Found problems: 119
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P2
Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that
$\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$.
[i]Proposed by Mirko Petrusevski[/i]
2004 VJIMC, Problem 2
Evaluate the sum
$$\sum_{n=0}^\infty\operatorname{arctan}\left(\frac1{1+n+n^2}\right).$$
2019 Jozsef Wildt International Math Competition, W. 53
Compute $$\lim \limits_{n \to \infty}\frac{1}{n}\sum \limits_{k=1}^n\frac{\sqrt[n+k+1]{n+1}-\sqrt[n+k]{n}}{\sqrt[n+k]{n+1}-\sqrt[n+k]{n}}$$
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$$(\alpha+3)\sum \limits_{cyc}(b_1-a_1)\left((b_2+b_3)^{\alpha+2}+(a_2+a_3)^{\alpha+2}-(a_2+b_3)^{\alpha+1}-(b_2+a_3)^{\alpha+1}\right)$$ $$\leq (\alpha+2)(\alpha+3)\sum \limits_{cyc}(b_1-a_1)(b_2-a_2)(b_3^{\alpha+1}-a_3^{\alpha+1})$$ $$+ (b_3 + b_2 + a_1)^{\alpha+3}+(b_3 + a_2 + a_1)^{\alpha+3}+(a_3 + b_2 + a_1)^{\alpha+3}+(a_3 + a_2 + b_1)^{\alpha+3}$$ $$-(b_3 + b_2 + b_1)^{\alpha+3}-(b_3 + a_2 + a_1)^{\alpha+3}-(a_3 + b_2 + b_1)^{\alpha+3}-(a_3 + a_2 + a_1)^{\alpha+3}$$
2015 BMT Spring, 7
Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.
2003 Kurschak Competition, 3
Prove that the following inequality holds with the exception of finitely many positive integers $n$:
\[\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.\]
2011 VJIMC, Problem 2
Let $k$ be a positive integer. Compute
$$\sum_{n_1=1}^\infty\sum_{n_2=1}^\infty\cdots\sum_{n_k=1}^\infty\frac1{n_1n_2\cdots n_k(n_1+n_2+\ldots+n_k+1)}.$$
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. 55
Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$
1968 IMO, 6
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
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 $S.$
1980 Austrian-Polish Competition, 4
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
1990 IMO Longlists, 2
Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$
2022 SAFEST Olympiad, 1
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2002 IMC, 9
For each $n\geq 1$ let
$$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$
Show that $a_{n}\cdot b_{n}$ is an integer.
2022 Germany Team Selection Test, 1
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2019 Jozsef Wildt International Math Competition, W. 61
If $a$, $b$, $c \in \mathbb{R}$ then$$\sum \limits_{cyc} \sqrt{(c+a)^2b^2+c^2a^2}+\sqrt{5}\left |\sum \limits_{cyc} \sqrt{ab}\right |\geq \sum \limits_{cyc}\sqrt{(ab+2bc+ca)^2+(b+c)^2a^2}$$
2016 IMC, 1
Let $(x_1,x_2,\ldots)$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}$. Prove that $$ \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2. $$
(Proposed by Gerhard J. Woeginger, The Netherlands)
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.
2017 Korea USCM, 1
$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value.
\[\sum_{A\in T} \text{det}(A)\]
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
1974 IMO Shortlist, 6
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
1978 Putnam, B6
Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ($h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$) satisfy $0 \leq c_{hk} \leq 1.$ Prove that
$$ \left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$
where each summation is over all admissible ordered pairs $(h,k).$
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.
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
[*] 73
[*] 107
[*] 119
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