Among all victims of zombie bites, $10\%$ are prescribed the experimental drug Undetenin to treat them. Overall, $4\%$ of the human population suffer an adverse reaction to Undetenin. Out of all the patients being treated with Undetenin, $2\%$ suffer an adverse reaction to the drug. What is the probability that a patient allergic to Undetenin is prescribed the drug?
Answer: $5\%$
For the upcoming semester, 100 math majors can take up to two out of five math electives. Suppose 22 will not take any math elective in the coming semester. Also,
- 7 will take Algebraic Number Theory and Galois Theory
- 12 will take Galois Theory and Hyperbolic Geometry
- 3 will take Hyperbolic Geometry and Cryptography
- 15 will take Cryptography and Topology
- 8 will take Topology and Algebraic Number Theory.
Everyone else will take only one math elective. Furthermore, 16 will take either Algebraic Number Theory or Cryptography, but not both. How many math majors will take exactly one of Galois Theory, Hyperbolic Geometry, or Topology?
A geometric sequence with at least three terms and a rational common ratio has first term $32^{16}$ and last term $625^{30}$. If the product of all possible values of the second term of this sequence is $16^{x} \cdot 625^{y}$, where $x$ and $y$ are integers, what are the values of $x$ and $y$ ?
Prove that for any real numbers $a_1, a_2, \dots, a_n$, $n \in \mathbb{N}$, there exists a real number $x$ such that the numbers $x + a_1, x + a_2, \dots, x + a_n$ are all irrational.
On a line, there are 51 positive integers whose sum is 100. Prove that, for all positive integers $k$, $1 \le k \le 99$, one can find either a succession of numbers on the line whose sum is $k$, or a succession of numbers whose sum is $100-k$.
Consider a cyclic quadrilateral $ABCD$ and let $M$ and $N$ be the midpoints of the diagonals $AC$ and $BD$, respectively. If $\angle AMB = \angle AMD$, prove that $\angle ANB = \angle BNC$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying the conditions:
(i) $f(f(x^2) + y + f(y)) = x^2 + 2f(y)$;
(ii) $x \le y$ implies $f(x) \le f(y)$;
for all real numbers $x$ and $y$.
Let $APD$ be an acute-angled triangle and let $B \in (AP)$, $C \in (PD)$ be two points. The diagonals of the quadrilateral $ABCD$ meet at the point $Q$. Denote by $H_1$ and $H_2$ the orthocenters of the triangles $APD$ and $BPC$ respectively. The circumcircles of the triangles $ABQ$ and $CDQ$ meet again at the point $X$ ($X \neq Q$) and the circumcircles of the triangles $ADQ$ and $BCQ$ meet again at the point $Y$ ($Y \neq Q$).
Show that if the line $H_1H_2$ passes through the point $X$, then it also passes through the point $Y$.
Let $ABC$ be an acute triangle, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively, and let $M, N, P$ be the midpoints of the sides $BC, CA, AB$, respectively. The circles $BDP$ and $CDN$ cross again at $X$, the circles $CEM$ and $AEP$ cross again at $Y$, and the circles $AFN$ and $BFM$ cross again at $Z$. Prove that the lines $AX, BY, CZ$ are concurrent.
Let $f_0$, $f_1$, $f_2$ and $f_3$ be polynomials in $\mathbb{R}[X]$ such that $f_k(1) = f_{k+1}(0)$, $k = 0, 1, 2, 3$ (indices are reduced modulo $4$). Show that there exists a polynomial $f$ in $\mathbb{R}[X, Y]$ such that $f(X, 0) = f_0(X)$, $f(1, Y) = f_1(Y)$, $f(1 - X, 1) = f_2(X)$, and $f(0, 1 - Y) = f_3(Y)$.
Is it possible to express every positive integer $n$ congruent to $9$ modulo $25$ in the form $n = \frac{a(a+1)}{2} + \frac{b(b+1)}{2} + \frac{c(c+1)}{2}$, where $a$, $b$, $c$ are non-negative integers that do not share parity?
Given a positive integer $k$, a loop of length $k$ in a graph is a list $v_1, e_1, v_2, e_2, \dots, v_k, e_k$, where the $v_i$ are (not necessarily distinct) vertices, the $e_i$ are (not necessarily distinct) edges, and each $e_i$ joins $v_i$ and $v_{i+1}$ (indices are reduced modulo $k$); the loop traces an edge $e$ if $e = e_i$ for some index $i$. Show that a connected graph with vertex set $V$ and edge set $E$ has a loop of length at most $|V| + |E| - 1$ tracing every edge of the graph.
Determine the largest value the expression $\sum_{1 \le i \lt j \le 4} (x_i + x_j) \sqrt{x_i x_j}$ may achieve, as $x_1, x_2, x_3, x_4$ run through the non-negative real numbers that add up to $1$. Determine also the $x_i$ at which the maximum is achieved.
Let $ABC$ be an acute triangle such that $AB \lt AC$. Let $I$ be the incentre of the triangle $ABC$, and let the incircle touch the side $BC$ at $D$. The line $AD$ crosses the circle $ABC$ again at $E$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the circular arc $BAC$. The line $EN$ crosses the circular arc $BIC$ at $P$. Show that the lines $AD$ and $MP$ are parallel.
Given an integer $n \ge 2$, colour red exactly $n$ cells of an infinite sheet of grid paper. A rectangular grid array is called *special* if it contains at least two red opposite corner cells; single red cells and 1-row or 1-column grid arrays whose end-cells are both red are special. Given a configuration of exactly $n$ red cells, let $N$ be the largest number of red cells a special rectangular grid array may contain. Determine the least value $N$ may take on over all possible configurations of exactly $n$ red cells.
Based on Mathematical Olympiad Rioplatense, 2010, Level 2
Let $ABC$ be an acute triangle, and let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. The lines $BC$ and $EF$ cross at $P$, and the line through $D$ and parallel to $EF$ crosses the lines $AC$ and $AB$ at $Q$ and $R$, respectively. Prove that the circle $PQR$ passes through the midpoint of the side $BC$.
A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.
Let $m$ and $n$ be positive integers, and let $A_1, \dots, A_m$ be pairwise disjoint $n$-element sets of positive integers such that no member of $A_i$ is divisible by one of $A_{i+1}$, whatever $i$ (indices are reduced modulo $m$). Determine the largest number of ordered pairs $(a, b)$, where $a$ and $b$ are members of distinct $A_i$'s, and $b$ is divisible by $a$.