Found problems: 85335
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.
2007 Romania National Olympiad, 2
Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if \[\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}\]
1994 National High School Mathematics League, 7
A directed line segment, starting point is $P(-1,1)$, finishing point is $Q(2,2)$. If line $l:x+my+m=0$ intersects $PQ$ at its extended line, then the range value of $m$ is________.