Found problems: 85335
2015 Bosnia and Herzegovina Junior BMO TST, 1
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers
2017 AIME Problems, 1
Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.
2009 Today's Calculation Of Integral, 481
For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$.
Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$
2020 LMT Fall, 30
$\triangle ABC$ has the property that $\angle ACB = 90^{\circ}$. Let $D$ and $E$ be points on $AB$ such that $D$ is on ray $BA$, $E$ is on segment $AB$, and $\angle DCA = \angle ACE$. Let the circumcircle of $\triangle CDE$ hit $BC$ at $F \ne C$, and $EF$ hit $AC$ and $DC$ at $P$ and $Q$, respectively. If $EP = FQ$, then the ratio $\frac{EF}{PQ}$ can be written as $a+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.
[i]Proposed by Kevin Zhao[/i]
1973 Swedish Mathematical Competition, 2
The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.
2019 IMEO, 6
Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\omega$. The internal and external bisectors of angle $\angle BAC$ intersect $BC$ at $D$ and $E$, respectively. Let $M$ be the point on segment $AC$ such that $MC = MB$. The tangent to $\omega$ at $B$ meets $MD$ at $S$. The circumcircles of triangles $ADE$ and $BIC$ intersect each other at $P$ and $Q$. If $AS$ meets $\omega$ at a point $K$ other than $A$, prove that $K$ lies on $PQ$.
[i]Proposed by Alexandru Lopotenco (Moldova)[/i]
2009 Kazakhstan National Olympiad, 6
Is there exist four points on plane, such that distance between any two of them is integer odd number?
May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:
2025 EGMO, 6
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?
[i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]
2024 China Team Selection Test, 15
$n>1$ is an integer. Let real number $x>1$ satisfy $$x^{101}-nx^{100}+nx-1=0.$$ Prove that for any real $0<a<b<1$, there exists a positive integer $m$ so that $a<\{x^m\}<b.$
[i]Proposed by Chenjie Yu[/i]
2007 Stanford Mathematics Tournament, 7
A boat is traveling upstream at 5 mph relative to the current flowing against it at 1 mph. If a tree branch 10 miles upstream from the boat falls into the current of the river, how many hours does it take to reach the boat?
2010 Junior Balkan Team Selection Tests - Romania, 2
Show that:
a) There is a sequence of non-zero natural numbers $a_1, a_2, ...$ uniquely determined, so that:
$n = \sum _ {d | n} a _ d$ for whatever $n \in N ^ {*}$ .
b) There is a sequence of non-zero natural numbers $b_1, b_2, ...$ uniquely determined, so that:
$n = \prod _ {d | n} b _ d$ for whatever $n \in N ^ {*}$ .
Note: The sum from a), respectively the product from b), are made after all the natural divisors $d$ of the number $n$ , including $1$ and $n$ .
2013 SDMO (Middle School), 3
Let $ABCD$ be a square, and let $\Gamma$ be the circle that is inscribed in square $ABCD$. Let $E$ and $F$ be points on line segments $AB$ and $AD$, respectively, so that $EF$ is tangent to $\Gamma$. Find the ratio of the area of triangle $CEF$ to the area of square $ABCD$.
2022 Grand Duchy of Lithuania, 2
During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got after the same number of points. Determine the maximum number of students could participate in the Olympics.
2005 Korea Junior Math Olympiad, 2
For triangle $ABC, P$ and $Q$ satisfy $\angle BPA + \angle AQC = 90^o$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \ne N$, however if $A$ is the only intersection $A = N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.
2020 Online Math Open Problems, 17
Let $ABC$ be a triangle with $AB=11,BC=12,$ and $CA=13$, let $M$ and $N$ be the midpoints of sides $CA$ and $AB$, respectively, and let the incircle touch sides $CA$ and $AB$ at points $X$ and $Y$, respectively. Suppose that $R,S,$ and $T$ are the midpoints of line segments $MN,BX,$ and $CY$, respectively. Then $\sin\angle{SRT}=\frac{k\sqrt{m}}{n}$, where $k,m,$ and $n$ are positive integers such that $\gcd(k,n)=1$ and $m$ is not divisible by the square of any prime. Compute $100k+10m+n$.
[i]Proposed by Tristan Shin[/i]
2009 China Team Selection Test, 3
Let $ X$ be a set containing $ 2k$ elements, $ F$ is a set of subsets of $ X$ consisting of certain $ k$ elements such that any one subset of $ X$ consisting of $ k \minus{} 1$ elements is exactly contained in an element of $ F.$ Show that $ k \plus{} 1$ is a prime number.
2022 Durer Math Competition Finals, 13
Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?
1997 Taiwan National Olympiad, 2
Given a line segment $AB$ in the plane, find all possible points $C$ such that in the triangle $ABC$, the altitude from $A$ and the median from $B$ have the same length.
1983 Tournament Of Towns, (049) 1
On sides $CB$ and $CD$ of square $ABCD$ are chosen points $M$ and $K$ so that the perimeter of triangle $CMK$ equals double the side of the square. Find angle $\angle MAK$.
2012 Putnam, 3
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that
(i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$
(ii) $ f(0)=1,$ and
(iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
Prove that $f$ is unique, and express $f(x)$ in closed form.
2019 Lusophon Mathematical Olympiad, 2
Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions:
1. $a + b + c = n$
2. $ax^2 + bx + c = 0$ has rational roots.
2016 Belarus Team Selection Test, 2
Points $B_1$ and $C_1$ are marked respectively on the sides $AB$ and $AC$ of an acute isosceles triangle $ABC$( $AB=AC$) such that $BB_1=AC_1$. The points $B,C$ and $S$ lie in the same half-plane with respect to the line $B_1C_1$ so that $\angle SB_1C_1=\angle SC_1B_1 = \angle BAC$
Prove that $B,C,S$ are colinear if and only if the triangle $ABC$ is equilateral.
2015 Middle European Mathematical Olympiad, 8
Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$.
2019 EGMO, 5
Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:
$\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$
$\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and
$\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$
(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)
2002 China Team Selection Test, 3
Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let:
\[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \}
\]
Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.