Found problems: 85335
1984 AMC 12/AHSME, 5
The largest integer $n$ for which $n^{200} < 5^{300}$ is
$\textbf{(A) }8\qquad
\textbf{(B) }9\qquad
\textbf{(C) }10\qquad
\textbf{(D) }11\qquad
\textbf{(E) }12$
2009 Saint Petersburg Mathematical Olympiad, 6
Some cities in country are connected by road, and from every city goes $\geq 2008$ roads. Every road is colored in one of two colors. Prove, that exists cycle without self-intersections ,where $\geq 504$ roads and all roads are same color.
2020 MMATHS, I6
Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors! How many possible values between $200$ and $1000$ are there for the total number of balls in the box?
[i]Proposed by Andrew Wu[/i]
1994 IMO Shortlist, 1
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
2020 Argentina National Olympiad, 4
Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.
ICMC 5, 3
A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry.
[i]Proposed by Ethan Tan[/i]
2000 Belarus Team Selection Test, 1.1
Find the minimal number of cells on a $5\times 7$ board that must be painted so that any cell which is not painted has exactly one neighboring (having a common side) painted cell.
1989 IMO Shortlist, 29
155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.
2003 South africa National Olympiad, 2
Given a parallelogram $ABCD$, join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$. $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$. Prove that $M$ and $N$ divide $BD$ into three equal parts.
2017 CCA Math Bonanza, TB2
Let $ABC$ be a triangle. $D$ and $E$ are points on line segments $BC$ and $AC$, respectively, such that $AD=60$, $BD=189$, $CD=36$, $AE=40$, and $CE=50$. What is $AB+DE$?
[i]2017 CCA Math Bonanza Tiebreaker Round #2[/i]
2019 AIME Problems, 10
There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.
2022 Middle European Mathematical Olympiad, 3
Let $n$ be a positive integer. There are $n$ purple and $n$ white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?
2024 Korea Junior Math Olympiad (First Round), 3.
Find the number of positive integers (m,n) which follows the following:
1) m<n
2) The sum of even numbers between 2m and 2n is 100 greater than the sum of odd numbers between 2m and 2n.
MathLinks Contest 2nd, 5.2
Let S be the set of positive integers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two rational numbers of the form $\frac{1}{k}$, where $k$ is a positive integer. Prove that $S$ cannot be written as the union of finitely many arithmetic progressions.
2016 Germany National Olympiad (4th Round), 3
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.
Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
2019 Stanford Mathematics Tournament, 1
Let $ABCD$ be a quadrilateral with $\angle DAB = \angle ABC = 120^o$. If $AB = 3$, $BC = 2$, and $AD = 4$, what is the length of $CD$?
Today's calculation of integrals, 859
In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$
Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$
1986 Dutch Mathematical Olympiad, 4
The lines $a$ and $b$ are parallel and the point $A$ lies on $a$. One chooses one circle $\gamma$ through A tangent to $b$ at $B$. $a$ intersects $\gamma$ for the second time at $T$. The tangent line at $T$ of $\gamma$ is called $t$.
Prove that independently of the choice of $\gamma$, there is a fixed point $P$ such that $BT$ passes through $P$.
Prove that independently of the choice of $\gamma$, there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$.
1991 Flanders Math Olympiad, 3
Given $\Delta ABC$ equilateral, with $X\in[A,B]$. Then we define unique points Y,Z so that $Y\in[B,C]$, $Z\in[A,C]$, $\Delta XYZ$ equilateral.
If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.
PEN K Problems, 10
Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(m+f(n))=f(f(m))+f(n).\]
Russian TST 2014, P1
Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$?
2015 All-Russian Olympiad, 4
We denote by $S(k)$ the sum of digits of a positive integer number $k$. We say that the positive integer $a$ is $n$-good, if there is a sequence of positive integers $a_0$, $a_1, \dots , a_n$, so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$.
Is it true that for any positive integer $n$ there exists a positive integer $b$, which is $n$-good, but not $(n + 1)$-good?
A. Antropov
2018 Online Math Open Problems, 15
Iris does not know what to do with her 1-kilogram pie, so she decides to share it with her friend Rosabel. Starting with Iris, they take turns to give exactly half of total amount of pie (by mass) they possess to the other person. Since both of them prefer to have as few number of pieces of pie as possible, they use the following strategy: During each person's turn, she orders the pieces of pie that she has in a line from left to right in increasing order by mass, and starts giving the pieces of pie to the other person beginning from the left. If she encounters a piece that exceeds the remaining mass to give, she cuts it up into two pieces with her sword and gives the appropriately sized piece to the other person.
When the pie has been cut into a total of 2017 pieces, the largest piece that Iris has is $\frac{m}{n}$ kilograms, and the largest piece that Rosabel has is $\frac{p}{q}$ kilograms, where $m,n,p,q$ are positive integers satisfying $\gcd(m,n)=\gcd(p,q)=1$. Compute the remainder when $m+n+p+q$ is divided by 2017.
[i]Proposed by Yannick Yao[/i]
2023 APMO, 5
There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows:
First, he chooses an endpoint of each segment as a “sink”. Then he places the present at
the endpoint of the segment he is at. The present moves as follows :
$\bullet$ If it is on a line segment, it moves towards the sink.
$\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink.
If the present reaches an endpoint, the friend on that endpoint can receive their present.
Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.
2015 Mathematical Talent Reward Programme, MCQ: P 8
In $\triangle A B C$, $A B=A C$ and $D$ is foot of the perpendicular from $C$ to $A B$ and $E$ the foot of the perpendicular from $B$ to $A C,$ then
[list=1]
[*] $BC^3>BD^3+BE^3$
[*] $BC^3 <BD^3+BE^3$
[*] $BC^3=BD^3+BE^3$
[*] None of these
[/list]