Found problems: 85335
1987 Traian Lălescu, 1.2
On side $ BC $ of the triangle $ ABC, $ consider points $ M,N, $ such that $ MN=\frac{1}{n}BC. $ Let $ L $ on side $ AC $ such that $ ML $ is parallel with $ AB, Q $ on side $ AB $ such that $ NQ $ is parallel with $ AC, $ and $ O $ is the intersection of $ NQ $ with $ ML. $ The parallel of $ BC $ through $ O $ intersects $ AB,BC $ in $ P, $ respectively, $ K. $
Determine the location of $ M $ so that the sum of the areas of the triangles $ OMN, OKL $ and $ OPQ $ is minimum, and calculate this minimum in function of $ n $ and the area of $ ABC. $
2024 239 Open Mathematical Olympiad, 3
a) (version for grades 10-11)
Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle.
b) (version for grades 8-9)
Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?
1900 Eotvos Mathematical Competition, 1
Let $a, b, c, d$ be fixed integers with $d$ not divisible by $5$. Assume that $m$ is an integer for which $$am3 +bm2 +cm+d$$ is divisible by $5$. Prove that there exists an integer $n$ for which $$dn3 +cn2 +bn+a$$ is also divisible by $5$.
1980 Czech And Slovak Olympiad IIIA, 4
Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.
2022 Kazakhstan National Olympiad, 6
Numbers from $1$ to $49$ are randomly placed in a $35 \times 35$ table such that number $i$ is used exactly $i$ times. Some random cells of the table are removed so that table falls apart into several connected (by sides) polygons. Among them, the one with the largest area is chosen (if there are several of the same largest areas, a random one of them is chosen). What is the largest number of cells that can be removed that guarantees that in the chosen polygon there is a number which occurs at least $15$ times?
1998 All-Russian Olympiad, 4
A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.
2015 Math Prize for Girls Problems, 4
A [i]binary palindrome[/i] is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeroes are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?
2016 Saudi Arabia BMO TST, 3
Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying
$$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$
2006 AMC 10, 25
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2021 AMC 10 Spring, 24
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?
$\textbf{(A)}\ \frac{8a^2}{(a+1)^2}\qquad\textbf{(B)}\ \frac{4a}{a+1}\qquad\textbf{(C)}\ \frac{8a}{a+1}\qquad\textbf{(D)}\ \frac{8a^2}{a^2+1}\qquad\textbf{(E)}\ \frac{8a}{a^2+1}$
1997 Romania Team Selection Test, 1
We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.
[i]Mircea Becheanu[/i]
2018 MIG, 16
A triangle with area $60\text{ units}^2$ has vertices with coordinates of $(-15,x)$, $(0,x)$, and $(25,0)$. Find the largest possible value of $x$.
$\textbf{(A) } {-}8\qquad\textbf{(B) } {-}4\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 16$
2023 Princeton University Math Competition, 2
2. Let $\Gamma_{1}$ and $\Gamma_{2}$ be externally tangent circles with radii $\frac{1}{2}$ and $\frac{1}{8}$, respectively. The line $\ell$ is a common external tangent to $\Gamma_{1}$ and $\Gamma_{2}$. For $n \geq 3$, we define $\Gamma_{n}$ as the smallest circle tangent to $\Gamma_{n-1}, \Gamma_{n-2}$, and $\ell$. The radius of $\Gamma_{10}$ can be expressed as $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a+b$.
1999 North Macedonia National Olympiad, 5
If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$
2019 LIMIT Category A, Problem 5
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$.
$\textbf{(A)}~\frac n2$
$\textbf{(B)}~n$
$\textbf{(C)}~n\pi$
$\textbf{(D)}~\frac{n\pi}2$
1970 Miklós Schweitzer, 8
Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2}
\;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\]
[i]P. Turan[/i]
2020 Taiwan TST Round 1, 1
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
2013 Saudi Arabia GMO TST, 1
An acute triangle $ABC$ is inscribed in circle $\omega$ centered at $O$. Line $BO$ and side $AC$ meet at $B_1$. Line $CO$ and side $AB$ meet at $C_1$. Line $B_1C_1$ meets circle $\omega$ at $P$ and $Q$. If $AP = AQ$, prove that $AB = AC$.
2014 Serbia JBMO TST, 2
Let $a,b,c,d$ be the natural numbers such that
$2^a+4^b+5^c=2014^d.$
Find all $(a,b,c,d).$
2025 Harvard-MIT Mathematics Tournament, 20
Compute the $100$th smallest positive multiple of $7$ whose digits in base $10$ are all strictly less than $3.$
2016 District Olympiad, 4
[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
[b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2018 IMO Shortlist, N2
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
2023 ELMO Shortlist, A6
Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\]
[i]Proposed by Luke Robitaille[/i]
1998 IberoAmerican, 1
There are representants from $n$ different countries sit around a circular table ($n\geq2$), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every $n$, the maximal number of people that can be sit around the table.