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Na slici je prikazan lanac sastavljen od $54$ jedinična kvadratića. Svaki kvadratić, osim dvaju rubnih, spojen je s dva susjedna u nasuprotnim vrhovima.

Svaki kvadratić smije se postaviti u bilo koji položaj u prostoru uz uvjet da ostane spojen sa susjednim kvadratićima u odgovarajućim vrhovima. Je li moguće taj lanac postaviti tako da tvori oplošje kocke dimenzija $3 \times 3 \times 3$?

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB$ parallel to $CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in the planes $P$ and $Q$ respectively.

Consider the cube $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ (with face $ABCD$ directly above face $A^{\prime}B^{\prime}C^{\prime}D^{\prime}$).

(a) Find the locus of the midpoints of segments $XY$, where $X$ is any point of $AC$ and $Y$ is any point of $B^{\prime}D^{\prime}$.

(b) Find the locus of points $Z$ which lie on the segments $XY$ of part (a) with $ZY=2XZ$.

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let $V_{1}$ be the volume of the cone and $V_{2}$ the volume of the cylinder.

(a) Prove that $V_{1}\neq V_{2}$.

(b) Find the smallest number $k$ for which $V_{1}=kV_{2}$, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.

Consider a plane $\varepsilon$ and three non-collinear points $A, B, C$ on the same side of $\varepsilon$; suppose the plane determined by these three points is not parallel to $\varepsilon$. In plane $\varepsilon$ take three arbitrary points $A', B', C'$. Let $L, M, N$ be the midpoints of segments $AA', BB', CC'$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L, M, N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\varepsilon$?

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, respectively, and edges $AA',BB',CC',DD'$ are parallel). The point $X$ moves at constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimeter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpoints of the segments $XY$.

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA,SB,SC,BC,CA,AB$, or to their extensions.

(a) Prove that the tetrahedron $SABC$ is regular.

(b) Prove conversely that for every regular tetrahedron five such spheres exist.

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting the segment $BC$.

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$ the centroid of $\triangle ABC$. Lines parallel to $DD_0$ are drawn through $A, B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_1, B_1$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result true if point $D_0$ is selected anywhere within $\triangle ABC$?

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\varepsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Prove: The sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

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

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

For each value of $k = 1, 2, 3, 4, 5$, find necessary and sufficient conditions on the number $a > 0$ so that there exists a tetrahedron with $k$ edges of length $a$, and the remaining $6 - k$ edges of length 1.

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?

Consider a convex polyhedron $P_1$ with nine vertices $A_1A_2, \ldots, A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i$ $(i = 2, 3, \ldots, 9)$. Prove that at least two of the polyhedra $P_1, P_2, \ldots, P_9$ have an interior point in common.

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y, Z, T$ are interior points of edges $BC$, $CD$, $DA$, respectively. Prove:

(a) If $\angle DAB + \angle BCD \neq \angle CDA + \angle ABC$, then among the polygonal paths, there is none of minimal length.

(b) If $\angle DAB + \angle BCD = \angle CDA + \angle ABC$, then there are infinitely many shortest polygonal paths, their common length being $2AC\sin(\alpha/2)$, where $\alpha = \angle BAC + \angle CAD + \angle DAB$.

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Determine whether or not there exists a finite set $M$ of points in space not lying in the same plane such that, for any two points $A$ and $B$ of $M$, one can select two other points $C$ and $D$ of $M$ so that lines $AB$ and $CD$ are parallel and not coincident.

A rectangular box can be filled completely with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, so that their edges are parallel to the edges of the box, one can fill exactly 40% of the box. Determine the possible dimensions of all such boxes.

$P$ is a given point inside a given sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U, V$, and $W$; $Q$ denotes the vertex diagonally opposite to $P$ in the parallelepiped determined by $PU$, $PV$, and $PW$. Find the locus of $Q$ for all such triads of rays from $P$.

A prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as top and bottom faces is given. Each side of the two pentagons and each of the line-segments $A_iB_j$ for all $i,j=1,\ldots,5,$ is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Show that all 10 sides of the top and bottom faces are the same color.

Given a plane $\pi,$ a point $P$ in this plane and a point $Q$ not in $\pi,$ find all points $R$ in $\pi$ such that the ratio $(QP+PA)/QR$ is a maximum.

Let $S$ be a finite set of points in three-dimensional space. Let $S_x, S_y, S_z$ be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that

$$|S|^2 \leq |S_x| \cdot |S_y| \cdot |S_z|,$$

where $|A|$ denotes the number of elements in the finite set $|A|$. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.)

Let $n$ be a positive integer. Consider

$$S = \{(x, y, z) : x, y, z \in \{0, 1, \ldots, n\}, x + y + z > 0\}$$

as a set of $(n + 1)^3 - 1$ points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.

Kugla polumjera $R$ presječena je s dvije paralelne ravnine tako da je središte kugle izvan sloja određenog tim ravninama. Neka su $P_1$ i $P_2$ površine presjeka, a $d$ međusobna udaljenost danih ravnina. Nađite površinu presjeka kugle ravninom koja je paralelna danim ravninama i jednako od njih udaljena.

Na dvije nasuprotne strane kocke dimenzija $1 \times 1 \times 1$ nalazi se po jedna točka, na druge dvije nasuprotne strane po dvije točke, a na preostale dvije strane po tri točke. Od osam takvih identičnih kocki napravljena je kocka dimenzija $2 \times 2 \times 2$. Matija je izbrojio ukupan broj točaka na svakoj od strana te kocke i zaključio "dobili smo šest uzastopnih prirodnih brojeva". Je li Matija u pravu? Obrazloži odgovor.

Štapić je kvadar dimenzija $1 \times 1 \times 2$, a posuda je tijelo dobiveno uklanjanjem kockice $1 \times 1 \times 1$ iz kvadra dimenzija $3 \times 3 \times 2$ na sredini jedne od dviju polovica $3 \times 3 \times 1$. Ako je dopušteno koristiti koliko god je potrebno štapića i posuda, koliko je najmanje takvih tijela potrebno za sastavljanje kocke dimenzija $303 \times 303 \times 303$ bez rupa i preklapanja? Tijela je dopušteno rotirati.

Odredi polumjer osnovke stošca čija je izvodnica duljine 1, tako da razlika površina njegovog plašta i njegove osnovke bude maksimalna.

U stožac osnovke polumjera 1 i visine duljine $2\sqrt{2}$ upisan je kvadar takav da jedna strana kvadra pripada osnovki stošca, a vrhovi suprotne strane pripadaju plaštu stošca.

Ako je strana kvadra koja pripada osnovki stošca kvadrat, koliko je najveće oplošje koje takav kvadar može imati?

Pet strana drvene kocke obojano je plavom bojom dok je jedna strana neobojana. Kocka je potom razrezana na sukladne manje kockice od kojih $649$ ima točno jednu plavu stranu. Koliko je manjih kockica koje imaju točno dvije plave strane?

Volumen kocke $ABCDA_1B_1C_1D_1$ jednak je $V$. Nađite volumen zajedničkog dijela tetraedara $AB_1CD_1$ i $A_1BC_1D$.

Pravilna četverostrana piramida presječena je ravninom koja prolazi jednim vrhom baze i okomita je na nasuprotni pobočni brid. Površina presjeka dvaput je manja od površine baze. Odredite prikloni kut pobočnog brida i baze.

Neka su u tetraedru $ABCD$ površine strana $ABD$, $ACD$, $BCD$ i $BCA$ redom jednake $S_{1}$, $S_{2}$, $Q_{1}$, $Q_{2}$, a prostorni kut između strana $ABD$ i $ACD$ jednak $\alpha$, odnosno $\beta$ između $BCD$ i $BCA$. Dokažite da je $$S_{1}^{2} + S_{2}^{2} - 2S_{1}S_{2}\cos\alpha = Q_{1}^{2} + Q_{2}^{2} - 2Q_{1}Q_{2}\cos\beta.$$

U stožac je upisana polusfera čija kružna baza leži u bazi stošca. Omjer oplošja stošca (uključujući i bazu) i oplošja polusfere (bez kružne baze) je $k = \dfrac{18}{5}$. Odredite vršni kut stošca.

Baza piramide $ABCDV$ je pravokutnik $ABCD$ čije su duljine stranica $|AB| = a$ i $|BC| = b$, a svi bočni bridovi su duljine $c$. Odredite površinu presjeka te piramide ravninom koja prolazi dijagonalom $\overline{BD}$ baze i paralelna je bočnom bridu $\overline{VA}$.

Kvadar je presječen ravninom tako da je presjek pravilni šesterokut. Dokažite da je to moguće samo ako je kvadar kocka.

Na dijagonalama $\overline{AB_1}$ i $\overline{CA_1}$ bočnih strana $ABB_1A_1$ i $CAA_1C_1$ trostrane prizme $ABCA_1B_1C_1$ dane su točke $E$ i $F$ takve da je $EF \parallel BC_1$. Nadite omjer duljina dužina $\overline{EF}$ i $\overline{BC_1}$.

Svi bridni kutovi pri vrhu $D$ tetraedra $ABCD$ jednaki su $\alpha$, a kutovi između dviju strana tetraedra kojima je jedan vrh $D$ jednaki su $\varphi$. Dokažite da postoji točno jedan kut $\alpha$ za koji je $\varphi = 2\alpha$.

Visine trostrane piramide sijeku se u jednoj točki. Dokažite da ta točka, težište jedne strane piramide, nožište visine na tu stranu i tri točke koje dijele preostale tri visine u omjeru $2:1$, počevši od vrha piramide, leže na istoj sferi.

Odredite skup svih točaka triedra takvih da je zbroj njihovih udaljenosti od strana triedra jednak zadanom pozitivnom broju $a$.

Bočni brid pravilne trostrane piramide je $b = 1$, a njezin obujam je $V = \dfrac{1}{6}$. Koliki je kut pri vrhu bočne strane?

Posuda oblika uspravnog stošca sadrži određenu količinu vode. Kada je stožac postavljen osnovkom na ravnu površinu vrhom prema gore, razina vode je $8$ cm ispod vrha stošca. Ako stožac preokrenemo, razina vode je $2$ cm ispod osnovke stošca.

Kolika je visina posude?

Sve točke prostora čija udaljenost od dužine $\overline{AB}$ iznosi najviše $3$ čine tijelo obujma $216\pi$. Odredi duljinu dužine $\overline{AB}$.

Kocka $ABCDA'B'C'D'$ stranice duljine $1$ presječena je sferom. Središte sfere je točka $S$ na dužini $\overline{AD}$ takva da je $|AS| = \sqrt{3} - 1$. Sfera prolazi točkama $C$ i $D'$, te siječe bridove $\overline{AB}$ i $\overline{AA'}$.

Odredi površinu onog dijela oplošja kocke koji se nalazi unutar te sfere.

Dan je valjak s visinom duljine $10$ cm. Na obodima njegovih osnovki su točke $A$ i $B$ takve da je $\overline{AB}$ paralelno s osi valjka. Spojimo li točke $A$ i $B$ najkraćom linijom koja jednom obilazi oko valjka po plaštu, njezina će duljina biti $15$ cm. Kolika je duljina najkraće linije koja dva puta obilazi oko valjka i spaja točke $A$ i $B$?

Lukas je odlučio napraviti snjegovića od tri kugle čiji su polumjeri $30$ cm, $26$ cm i $18$ cm. Dvije veće kugle prerezao je tako da oba presjeka budu krugovi polumjera $24$ cm, te je odbacio manje dijelove, a veće dijelove stavio jedan na drugi, spajajući ih duž tog kruga. Na kraju je na vrh položio najmanju kuglu. Kolika je ukupna visina Lukasovog snjegovića?

Dan je tetraedar kojem je jedan brid duljine $3$, a svi ostali duljine $2$. Odredi obujam tog tetraedra.

Četiri sfere polumjera $R$ leže na bazi stošca tako da svaka dodiruje dvije od preostalih sfera te plašt stošca. Peta sfera istog polumjera dodiruje prve četiri sfere i plašt stošca. Odredi volumen tog stošca.

Baza piramide je pravilni $n$-terokut. Svaka stranica baze obojena je crnom bojom, dok su svaka dijagonala baze i svaki pobočni brid piramide obojeni ili crvenom ili plavom bojom. Odredi najmanji prirodni broj $n \geqslant 4$ za koji nužno postoji trokut čiji vrhovi su vrhovi piramide i kojemu su sve tri stranice jednake boje.