Found problems: 119
2010 VTRMC, Problem 7
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
2016 Mathematical Talent Reward Programme, MCQ: P 2
Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals
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[*] 4
[*] 6
[*] 12
[*] 24
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2011 Putnam, A2
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$
2019 Jozsef Wildt International Math Competition, W. 26
Let $n \in \mathbb{N}$, $n \geq 2$, $a_1, a_2, \cdots , a_n \in \mathbb{R}$ and $a_n = max \{a_1, a_2,\cdots , a_n\}$
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[*]If $t_k$, $t'_k \in \mathbb{R}$, $k \in \{1, 2,\cdots , n\}$ , $t_k \leq t'_k$, for any $k \in \{1, 2, \cdots, n - 1\}$ and $$\sum \limits_{k=1}^nt_k=\sum \limits_{k=1}^nt'_k$$Prove that $$\sum \limits_{k=1}^nt_ka_k\geq \sum \limits_{k=1}^nt'_ka_k$$
[*] If $b_k$, $c_k \in \mathbb{R}^*_+$, $k \in \{1, 2,\cdots , n\}$ , $b_k \leq c_k$ for any $k \in \{1, 2,\cdots, k - 1\}$ and $$b_1b_2\cdots b_n=c_1c_2\cdots c_n$$Prove that $$\prod \limits_{k=1}^n b_k^{a_k}\geq \prod \limits_{k=1}^nc_k^{a_k}$$
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2019 Jozsef Wildt International Math Competition, W. 54
Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$
2021 Harvard-MIT Mathematics Tournament., 4
Let $k$ and $n$ be positive integers and let
$$S=\{(a_1,\ldots,a_k)\in \mathbb{Z}^{k}\;|\; 0\leq a_k\leq\cdots\leq a_1 \leq n,a_1+\cdots+a_k=n\}$$.
Determine, with proof, the value of
$$\sum_{(a_1,\ldots,a_k)\in S}\binom{n}{a_1}\binom{a_1}{a_2}\cdots\binom{a_{k-1}}{a_k}$$
in terms of $k$ and $n$, where the sum is over all $k$-tuples in $S$.
2021 Harvard-MIT Mathematics Tournament., 5
Let $n$ be the product of the first $10$ primes, and let
$$S=\sum_{xy\mid n} \varphi(x) \cdot y,$$
where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $xy$ divides $n$. Compute $\tfrac{S}{n}.$
2017 Mathematical Talent Reward Programme, MCQ: P 8
How many finite sequances $x_1,x_2,\cdots,x_m$ are there such that $x_i=1$ or 2 and $\sum \limits_{i=1}^mx_i=10$ ?
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[*] 89
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[*] 119
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1989 IMO Shortlist, 11
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$
1996 VJIMC, Problem 2
Let $\{x_n\}^\infty_{n=0}$ be the sequence such that $x_0=2$, $x_1=1$ and $x_{n+2}$ is the remainder of the number $x_{n+1}+x_n$ divided by $7$. Prove that $x_n$ is the remainder of the number
$$4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k$$
1967 IMO Longlists, 44
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
2019 Jozsef Wildt International Math Competition, W. 33
Let $0 < \frac{1}{q} \leq \frac{1}{p} < 1$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u^2_k > a^p_k$ and $v_k > b^q_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1\leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$ , then $$\left(\sum \limits_{k=1}^n\left(l^p\left(u_k+v_k\right)^2-\left(a_k+b_k\right)^p\right)\right)^{\frac{1}{p}}\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^p\right)\right)^{\frac{1}{p}}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^p\right)\right)^{\frac{1}{p}}$$where $$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
2021 Indonesia TST, C
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
2014 VJIMC, Problem 3
Let $k$ be a positive even integer. Show that
$$\sum_{n=0}^{k/2}(-1)^n\binom{k+2}n\binom{2(k-n)+1}{k+1}=\frac{(k+1)(k+2)}2.$$
II Soros Olympiad 1995 - 96 (Russia), 11.7
Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.
2010 ELMO Shortlist, 1
For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$.
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[*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?
[*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list]
[i]Brian Hamrick.[/i]
1986 Miklós Schweitzer, 7
Prove that the series $\sum_p c_p f(px)$, where the summation is over all primes, unconditionally converges in $L^2[0,1]$ for every $1$-periodic function $f$ whose restriction to $[0,1]$ is in $L^2[0,1]$ if and only if $\sum_p |c_p|<\infty$. ([i]Unconditional convergence[/i] means convergence for all rearrangements.) [G. Halasz]
2020 Jozsef Wildt International Math Competition, W51
Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that
$$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$
Show that:
$$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$
[i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]
1966 IMO Shortlist, 29
A given natural number $N$ is being decomposed in a sum of some consecutive integers.
[b]a.)[/b] Find all such decompositions for $N=500.$
[b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only?
[b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$
[b]Note by Darij:[/b] The $0$ is not considered as a natural number.
2019 Jozsef Wildt International Math Competition, W. 63
If $b_k \geq a_k \geq 0$ $(k = 1, 2, 3)$ and $\alpha \geq 1$ then$$(\alpha+3)\sum \limits_{cyc}(b_1-a_1)\left((b_2+b_3)^{\alpha+2}+(a_2+a_3)^{\alpha+2}-(a_2+b_3)^{\alpha+1}-(b_2+a_3)^{\alpha+1}\right)$$ $$\leq (\alpha+2)(\alpha+3)\sum \limits_{cyc}(b_1-a_1)(b_2-a_2)(b_3^{\alpha+1}-a_3^{\alpha+1})$$ $$+ (b_3 + b_2 + a_1)^{\alpha+3}+(b_3 + a_2 + a_1)^{\alpha+3}+(a_3 + b_2 + a_1)^{\alpha+3}+(a_3 + a_2 + b_1)^{\alpha+3}$$ $$-(b_3 + b_2 + b_1)^{\alpha+3}-(b_3 + a_2 + a_1)^{\alpha+3}-(a_3 + b_2 + b_1)^{\alpha+3}-(a_3 + a_2 + a_1)^{\alpha+3}$$
2006 VTRMC, Problem 5
Let $\{a_n\}$ be a monotonically decreasing sequence of positive real numbers with limit $0$. Let $\{b_n\}$ be a rearrangement of the sequence such that for every non-negative integer $m$, the terms $b_{3m+1}$, $b_{3m+2}$, $b_{3m+3}$ are a rearrangement of the terms $a_{3m+1}$, $a_{3m+2}$, $a_{3m+3}$. Prove or give a counterexample to the following statement: the series $\sum_{n=1}^\infty(-1)^nb_n$ is convergent.
2002 USA Team Selection Test, 2
Let $p>5$ be a prime number. For any integer $x$, define
\[{f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}\]
Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is divisible by $p^3$.
1957 Putnam, B4
Let $a(n)$ be the number of representations of the positive integer $n$ as an ordered sum of $1$'s and $2$'s. Let $b(n)$ be the number of representations of the positive integer $n$ as an ordered sum of integers greater than $1.$ Show that $a(n)=b(n+2)$ for each $n$.
2020 Jozsef Wildt International Math Competition, W28
For positive integers $j\le n$, prove that
$$\sum_{k=j}^n\binom{2n}{2k}\binom kj=\frac{n\cdot4^{n-j}}j\binom{2n-j-1}{j-1}.$$
[i]Proposed by Ángel Plaza[/i]
2019 Harvard-MIT Mathematics Tournament, 7
Find the value of
\[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]