Found problems: 85335
1983 USAMO, 5
Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.
2017 Hanoi Open Mathematics Competitions, 15
Show that an arbitrary quadrilateral can be divided into nine isosceles triangles.
2021 Thailand TST, 2
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.
1996 Bosnia and Herzegovina Team Selection Test, 4
Solve the functional equation $$f(x+y)+f(x-y)=2f(x)\cos{y}$$ where $x,y \in \mathbb{R}$ and $f : \mathbb{R} \rightarrow \mathbb{R}$
1953 AMC 12/AHSME, 36
Determine $ m$ so that $ 4x^2\minus{}6x\plus{}m$ is divisible by $ x\minus{}3$. The obtained value, $ m$, is an exact divisor of:
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 64$
2017 NIMO Problems, 2
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$.
Suppose that $ABC$ is an equilateral triangle of side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. Then the area of $AMNPQ$ is $n-p\sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n+10p+q$.
[i]Proposed by Michael Ren[/i]
2017 IMO Shortlist, N5
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
2018 Ecuador Juniors, 6
What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 2
Let $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$
(1) Find the maximum value of $f(x,\ y)$ for $0\leq x\leq 1,\ 0\leq y\leq 1.$
(2) Find the maximum value of $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$
2023 Greece National Olympiad, 1
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system:
$\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\
x+y+z+w= 48
\end{cases}$
1990 Irish Math Olympiad, 1
Given a natural number $n$, calculate the number of rectangles in the plane, the coordinates of whose vertices are integers in the range $0$ to $n$, and whose sides are parallel to the axes.
2022 Malaysia IMONST 2, 1
Given a polygon $ABCDEFGHIJ$.
How many diagonals does the polygon have?
Russian TST 2014, P3
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $f(0) = 0$ and for any real numbers $x, y$ the following equality holds \[f(x^2+yf(x))+f(y^2+xf(y))=f(x+y)^2.\]
1998 Polish MO Finals, 2
The points $D, E$ on the side $AB$ of the triangle $ABC$ are such that $\frac{AD}{DB}\frac{AE}{EB} = \left(\frac{AC}{CB}\right)^2$. Show that $\angle ACD = \angle BCE$.
2009 ITAMO, 2
$ABCD$ is a square with centre $O$. Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$. Show that $OM$ and $BK$ are perpendicular to each other.
2023 AMC 12/AHSME, 4
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?
$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$
1988 IMO Longlists, 57
$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as
\[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|.
\]
$ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.
1978 IMO, 2
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
2016 Harvard-MIT Mathematics Tournament, 13
A right triangle has side lengths $a$, $b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle.
2017 Turkey Team Selection Test, 4
Each two of $n$ students, who attended an activity, have different ages. It is given that each student shook hands with at least one student, who did not shake hands with anyone younger than the other. Find all possible values of $n$.
2020 MIG, 20
John can purchase pieces of gum in packs of $4$, $14$, and $20$ pieces. Given that he purchases at least one of each kind of pack, what is the positive difference between the greatest and least number of packs he can purchase to end up with exactly $86$ pieces of gum?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2008 Moldova National Olympiad, 12.1
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
2016 Finnish National High School Mathematics Comp, 3
From the foot of one altitude of the acute triangle, perpendiculars are drawn on the other two sides, that meet the other sides at $P$ and $Q$. Show that the length of $PQ$ does not depend on which of the three altitudes is selected.
2004 Tournament Of Towns, 3
What is the maximal number of knights that can be placed on the usual 8x8 chessboard so that each of then threatens at most 7 others?
1982 IMO Longlists, 39
Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$.