Found problems: 85335
2017 Peru MO (ONEM), 2
Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board?
Clarification: Two squares are [i]neighbors [/i] if they have a common side.
2022 VN Math Olympiad For High School Students, Problem 3
Given a positive integer $N$.
Prove that: there are infinitely elements of the [i]Fibonacci[/i] sequence that are divisible by $N$.
2019 Putnam, A3
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
2018 PUMaC Algebra B, 1
Find the sum of the solutions to $\dfrac{1}{1+\dfrac{1}{|x-25|}}=\frac{49}{50}$.
2023 Purple Comet Problems, 10
The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]
2005 MOP Homework, 6
Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.
1993 Greece National Olympiad, 3
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n$.
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array} \]
In the newspaper story covering the event, it was reported that
(a) the winner caught 15 fish;
(b) those who caught 3 or more fish averaged 6 fish each;
(c) those who caught 12 or fewer fish averaged 5 fish each.
What was the total number of fish caught during the festival?
2025 Serbia Team Selection Test for the BMO 2025, 4
Let $a_1, a_2, \ldots, a_8$ be real numbers. Prove that
$$\sum_{i=1}^{8} \left( a_i^2 + a_i a_{i+2} \right) \geq \sum_{i=1}^{8} \left( a_i a_{i+1} + a_i a_{i+3} \right),$$
where the indices are taken modulo 8, i.e., $a_9 = a_1$, $a_{10} = a_2$, and $a_{11} = a_3$. In which cases does equality hold?
[i]Proposed by Vukašin Pantelić and Andrija Živadinović[/i]
MathLinks Contest 2nd, 5.1
For which positive integers $n \ge 4$ one can find n points in the plane, no three collinear, such that for each triangle formed with three of the $n$ points which are on the convex hull, exactly one of the $n - 3$ remaining points belongs to its interior.
1999 IMO Shortlist, 4
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
2025 239 Open Mathematical Olympiad, 1
The numbers from $1$ to $2n$ are arranged in a certain order in the cell of the strip $1 \times 2n$. Let's call a [i]flip[/i] an operation that takes one cell from the left half of the strip and one cell from the right half of the strip, after which it swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. Prove that if all $n^2$ possible flips are performed in any order, then all numbers from $1$ to $n$ will be written in the left, and all numbers from $n + 1$ up to $2n$ — in the right half of the strip.
2014 Costa Rica - Final Round, 3
Find all possible pairs of integers $ a$ and $ b$ such that $ab = 160 + 90 (a,b)$, where $(a, b)$ is the greatest common divisor of $ a$ and $ b$.
1991 Tournament Of Towns, (313) 3
Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$
Prove that angle $C$ is obtuse.
(Sergey Berlov)
2006 Princeton University Math Competition, 6
I have a set $A$ containing $n$ distinct integers. This set has the property that if $a,b \in A$, then $12 \nmid |a+b|$ and $12 \nmid |a-b|$. What is the largest possible value of $n$?
2017 Thailand TSTST, 3
Let $a, b, c \in\mathbb{R}^+$. Prove that $$\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}.$$
2008 Postal Coaching, 2
Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.
2007 Gheorghe Vranceanu, 1
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $
Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $
1987 IMO Longlists, 57
The bisectors of the angles $B,C$ of a triangle $ABC$ intersect the opposite sides in $B', C'$ respectively. Prove that the straight line $B'C'$ intersects the inscribed circle in two different points.
2010 Purple Comet Problems, 26
In the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry of the parabola is a line with slope $\tfrac{r}{s}$ where r and s are relatively prime positive integers. Find $r + s$.
2005 China Girls Math Olympiad, 4
Determine all positive real numbers $ a$ such that there exists a positive integer $ n$ and sets $ A_1, A_2, \ldots, A_n$ satisfying the following conditions:
(1) every set $ A_i$ has infinitely many elements;
(2) every pair of distinct sets $ A_i$ and $ A_j$ do not share any common element
(3) the union of sets $ A_1, A_2, \ldots, A_n$ is the set of all integers;
(4) for every set $ A_i,$ the positive difference of any pair of elements in $ A_i$ is at least $ a^i.$
2025 All-Russian Olympiad, 11.6
$100$ ones are written in a circle. Petya and Vasya take turns making \( 10^{10} \) moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by $2$. Vasya chooses $10$ consecutive numbers and increases each by $1$. They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays. \\
2005 Today's Calculation Of Integral, 86
Prove
\[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]
2021 Estonia Team Selection Test, 3
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
2020 Sharygin Geometry Olympiad, 1
Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$.
2017 USA Team Selection Test, 2
Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus.
[i]Danielle Wang and Evan Chen[/i]