Found problems: 85335
2008 India Regional Mathematical Olympiad, 5
Three nonzero real numbers $ a,b,c$ are said to be in harmonic progression if $ \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}$. Find all three term harmonic progressions $ a,b,c$ of strictly increasing positive integers in which $ a \equal{} 20$ and $ b$ divides $ c$.
[17 points out of 100 for the 6 problems]
2008 Romania National Olympiad, 1
Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.
2020 Dürer Math Competition (First Round), P2
How many ways can you fill a table of size $n\times n$ with integers such that each cell contains the total number of even numbers in its row and column other than itself?
Two tables are different if they differ in at least one cell.
2009 Iran Team Selection Test, 4
Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition:
\[ p \mid uv - 1 \]
we always have $p \mid f(u)f(v) - 1$.
1962 AMC 12/AHSME, 16
Given rectangle $ R_1$ with one side $ 2$ inches and area $ 12$ square inches. Rectangle $ R_2$ with diagonal $ 15$ inches is similar to $ R_1.$ Expressed in square inches the area of $ R_2$ is:
$ \textbf{(A)}\ \frac92 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ \frac{135}{2} \qquad
\textbf{(D)}\ 9 \sqrt{10} \qquad
\textbf{(E)}\ \frac{27 \sqrt{10}}{4}$
2008 Cuba MO, 6
We have an isosceles triangle $ABC$ with base $BC$. Through vertex $A$ draw a line $r$ parallel to $BC$. The points $P, Q$ are located on the perpendicular bisectors of $AB$ and $AC$ respectively, such that $PQ\perp BC$. They are points $M$ and $N$ on the line $r$ such that $\angle APM = \angle AQN = 90^o$. Prove that $$\frac{1}{AM} + \frac{1}{AN}\le \frac{2}{ AB}$$
2021 Greece Junior Math Olympiad, 4
Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.
1991 AIME Problems, 5
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?
2002 Czech and Slovak Olympiad III A, 6
Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality
\[f(xf(y))=f(xy)+x\]
1994 Abels Math Contest (Norwegian MO), 4b
Finitely many cities are connected by one-way roads. For any two cities it is possible to come from one of them to the other (with possible transfers), but not necessarily both ways. Prove that there is a city which can be reached from any other city, and that there is a city from which any other city can be reached.
1990 AMC 12/AHSME, 23
If $x,y>0$, $\log_yx+\log_xy=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$
$ \textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 13\sqrt{3} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36 $
2012 USAJMO, 4
Let $\alpha$ be an irrational number with $0<\alpha < 1$, and draw a circle in the plane whose circumference has length $1$. Given any integer $n\ge 3$, define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b\le n$.
2024 IFYM, Sozopol, 1
Let \( n \geq 2 \) be a positive integer. Find all \( n \)-tuples \( (a_1, \ldots, a_n) \) of complex numbers such that the numbers \( a_1 - 2a_2 \), \( a_2 - 2a_3 \), $\ldots$ , \( a_{n-1} - 2a_n \), \( a_n - 2a_1 \) form a permutation of the numbers \( a_1, \ldots, a_n \).
2018 Abels Math Contest (Norwegian MO) Final, 4
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.
2023 Simon Marais Mathematical Competition, A3
For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$ where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.
2023 MMATHS, 11
Suppose we have sequences $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ and the function $f(x)=\tfrac{1}{x}$ such that for all $n$ we have:
[list]
[*]$a_{n+1} = f(f(a_n+b_n)-f(f(a_n)+f(b_n))$
[*]$a_{n+2} = f(1-a_n) - f(1+a_n)$
[*]$b_{n+2} = f(1-b_n) - f(1+b_n)$
[/list]
Given that $a_0=\tfrac{1}{6}$ and $b_0=\tfrac{1}{7},$ then $b_5=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the sum of the prime factors of $mn.$
Geometry Mathley 2011-12, 5.3
Let $ABC$ be an acute triangle, not being isoceles. Let $\ell_a$ be the line passing through the points of tangency of the escribed circles in the angle $A$ with the lines $AB, AC$ produced. Let $d_a$ be the line through $A$ parallel to the line that joins the incenter $I$ of the triangle $ABC$ and the midpoint of $BC$. Lines $\ell_b, d_b, \ell_c, d_c$ are defined in the same manner. Three lines $\ell_a, \ell_b, \ell_c$ intersect each other and these intersections make a triangle called $MNP$. Prove that the lines $d_a, d_b$ and $d_c$ are concurrent and their point of concurrency lies on the Euler line of the triangle $MNP$.
Lê Phúc Lữ
1980 All Soviet Union Mathematical Olympiad, 300
The $A$ set consists of integers only. Its minimal element is $1$ and its maximal element is $100$. Every element of $A$ except $1$ equals to the sum of two (may be equal) numbers being contained in $A$. What is the least possible number of elements in $A$?
LMT Speed Rounds, 2016.2
Mike rides a bike for $30$ minutes, traveling $8$ miles. He started riding at $20$ miles per hour, but by the end of his journey he was only traveling at $10$ miles per hour. What was his average speed, in miles per hour?
[i]Proposed by Nathan Ramesh
1998 National Olympiad First Round, 30
Let $ m\equal{}\left(abab\right)$ and $ n\equal{}\left(cdcd\right)$ be four-digit numbers in decimal system. If $ m\plus{}n$ is a perfect square, what is the largest value of $ a\cdot b\cdot c\cdot d$?
$\textbf{(A)}\ 392 \qquad\textbf{(B)}\ 420 \qquad\textbf{(C)}\ 588 \qquad\textbf{(D)}\ 600 \qquad\textbf{(E)}\ 750$
2018 China Team Selection Test, 3
In isosceles $\triangle ABC$, $AB=AC$, points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$, $DF\parallel AC$. The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$. Let $DE$ meet $\omega_2$ at $K\neq E$. Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$. Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.
2003 All-Russian Olympiad Regional Round, 9.8
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.
2018 CMIMC Team, 4-1/4-2
Define an integer $n \ge 0$ to be \textit{two-far} if there exist integers $a$ and $b$ such that $a$, $b$, and $n + a + b$ are all powers of two. If $N$ is the number of two-far integers less than 2048, find the remainder when $N$ is divided by 100.
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
1999 AMC 12/AHSME, 19
Consider all triangles $ ABC$ satisfying the following conditions: $ AB \equal{} AC$, $ D$ is a point on $ \overline{AC}$ for which $ \overline{BD} \perp \overline{AC}$, $ AD$ and $ CD$ are integers, and $ BD^2 \equal{} 57$. Among all such triangles, the smallest possible value of $ AC$ is
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 13$
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;
pair B = (0,0);
pair C = (5,0);
pair A = (2.5,7.5);
pair D = foot(B,A,C);
dot(A);dot(B);dot(C);dot(D);
label("$A$", A, N);label("$B$", B, SW);label("$C$", C, SE);label("$D$", D, NE);
draw(A--B--C--cycle);draw(B--D);[/asy]
2017 Estonia Team Selection Test, 4
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:
a) which vertex of $BCF$ is its apex,
b) the size of $\angle BAC$