point $P$ for which $\angle APC = 90^o$.

2014 Polish Junor finals p5

Is there a convex polyhedron that at each vertex at least four edges meet and which we can cut with a plane, obtaining a triangle in cross-section? Justify your answer.

2015 Polish Junor finals p3

A convex quadrilateral $ABCD$ is given in which $\angle DAB + \angle ABC = 90^o$. Point $M$ is the midpoint of the side $CD$. Knowing the lengths of the sections $AD$ and $BC$, which are $a$ and $b$, respectively, calculate the value of $[ABM] - [DAM] - [BCM]$.

Caution: The symbol $[F]$ marks the area of figure $F$.

2015 Polish Junor finals p5

Is there a convex polyhedron whose exactly one face is not a regular polygon? Justify your answer.

2016 Polish Junor finals p3

An equilateral triangle $ABC$ is given. Point $P$ lies on the shorter arc $AB$ of the circle circumscribed around the triangle. Point $M$ is midpoints of segment $AC$. Point $Q$ is symmetrical to point $P$ with respect to the point $M$. Prove that $BQ = PQ$.

2016 Polish Junor finals p5

Is there a convex polyhedron in which each edge is a side of a heptagonal face of this polyhedron? Justify your answer.

2017 Polish Junor finals p2

Point $D$ is on the side $AB$ of $ABC$. Point $E$ lies on the line segment $CD$. Show that if the sum of the areas of the triangles $ACE$ and $BDE$ is equal to half the area of triangle $ABC$, then point $D$ is the midpoint of side $AB$ or point $E$ is midpoint of segment $CD$.

2017 Polish Junor finals p4

In a convex hexagon $ABCDEF$, the internal angles at the vertices $B, C, E, F$ are equal. Moreover, equality is satisfied $AB + DE = AF + CD$. Show that the line $AD$ and the perpendicular bisectors of segments $BC$ and $EF$ have a point common.

2018 Polish Junior finals p2

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ in which $AB + CD = AD$. Diagonals $AC$ and $BD$ intersect in point $E$. Line passing through point $E$ and parallel to bases of trapezium cuts $AD$ in point $F$. Prove that $\sphericalangle BFC = 90 ^{\circ}$.

2018 Polish Junior finals p5

Point $M$ is middle of side $AB$ of equilateral triangle $ABC$. Points $D$ and $E$ lie on segments $AC$ and $BC$, respectively and $\angle DME = 60 ^{\circ}$. Prove that, $AD + BE = DE + \frac{1}{2}AB$.

2019 Polish Junior finals p2

Let $ABCD$ be the isosceles trapezium with bases $AB$ and $CD$, such that $AC = BC$. The point $M$ is the midpoint of side $AD$. Prove that $\sphericalangle ACM = \sphericalangle CBD$.

2019 Polish Junior finals p4

The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that $\sphericalangle EAD = \sphericalangle AED \quad \text{and} \quad \sphericalangle ECB = \sphericalangle CEB.$ Show that $AC + BC > AB + CE$.

final round didn't take place in 2020

2021 Polish Junior finals p2

Point $M$ is the midpoint of the hypotenuse $AB$ of a right angled triangle $ABC$. Points $P$ and

$Q$ lie on segments $AM$ and $MB$ respectively and $PQ=CQ$. Prove that $AP\leq 2\cdot MQ$. 2021 Polish Junior finals p4

On side $AB$ of a scalene triangle $ABC$ there are points $M$, $N$ such that $AN=AC$ and

$BM=BC$. The line parallel to $BC$ through $M$ and the line parallel to $AC$ through $N$

intersect at $S$. Prove that $\measuredangle{CSM} = \measuredangle{CSN}$.

source: https://omj.edu.pl/zadania