Found problems: 85335
2008 Junior Balkan Team Selection Tests - Romania, 2
Let $ a,b,c$ be positive reals with $ ab \plus{} bc \plus{} ca \equal{} 3$. Prove that:
\[ \frac {1}{1 \plus{} a^2(b \plus{} c)} \plus{} \frac {1}{1 \plus{} b^2(a \plus{} c)} \plus{} \frac {1}{1 \plus{} c^2(b \plus{} a)}\le \frac {1}{abc}.
\]
1999 AMC 8, 1
$ (6?3)+4-(2-1) = 5. $ To make this statement true, the question mark between the 6 and the 3 should be replaced by
$ \text{(A)}\div\qquad\text{(B)}\ \times\qquad\text{(C)}+\qquad\text{(D)}\ -\qquad\text{(E)}\ \text{None of these} $
2021 Brazil Team Selection Test, 4
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
1991 Greece National Olympiad, 4
Find all positive intger solutions of $3^x+29=2^y$.
2012 Romania National Olympiad, 1
[color=darkred]The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .[/color]
2015 Azerbaijan JBMO TST, 3
There is a triangle $ABC$ that $AB$ is not equal to $AC$.$BD$ is interior bisector of $\angle{ABC}$($D\in AC$) $M$ is midpoint of $CBA$ arc.Circumcircle of $\triangle{BDM}$ cuts $AB$ at $K$ and $J,$ is symmetry of $A$ according $K$.If $DJ\cap AM=(O)$, Prove that $J,B,M,O$ are cyclic.
2002 Bundeswettbewerb Mathematik, 4
Consider a $12$-gon with sidelengths $1$, $2$, $3$, $4$, ..., $12$.
Prove that there are three consecutive sides in this $12$-gon, whose lengths have a sum $> 20$.
2008 Korea Junior Math Olympiad, 7
Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ :
$$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$
2008 Purple Comet Problems, 2
A canister contains two and a half cups of flour. Greg and Sally have a brownie recipe which calls for one and one third cups of flour. Greg and Sally want to make one and a half recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?
2006 National Olympiad First Round, 14
How many four digit perfect square numbers are there in the form $AABB$ where $A,B \in \{1,2,\dots, 9\}$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None of above}
$
2010 All-Russian Olympiad Regional Round, 9.5
Dunno wrote down $11$ natural numbers in a circle. For every two adjacent numbers, he calculated their difference. As a result among the differences found there were four units, four twos and three threes. Prove that Dunno made a mistake somewhere an error.
2008 Junior Balkan Team Selection Tests - Moldova, 9
Find all triplets $ (x,y,z)$, that satisfy:
$ \{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\
y^2 - 2y - 2x = - 14 \
z^2 - 4y - 4z = - 18 \end{array}$
1960 IMO, 4
Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.
2022 HMNT, 25
In convex quadrilateral $ABCD$ with $AB = 11$ and $CD = 13,$ there is a point $P$ for which $\triangle{ADP}$ and $\triangle{BCP}$ are congruent equilateral triangles. Compute the side length of these triangles.
2014 Harvard-MIT Mathematics Tournament, 3
$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.
2018 Hanoi Open Mathematics Competitions, 11
Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $.
2025 Harvard-MIT Mathematics Tournament, 7
The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.
2003 Romania National Olympiad, 3
The real numbers $ a,b$ fulfil the conditions
(i) $ 0<a<a\plus{}\frac12\le b$;
(ii) $ a^{40}\plus{}b^{40}\equal{}1$.
Prove that $ b$ has the first 12 digits after the decimal point equal to 9.
[i]Mircea Fianu[/i]
2016 Harvard-MIT Mathematics Tournament, 9
Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.
2022 Sharygin Geometry Olympiad, 9.2
Let circles $s_1$ and $s_2$ meet at points $A$ and $B$. Consider all lines passing through $A$ and meeting the circles for the second time at points $P_1$ and $P_2$ respectively. Construct by a compass and a ruler a line such that $AP_1.AP_2$ is maximal.
2016 Azerbaijan Team Selection Test, 2
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2007 Stanford Mathematics Tournament, 21
Convert the following decimal to a common fraction in lowest terms: $ 0.92007200720072007...$ (or $ 0.9\overline{2007}$).
2010 Balkan MO Shortlist, C4
Integers are written in the cells of a table $2010 \times 2010$. Adding $1$ to all the numbers in a row or in a column is called a [i]move[/i]. We say that a table is [i]equilibrium[/i] if one can obtain after finitely many moves a table in which all the numbers are equal.
[list=a]
[*]Find the largest positive integer $n$, for which there exists an [i]equilibrium[/i] table containing the numbers $2^0, 2^1, \ldots , 2^n$.
[*] For this $n$, find the maximal number that may be contained in such a table.
[/list]
2004 Purple Comet Problems, 3
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]
1988 AMC 12/AHSME, 11
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated. Which city had the greatest percentage increase in population from 1970 to 1980?
[asy]
size(300);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(5,0), B=(7,0), C=(10,0), D=(13,0), E=(16,0);
pair F=(4,3), G=(5,3), H=(7,3), I=(10,3), J=(12,3);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(I);
dot(J);
draw((0,0)--(18,0)^^(0,3)--(18,3));
draw((0,0)--(0,.5)^^(5,0)--(5,.5)^^(10,0)--(10,.5)^^(15,0)--(15,.5));
draw((0,3)--(0,2.5)^^(5,3)--(5,2.5)^^(10,3)--(10,2.5)^^(15,3)--(15,2.5));
draw((1,0)--(1,.2)^^(2,0)--(2,.2)^^(3,0)--(3,.2)^^(4,0)--(4,.2)^^(6,0)--(6,.2)^^(7,0)--(7,.2)^^(8,0)--(8,.2)^^(9,0)--(9,.2)^^(10,0)--(10,.2)^^(11,0)--(11,.2)^^(12,0)--(12,.2)^^(13,0)--(13,.2)^^(14,0)--(14,.2)^^(16,0)--(16,.2)^^(17,0)--(17,.2)^^(18,0)--(18,.2));
draw((1,3)--(1,2.8)^^(2,3)--(2,2.8)^^(3,3)--(3,2.8)^^(4,3)--(4,2.8)^^(6,3)--(6,2.8)^^(7,3)--(7,2.8)^^(8,3)--(8,2.8)^^(9,3)--(9,2.8)^^(10,3)--(10,2.8)^^(11,3)--(11,2.8)^^(12,3)--(12,2.8)^^(13,3)--(13,2.8)^^(14,3)--(14,2.8)^^(16,3)--(16,2.8)^^(17,3)--(17,2.8)^^(18,3)--(18,2.8));
label("A", A, S);
label("B", B, S);
label("C", C, S);
label("D", D, S);
label("E", E, S);
label("A", F, N);
label("B", G, N);
label("C", H, N);
label("D", I, N);
label("E", J, N);
label("1970", (0,3), W);
label("1980", (0,0), W);
label("0", (0,1.5));
label("50", (5,1.5));
label("100", (10,1.5));
label("150", (15,1.5));
label("Population", (21,2));
label("in thousands", (21.4,1));[/asy]
$ \textbf{(A)}\ A\qquad\textbf{(B)}\ B\qquad\textbf{(C)}\ C\qquad\textbf{(D)}\ D\qquad\textbf{(E)}\ E $