Found problems: 13
Russian TST 2016, P2
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1.$ Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8.$
2015 Putnam, A4
For each real number $x,$ let \[f(x)=\sum_{n\in S_x}\frac1{2^n}\] where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even.
What is the largest real number $L$ such that $f(x)\ge L$ for all $x\in [0,1)$?
(As usual, $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$
2015 Putnam, B5
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all $i,j$ in $\{1,2,\dots,n\}.$ Show that for $n\ge 2,$ the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on $n,$ and find its value.
2015 Putnam, A2
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$
2015 Putnam, A6
Let $n$ be a positive integer. Suppose that $A,B,$ and $M$ are $n\times n$ matrices with real entries such that $AM=MB,$ and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX)=\det(B-XM)$ for every $n\times n$ matrix $X$ with real entries.
2015 Putnam, B2
Given a list of the positive integers $1,2,3,4,\dots,$ take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16.$ Continue in this way, crossing off the three smallest remaining numbers and their sum and consider the sequence of sums produced: $6,16,27, 36, \dots.$ Prove or disprove that there is some number in this sequence whose base 10 representation ends with $2015.$
2015 Putnam, A1
Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1.$ Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB.$
2015 Putnam, A5
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1.$ Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8.$
2015 Putnam, B4
Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express \[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\] as a rational number in lowest terms.
2015 Putnam, B6
For each positive integer $k,$ let $A(k)$ be the number of odd divisors of $k$ in the interval $\left[1,\sqrt{2k}\right).$ Evaluate: \[\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}k.\]
2015 Putnam, B3
Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$
2015 Putnam, B1
Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeros.
2015 Putnam, A3
Compute \[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\] Here $i$ is the imaginary unit (that is, $i^2=-1$).