Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

Click for solution I think it is $17$, however I havent checked all the cases, what one should try is to look what pairs $(a,b)$ modulo 16, give $3^a-2^b$ a multiple of 17, and show using factorizations, modulo 5, modulo 11, modulo 16, or more, to show they have more factors or the equivalence is not truth!

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

Click for solution All the triangles $MPN$ are directly similar, since $\angle MPN$ and $\angle PMN$ are constant. $\frac{AM}{AN}$ is constant, so all the triangles $MPA$ are directly similar. This means that $P$ moves on a circle which is the image of the large circle through a spiral similarity $\mathcal S$ centered at $A$, and the actual locus is the image of the large arc $AB$ through $\mathcal S$.

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

Click for solution To finish things off, here's a construction with $T=7$ and 14 questions overall Label the contestants by points $(i,j)$, $1 \leq i \leq 7$, $1 \leq j \leq 3$ Let one question be solved by all contestants of the form $(i, ai+b)$ for each $1 \leq a \leq 3$, $1 \leq b \leq 3$. The 10th question is solved by all contestants where $i \in \{2, 5\}$ The 11th question is solved by all contestants where $i \in \{3, 6\}$ The 12th question is solved by all contestants where $i \in \{1, 4\}$ The 13th question is solved by all contestants where $i \in \{1, 7\}$ The 14th question is solved by all contestants where $i \in \{4,7\}$

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} + \_x^{2n - 1} + \_x^{2n - 2} + \ldots + \_x + 1 = 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

Click for solution Beautiful problem. I think that points $(1+cos(\frac{\pi i}{2n}))/2$ define such a partition for polynomials of degree less than $n$.