Found problems: 85335
1984 IMO Longlists, 25
Prove that the product of five consecutive positive integers cannot be the square of an integer.
2012 Korea Junior Math Olympiad, 8
Let there be $n$ students, numbered $1$ through $n$. Let there be $n$ cards with numbers $1$ through $n$ written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be $p_n$.
Prove that $p_n - p_{n-1}=0$ if $n$ is odd, and
prove that $p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}$ if $n = 2k$.
2024 Auckland Mathematical Olympiad, 4
The altitude $AH$ and the bisector $CL$ of triangle $ABC$ intersect at point $O$. Find the angle $BAC$, if it is known that the difference between angle $COH$ and half of angle $ABC$ is $46$.
2012 NIMO Problems, 3
Let
\[
S = \sum_{i = 1}^{2012} i!.
\]
The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$.
[i]Proposed by Lewis Chen[/i]
2016 ASDAN Math Tournament, 5
Let $f(x)$ be a real valued function. Recall that if the inverse function $f^{-1}(x)$ exists, then $f^{-1}(x)$ satisfies $f(f^{-1}(x))=f^{-1}(f(x))=x$. Given that the inverse of the function $f(x)=x^3-12x^2+48x-60$ exists, find all real $a$ that satisfy $f(a)=f^{-1}(a)$.
1988 Greece Junior Math Olympiad, 1
i) Simplify $\left(a-\frac{4ab}{a+b}+b\right): \left(\frac{a}{a+b}-\frac{b}{b-a}-\frac{2ab}{a^2-b^2}\right)$
ii) Simplify $\frac{2x^2-(3a+b)x+a^2+ab}{2x^2-(a+3b)x+ab+b^2}$
2010 Albania National Olympiad, 5
All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission.
[b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions.
[b](b)[/b]Prove that this number is achievable.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.
2017 Purple Comet Problems, 20
A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a sphere as shown. The volume of the sphere is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[center][img]https://snag.gy/92ikv3.jpg[/img][/center]
2011 Laurențiu Duican, 3
Prove that for a triangle $ ABC $ with $ \angle BAC \ge 90^{\circ } , $ having circumradius $ R $ and inradius $ r, $ the following inequality holds:
$$ R\sin A>2r $$
[i]Romeo Ilie[/i]
2014 IMO, 3
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
1971 Canada National Olympiad, 7
Let $n$ be a five digit number (whose first digit is non-zero) and let $m$ be the four digit number formed from $n$ by removing its middle digit. Determine all $n$ such that $n/m$ is an integer.
1985 Iran MO (2nd round), 4
Let $x$ and $y$ be two real numbers. Prove that the equations
\[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\]
Holds if and only if at least one of $x$ or $y$ be integer.
1995 Tuymaada Olympiad, 1
Give a geometric proof of the statement that the fold line on a sheet of paper is straight.
2021 Dutch IMO TST, 1
Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.
2012 Thailand Mathematical Olympiad, 11
Let $\vartriangle ABC$ be an acute triangle, and let $P$ be the foot of altitude from $C$ to $AB$. Let $\omega$ be the circle with diameter $BC$. The tangents from $A$ to $\omega$ are drawn touching $\omega$ at $D$ and $E$. Lines $AD$ and $AE$ intersect line $BC$ at $M$ and $N$ respectively, so that $B$ lies between $M$ and $C$. Let $CP$ intersect $DE$ at $Q, ME$ intersect $ND$ at $R$, and let $QR$ intersect $BC$ at $S$. Show that $QS$ bisects $\angle DSE$
2000 Bulgaria National Olympiad, 1
In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called [i]good[/i] if $0\leq x_i \leq 83$, $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$. Find the largest positive integer $n$ such that, for any good set, the interior and boundary of some unit square contains exactly $n$ of the points in the set on its interior or its boundary.
2009 CIIM, Problem 5
Let $f:\mathbb{R} \to \mathbb{R}$, such that
i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$, exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$
ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$, exists $x,y \in \mathbb{R}$ with $ b - \epsilon < x < b < y < b + \epsilon$, such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$
Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$.
2019 Saudi Arabia JBMO TST, 3
Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$.
Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.
2024 ELMO Shortlist, C5
Let $\mathcal{S}$ be a set of $10$ points in a plane that lie within a disk of radius $1$ billion. Define a $move$ as picking a point $P \in \mathcal{S}$ and reflecting it across $\mathcal{S}$'s centroid. Does there always exist a sequence of at most $1500$ moves after which all points of $\mathcal{S}$ are contained in a disk of radius $10$?
[i]Advaith Avadhanam[/i]
1918 Eotvos Mathematical Competition, 1
Let $AC$ be the longer of the two diagonals of the parallelogram $ABCD$. Drop perpendiculars from $C$ to $AB$ and $AD$ extended. If $E$ and $F$ are the feet of these perpendiculars, prove that $$AB \cdot AE + AD \cdot AF = (AC)^2.$$
2014 PUMaC Algebra B, 8
Given that $x_{n+2}=\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\sum_{n=0}^\infty\dfrac{x_{3n}}{2^n}=\dfrac pq$ for some positive integers $p$, $q$ with $GCD(p,q)=1$. Find $p+q$.
2005 IMO Shortlist, 2
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
1991 Poland - Second Round, 1
The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that
$$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$
2010 Turkey Team Selection Test, 3
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
Fractal Edition 1, P3
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following two conditions:
\[
\left\{
\begin{array}{ll}
\mbox{If } f(0) = 0, \mbox{ then } f(x) \neq 0 \mbox{ for any non-zero } x. \\
\\
f(x + y)f(y + z)f(z + x) = f(x + y + z)f(xy + yz + zx) - f(x)f(y)f(z) \quad \forall x, y, z \in \mathbb{R}.
\end{array}
\right.
\]