Found problems: 85335
1967 IMO Longlists, 16
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.
2024 New Zealand MO, 6
Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.
1996 Moldova Team Selection Test, 3
In triangle $ABC$ medians from $B$ and $C$ are perpendicular. Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$
2008 Mathcenter Contest, 2
Find all the functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the functional equation $$f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)$$ for every $x,y\in\mathbb{R}$ and $f(2008) =f(-2008)$
[i](nooonuii)[/i]
2007 Thailand Mathematical Olympiad, 17
Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.
2021 Baltic Way, 3
Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying
\[a_{n+1}^2=1+(n+2021)a_n\]
for all $n \ge 1$.
Kyiv City MO 1984-93 - geometry, 1991.8.5
The diagonals of the convex quadrilateral $ABCD$ are mutually perpendicular. Through the midpoint of the sides $AB$ and $AD$ draw lines, which are perpendicular to the opposite sides. Prove that they intersect on line $AC$.
2016 Junior Balkan Team Selection Tests - Romania, 1
Triangle $\triangle{ABC}$,O=circumcenter of (ABC),OA=R,the A-excircle intersect (AB),(BC),(CA) at points F,D,E.
If the A-excircle has radius R prove that $OD\perp EF$
2005 Today's Calculation Of Integral, 59
Evaluate
\[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]
2010 AMC 12/AHSME, 5
Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$?
$ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$
2012 Tournament of Towns, 4
A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.
PEN O Problems, 43
Is it possible to find a set $A$ of eleven positive integers such that no six elements of $A$ have a sum which is divisible by $6$?
MathLinks Contest 1st, 3
Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$. Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$.
LMT Guts Rounds, 2020 F13
Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S.$
[i]Proposed by Alex Li[/i]
2016 Hong Kong TST, 3
Let $ABC$ be a triangle such that $AB \neq AC$. The incircle with centre $I$ touches $BC$ at $D$. Line $AI$ intersects the circumcircle $\Gamma$ of $ABC$ at $M$, and $DM$ again meets $\Gamma$ at $P$. Find $\angle API$
2005 Harvard-MIT Mathematics Tournament, 7
Let $x$ be a positive real number. Find the maximum possible value of \[\frac{x^2+2-\sqrt{x^4+4}}{x}.\]
2024 IFYM, Sozopol, 3
The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \).
a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \).
b) Does there exist a non-constant bounded sequence with the above properties?
(A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)
2001 India Regional Mathematical Olympiad, 4
Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.
2025 Austrian MO National Competition, 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.
[i](Karl Czakler)[/i]
1972 Poland - Second Round, 6
Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.
2012 Today's Calculation Of Integral, 773
For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.
PEN A Problems, 23
(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.
2017 Serbia Team Selection Test, 5
Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which:
(i) $x_1+ \dots +x_n=0$, and
(ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$).
Prove that
a) $C_n\geq 2$, and
b) $C_n=2$ if and only $2 \mid n$.
1985 Putnam, A6
If $p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}$ is a polynomial with real coefficients $a_{i},$ then set
$$
\Gamma(p(x))=a_{0}^{2}+a_{1}^{2}+\cdots+a_{m}^{2}.
$$
Let $F(x)=3 x^{2}+7 x+2 .$ Find, with proof, a polynomial $g(x)$ with real coefficients such that
(i) $g(0)=1,$ and
(ii) $\Gamma\left(f(x)^{n}\right)=\Gamma\left(g(x)^{n}\right)$
for every integer $n \geq 1.$
2021 Czech and Slovak Olympiad III A, 5
We call a string of characters [i]neat [/i] when it has an even length and its first half is identical to the other half (eg. [i]abab[/i]). We call a string [i]nice [/i] if it can be split on several neat strings (e.g. [i]abcabcdedef [/i]to [i]abcabc[/i], [i]dede[/i], and [i]ff[/i]). By string [i]reduction[/i] we call an operation in which we wipe two identical adjacent characters from the string (e.g. the string [i]abbac[/i] can be reduced to [i]aac[/i] and further to [i]c[/i]). Prove any string containing each of its characters in even numbers can be obtained by a series of reductions from a suitable nice string.
(Martin Melicher)