Found problems: 85335
2009 Iran MO (3rd Round), 7
A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge.
Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces.
Time allowed for this problem was 1 hour.
2008 ISI B.Stat Entrance Exam, 4
Suppose $P$ and $Q$ are the centres of two disjoint circles $C_1$ and $C_2$ respectively, such that $P$ lies outside $C_2$ and $Q$ lies outside $C_1$. Two tangents are drawn from the point $P$ to the circle $C_2$, which intersect the circle $C_1$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $C_1$, which intersect the circle $C_2$ at points $M$ and $N$. Show that $AB=MN$
2020 Malaysia IMONST 1, 18
In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest
side has length $12$. What is the length of the altitude (height) of the triangle that
is perpendicular to the longest side?
2015 Saint Petersburg Mathematical Olympiad, 4
$ABCD$ is convex quadrilateral. Circumcircle of $ABC$ intersect $AD$ and $DC$ at points $P$ and $Q$. Circumcircle of $ADC$ intersect $AB$ and $BC$ at points $S$ and $R$. Prove that if $PQRS$ is parallelogram then $ABCD$ is parallelogram
1980 AMC 12/AHSME, 24
For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$?
$\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$
2012 China Team Selection Test, 1
Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$.
$S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.
2008 Ukraine Team Selection Test, 8
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
2016 Latvia Baltic Way TST, 11
Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)
1994 China Team Selection Test, 3
For any 2 convex polygons $S$ and $T$, if all the vertices of $S$ are vertices of $T$, call $S$ a sub-polygon of $T$.
[b]I. [/b]Prove that for an odd number $n \geq 5$, there exists $m$ sub-polygons of a convex $n$-gon such that they do not share any edges, and every edge and diagonal of the $n$-gon are edges of the $m$ sub-polygons.
[b]II.[/b] Find the smallest possible value of $m$.
2014 AMC 10, 14
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$?
$ \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 $
2017 China Western Mathematical Olympiad, 1
Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$. Show that $p<2n$.
2024 Thailand TST, 3
Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\] for every $k=1,2,\dots,N$.
2022 CMIMC Integration Bee, 5
\[\int \frac{1}{(1+x)\sqrt{x}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2021 Philippine MO, 3
Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies
• $f(p) = 1$ for all primes $p$, and
• $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $.
For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?
2012 JHMT, 9
Let $ABC$ be a triangle with incircle $O$ and side lengths $5, 8$, and $9$. Consider the other tangent line to $O$ parallel to $BC$, which intersects $AB$ at $B_a$ and $AC$ at $C_a$. Let $r_a$ be the inradius of triangle $AB_aC_a$, and define $r_b$ and $r_c$ similarly. Find $r_a + r_b + r_c$.
2008 Purple Comet Problems, 17
$24! = 620,448,401,733,239,439,360,000$ ends in four zeros, and $25!=15,511,210,043,330,985,984,000,000$ ends in six zeros. Thus, there is no integer $n$ such that $n!$ ends in exactly five zeros. Let $S$ be the set of all $k$ such that for no integer n does $n!$ end in exactly $k$ zeros. If the numbers in $S$ are listed in increasing order, 5 will be the first number. Find the 100th number in that list.
1997 South africa National Olympiad, 3
Find all solutions $x,y \in \mathbb{Z}$, $x,y \geq 0$, to the equation \[ 1 + 3^x = 2^y. \]
Geometry Mathley 2011-12, 14.2
The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$.
Đỗ Thanh Sơn
2024 Romania National Olympiad, 1
Solve over the real numbers the equation $$3^{\log_5(5x-10)}-2=5^{-1+\log_3x}.$$
1998 Baltic Way, 13
In convex pentagon $ABCDE$, the sides $AE,BC$ are parallel and $\angle ADE=\angle BDC$. The diagonals $AC$ and $BE$ intersect at $P$. Prove that $\angle EAD=\angle BDP$ and $\angle CBD=\angle ADP$.
2024 AMC 10, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2011 Sharygin Geometry Olympiad, 4
Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.
2001 AMC 10, 12
Suppose that $ n$ is the product of three consecutive integers and that $ n$ is divisible by $ 7$. Which of the following is not necessarily a divisor of $ n$?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 42$
2019 MIG, 20
Given that two real numbers $x$ and $y$ satisfy $x^2 - 6xy + 9y^2 + |x - 3| = 0$, calculate $x + y$.
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }16\qquad\textbf{(E) }\text{impossible to determine}$
VMEO III 2006, 12.1
Given a triangle $ABC$ and a point $K$ . The lines $AK$,$BK$,$CK$ hit the opposite side of the triangle at $D,E,F$ respectively. On the exterior of $ABC$, we construct three pairs of similar triangles: $BDM$,$DCN$ on $BD$,$DC$, $CEP$,$EAQ$ on $CE$,$EA$, and $AFR$,$FBS$ on $AF$, $FB$. The lines $MN$,$PQ$,$RS$ intersect each other form a triangle $XYZ$. Prove that $AX$,$BY$,$CZ$ are concurrent.