Found problems: 85335
1996 Tournament Of Towns, (505) 2
For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$?
(NB Vassiliev)
2010 Indonesia TST, 3
Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\]
[i]Fajar Yuliawan, Bandung[/i]
2016 Bosnia And Herzegovina - Regional Olympiad, 1
If $\mid ax^2+bx+c \mid \leq 1$ for all $x \in [-1,1]$ prove that:
$a)$ $\mid c \mid \leq 1$
$b)$ $\mid a+c \mid \leq 1$
$c)$ $a^2+b^2+c^2 \leq 5$
2012 Stanford Mathematics Tournament, 2
Find all real values of $x$ such that $(\frac{1}{5}(x^2-10x+26))^{x^2-6x+5}=1$
2023 Sharygin Geometry Olympiad, 21
Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.
2006 Junior Balkan Team Selection Tests - Romania, 2
Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that \[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \] then $M$ is the centroid of the triangle.
2025 Abelkonkurransen Finale, 4b
Determine the largest real number \(C\) such that
$$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$
for all real numbers \(x,y,z\neq 0\) satisfying the equation
$$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$
2013 Bulgaria National Olympiad, 4
Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$.
Prove that:
\[\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0\]
[i]Proposed by Nikolay Nikolov[/i]
2021 All-Russian Olympiad, 7
Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.
2017 Yasinsky Geometry Olympiad, 4
Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.
1970 Canada National Olympiad, 10
Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.
2014 BMT Spring, 2
A mathematician is walking through a library with twenty-six shelves, one for each letter of the alphabet. As he walks, the mathematician will take at most one book off each shelf. He likes symmetry, so if the letter of a shelf has at least one line of symmetry (e.g., M works, L does not), he will pick a book with probability $\frac12$. Otherwise he has a $\frac14$ probability of taking a book. What is the expected number of books that the mathematician will take?
1983 AIME Problems, 10
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?
2012 VJIMC, Problem 3
Let $(A,+,\cdot)$ be a ring with unity, having the following property: for all $x\in A$ either $x^2=1$ or $x^n=0$ for some $n\in\mathbb N$. Show that $A$ is a commutative ring.
2009 Today's Calculation Of Integral, 432
Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.
1999 Czech and Slovak Match, 5
Find all functions $f: (1,\infty)\text{to R}$ satisfying
$f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$.
[hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution.
I have also solved by using this method.By finding $f(x^5)$ in two ways
I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide]
2010 HMNT, 9
Newton and Leibniz are playing a game with a coin that comes up heads with probability $p$. They take turns flipping the coin until one of them wins with Newton going first. Newton wins if he flips a heads and Leibniz wins if he flips a tails. Given that Newton and Leibniz each win the game half of the time, what is the probability $p$?
2024 AMC 12/AHSME, 24
What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
$
\textbf{(A) }2\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }5\qquad
\textbf{(E) }6\qquad
$
2018 Indonesia MO, 2
Let $\Gamma_1, \Gamma_2$ be circles that touch at a point $A$, and $\Gamma_2$ is inside $\Gamma_1$. Let $B$ be on $\Gamma_2$, and let $AB$ intersect $\Gamma_1$ on $C$. Let $D$ be on $\Gamma_1$ and $P$ be on the line $CD$ (may be outside of the segment $CD$). $BP$ intersects $\Gamma_2$ at $Q$. Prove that $A,D,P,Q$ lie on a circle.
2019 Jozsef Wildt International Math Competition, W. 21
Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.
2001 China Team Selection Test, 2.2
Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).
1987 USAMO, 4
Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Now consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$. For what ratio $XB/BY$ will the segment $XY$ have minimal length?
2020 LMT Fall, 20
Cyclic quadrilateral $ABCD$ has $AC=AD=5, CD=6,$ and $AB=BC.$ If the length of $AB$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,c$ are relatively prime positive integers and $b$ is square-fre,e evaluate $a+b+c.$
[i]Proposed by Ada Tsui[/i]
2020-IMOC, G3
Triangle $ABC$ has incenter $I$ and circumcenter $O$. $AI, BI, CI$ intersect the circumcircle of $ABC$ again at $M_A, M_B, M_C$, respectively. Show that the Euler line of $BIC$ passes through the circumcenter of $OM_BM_C$.
(houkai)
2017 Hanoi Open Mathematics Competitions, 13
Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$.
Determine the smallest value of $M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}$.