- Same as 10.1.
- Is it true that for all nonzero integers $a,b$ the system
$$\begin{cases} \tan(13x)\tan(ay)=1 \\ \tan(21x)\tan(by)=1\end{cases$$
has at least one solution?
- $n$ coins are given of pairwise different masses and $n$ pan scales, $n> 2$. At each weighing, it is allowed to choose some scales, put one coin in their bowl, look at the balance and then remove the coins back. Some of the scales (it is not known which ones) are corrupted and can produce randomly both correct and direct results. For what is the least amount of weighing you can obviously find the heaviest coin?
- Given a tetrahedron $ABCD$, sphere $\omega_A$ touches face $BCD$, and also the planes of the other three faces outside the face itself (i.e. $\omega_A$ is the $A$-exsphere). Analogously define $\omega B$. Let point $K$ be the tangency point between $\omega_A$ with plane $ACD$, and $L$ be the tangency point between $\omega_B$ and $plane $BCD$. Select points $X,Y$ on the the continuations of lines $AK,BL$ (through $K,L$ respectively) such that $\angle CKD = \angle CXD+\angle CBD$ and $\angle CLD = \angle CYD + \angle CAD$. Prove that, $X,Y$ are equidistant from the line $CD$.
5. The radii of concentric circles $\omega_0,\omega_1,…,\omega_4$ form a geometric progression (in that order) with common ratio $q$. For what is the largest $q$ such that you can draw an unclosed polyline $A_0A_1…A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for $i=0,1,2,3,4$?
6. Let $D$ be a point on side $AC$ of isosceles triangle $ABC$ (with base $BC$). Let $K$ be a point on the minor arc $CD$ along the circumcircle of $BCD$. Ray $CK$ intersects line parallel to $BC$ passing through $A$ at point $T$. Let $M$ be the midpoint of $DT$. Prove that $\angle AKT = \angle CAM$.
- 10.8
- Let $n$ be some positive integer. Assemble a $3\times 3\times 3$ cube out of 26 white (unit) cubes and 1 black (unit) cube. Put $n^3$ copies of this cube together to form a big cube with edge length $3n$. What is the minimum number of white cubes that must be colored red such that each white cube shares a common vertex with some red cube?