Found problems: 85335
1990 Polish MO Finals, 2
Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$
Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.
1994 Irish Math Olympiad, 4
Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.
2013 ELMO Shortlist, 9
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
2020 Putnam, B4
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_0=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n$. Define
\[
q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j},
\]
and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n$. Evaluate $M(2020)$.
2021 Macedonian Mathematical Olympiad, Problem 1
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$
For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
2012 Vietnam Team Selection Test, 1
Consider a circle $(O)$ and two fixed points $B,C$ on $(O)$ such that $BC$ is not the diameter of $(O)$. $A$ is an arbitrary point on $(O)$, distinct from $B,C$. Let $D,J,K$ be the midpoints of $BC,CA,AB$, respectively, $E,M,N$ be the feet of perpendiculars from $A$ to $BC$, $B$ to $DJ$, $C$ to $DK$, respectively. The two tangents at $M,N$ to the circumcircle of triangle $EMN$ meet at $T$. Prove that $T$ is a fixed point (as $A$ moves on $(O)$).
2021 MOAA, 19
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
2004 Unirea, 1
Let $a,b,c$ be real numbers. Show that $\sqrt[3]{a} + \sqrt[3]{b} +\sqrt[3]{c} = \sqrt[3]{a+b+c}$ if and only if $ a^3 + b^3 + c^3 = (a + b + c)^3 $
1988 ITAMO, 6
The edge lengths of the base of a tetrahedron are $a,b,c$, and the lateral edge lengths are $x,y,z$. If $d$ is the distance from the top vertex to the centroid of the base, prove that $x+y+z \le a+b+c+3d$.
2005 May Olympiad, 5
a) In each box of a $7\times 7$ board one of the numbers is written: $1, 2, 3, 4, 5, 6$ or $7$ of so that each number is written in seven different boxes. Is it possible that in no row and no column are consecutive numbers written?
b) In each box of a $5\times 5$ board one of the numbers is written: $1, 2, 3, 4$ or $5$ of so that each one is written in five different boxes. Is it possible that in no row and in no column are consecutive numbers written?
2011 China Team Selection Test, 2
Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$.
Show that for all positive integers $r$,
\[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]
2005 Romania Team Selection Test, 2
Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$. Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$.
2020 Online Math Open Problems, 2
Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased.
[i]Proposed by Ankan Bhattacharya[/i]
2015 Bosnia And Herzegovina - Regional Olympiad, 3
In parallelogram $ABCD$ holds $AB=BD$. Let $K$ be a point on $AB$, different from $A$, such that $KD=AD$. Let $M$ be a point symmetric to $C$ with respect to $K$, and $N$ be a point symmetric to point $B$ with respect to $A$. Prove that $DM=DN$
1998 National Olympiad First Round, 25
In triangle $ ABC$ with $ \left|BC\right|>\left|BA\right|$, $ D$ is a point inside the triangle such that $ \angle ABD\equal{}\angle DBC$, $ \angle BDC\equal{}150{}^\circ$ and $ \angle DAC\equal{}60{}^\circ$. What is the measure of $ \angle BAD$?
$\textbf{(A)}\ 45 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 80$
2007 Czech-Polish-Slovak Match, 6
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $\angle PAB+\angle PDC \leq 90^{\circ}$ and $\angle PBA+\angle PCD \leq 90^{\circ}.$ Prove that $AB+CD\geq BC+AD.$
2011 South East Mathematical Olympiad, 4
Let $O$ be the circumcenter of triangle $ABC$ , a line passes through $O$ intersects sides $AB,AC$ at points $M,N$ , $E$ is the midpoint of $MC$ , $F$ is the midpoint of $NB$ , prove that : $\angle FOE= \angle BAC$
2021 Azerbaijan EGMO TST, 1
Let $n$ be an even positive integer. There are $n$ real numbers written on the blackboard. In every step, we choose two numbers, erase them, and replace each of them with their product. Show that for any initial $n$-tuple it is possible to obtain $n$ equal numbers on the blackboard after a finite number of steps.
1996 Argentina National Olympiad, 3
The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.
1972 Bulgaria National Olympiad, Problem 3
Prove the equality:
$$\sum_{k=1}^{n-1}\frac1{\sin^2\frac{(2k+1)\pi}{2n}}=n^2$$
where $n$ is a natural number.
[i]H. Lesov[/i]
1988 Austrian-Polish Competition, 5
Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.
2010 Mediterranean Mathematics Olympiad, 1
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[
x^{2}-yz-zu-yu=a\]
\[
y^{2}-zu-ux-xz=b\]
\[
z^{2}-ux-xy-yu=c\]
\[
u^{2}-xy-yz-zx=d\]
1988 AIME Problems, 3
Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.
2006 Romania National Olympiad, 3
We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$, for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$.
(a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions.
(b) Prove that $PD=r$.
[i]Virgil Nicula[/i]
2021 AMC 10 Fall, 16
The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) }$ the $y$-axis $\qquad \textbf{(B) }$ the line $x = 1$ $\qquad \textbf{(C) }$ the origin $\qquad \textbf{(D) }$ the point $\left(\dfrac12, 0\right)$ $\qquad \textbf{(E) }$ the point $(1,0)$