Found problems: 85335
2014 Contests, 2
An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.
2006 Sharygin Geometry Olympiad, 10
At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?
2017 AIME Problems, 5
A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.
2024 IFYM, Sozopol, 7
A set \( S \) of two or more positive integers is called [i]almost closed under addition[/i] if the sum of any two distinct elements of \( S \) also belongs to \( S \). Let \( P(x) \) be a polynomial with integer coefficients for which there exists an almost closed under addition set \( S \), such that for any two distinct \( a \) and \( b \) from \( S \), the numbers \( P(a) \) and \( P(b) \) are coprime. Prove that \( P \) is a constant.
2016 USA TSTST, 5
In the coordinate plane are finitely many [i]walls[/i]; which are disjoint line segments, none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues moving. (Thus the bulldozer always moves parallel to the axes.)
Prove that it is impossible for the bulldozer to hit both sides of every wall.
[i]Proposed by Linus Hamilton and David Stoner[/i]
Ukrainian TYM Qualifying - geometry, 2020.12
On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that:
a) the points $D, K, H, J, F$ lie on the same circle;
b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.
2013 Princeton University Math Competition, 4
Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).
Note: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.
2019 Korea Winter Program Practice Test, 1
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.
2022-IMOC, G5
$P$ is a point inside $ABC$. $BP$, $CP$ intersect $AC, AB$ at $E, F$, respectively. $AP$ intersect $\odot (ABC)$ again at X. $\odot (ABC)$ and $\odot (AEF)$ intersect again at $S$. $T$ is a point on $BC$ such that $P T \parallel EF$. Prove that $\odot (ST X)$ passes through the midpoint of $BC$.
[i]proposed by chengbilly[/i]
2019 Oral Moscow Geometry Olympiad, 2
The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?
2014 Math Prize For Girls Problems, 14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?
2015 British Mathematical Olympiad Round 1, 6
A positive integer is called [i]charming[/i] if it is equal to $2$ or is of the form $3^{i}5^{j}$ where $i$ and $j$ are non-negative integers. Prove that every positive integer can be written as a sum of different charming numbers.
2019 Korea National Olympiad, 2
Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.
2023 JBMO Shortlist, A7
Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and
$$a_{n+1}=a_n+\frac{1}{a_n^2}$$
for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than
$$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$
How many digits does $x$ have?
[i]Proposed by Miroslav Marinov, Bulgaria[/i]
2019 Brazil National Olympiad, 5
(a) Prove that given constants $a,b$ with $1<a<2<b$, there is no partition of the set of positive integers into two subsets $A_0$ and $A_1$ such that: if $j \in \{0,1\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
(b) Find all pairs of real numbers $(a,b)$ with $1<a<2<b$ for which the following property holds: there exists a partition of the set of positive integers into three subsets $A_0, A_1, A_2$ such that if $j \in \{0,1,2\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
2021 USAJMO, 5
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)
Given this information, find all possible values for the number of elements of $S$.
2013 Stanford Mathematics Tournament, 3
Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?
1995 Czech And Slovak Olympiad IIIA, 6
Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$
has three distinct real roots which are sides of a right triangle.
Ukraine Correspondence MO - geometry, 2011.11
In a quadrilateral $ABCD$, the diagonals are perpendicular and intersect at the point $S$. Let $K, L, M$, and $N$ be points symmetric to $S$ with respect to the lines $AB, BC, CD$, and $DA$, respectively, $BN$ intersects the circumcircle of the triangle $SKN$ at point $E$, and $BM$ intersects circumscribed the circle of the triangle $SLM$ at the point $F$. Prove that the quadrilateral $EFLK$ is cyclic .
2009 China Team Selection Test, 1
Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$
2013 AMC 10, 11
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$?
$ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
2009 AIME Problems, 2
There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that
\[ \frac {z}{z \plus{} n} \equal{} 4i.
\]Find $ n$.
2011 Morocco TST, 2
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$.
[i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]
2022 Purple Comet Problems, 26
Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2017 Saudi Arabia JBMO TST, 4
Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)