Found problems: 85335
2002 AMC 12/AHSME, 1
Which of the following numbers is a perfect square?
$\textbf{(A) }4^45^56^6\qquad\textbf{(B) }4^45^66^5\qquad\textbf{(C) }4^55^46^6\qquad\textbf{(D) }4^65^46^5\qquad\textbf{(E) }4^65^56^4$
2020 Taiwan TST Round 1, 2
Let point $H$ be the orthocenter of a scalene triangle $ABC$. Line $AH$ intersects with the circumcircle $\Omega$ of triangle $ABC$ again at point $P$. Line $BH, CH$ meets with $AC,AB$ at point $E$ and $F$, respectively. Let $PE, PF$ meet $\Omega$ again at point $Q,R$, respectively. Point $Y$ lies on $\Omega$ so that lines $AY,QR$ and $EF$ are concurrent. Prove that $PY$ bisects $EF$.
1977 Yugoslav Team Selection Test, Problem 3
Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that:
(a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$.
(b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.
2010 Dutch IMO TST, 3
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer.
Prove that $M(a,b)$ is a square.
(b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.
2002 Bulgaria National Olympiad, 4
Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$, $T_2$, and $T_3$ be the tangency points of the incircle with the sides $BC$, $CA$ and $AB$, respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$, where $O$ is the circumcenter of triangle $ABC$.
[i]Proposed by Georgi Ganchev[/i]
2008 Korea Junior Math Olympiad, 6
If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we defi ne $f_s(n) = d_1^s+d_2^s+..+d_k^s$.
For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$.
Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.
2014 Singapore Junior Math Olympiad, 1
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.
2023 Regional Olympiad of Mexico West, 6
There are $2023$ guinea pigs placed in a circle, from which everyone except one of them, call it $M$, has a mirror that points towards one of the $2022$ other guinea pigs. $M$ has a lantern that will shoot a light beam towards one of the guinea pigs with a mirror and will reflect to the guinea pig that the mirror is pointing and will keep reflecting with every mirror it reaches. Isaías will re-direct some of the mirrors to point to some other of the $2023$ guinea pigs. In the worst case scenario, what is the least number of mirrors that need to be re-directed, such that the light beam hits $M$ no matter the starting point of the light beam?
1989 Vietnam National Olympiad, 3
Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.
2017 CCA Math Bonanza, L2.1
Adam and Mada are playing a game of one-on-one basketball, in which participants may take $2$-point shots (worth $2$ points) or $3$-point shots (worth $3$ points). Adam makes $10$ shots of either value while Mada makes $11$ shots of either value. Furthermore, Adam made the same number of $2$-point shots as Mada made $3$-point shots. At the end of the game, the two basketball players realize that they have the exact same number of points! How many total points were scored in the game?
[i]2017 CCA Math Bonanza Lightning Round #2.1[/i]
2019 CCA Math Bonanza, I9
Isosceles triangle $\triangle{ABC}$ has $\angle{BAC}=\angle{ABC}=30^\circ$ and $AC=BC=2$. If the midpoints of $BC$ and $AC$ are $M$ and $N$, respectively, and the circumcircle of $\triangle{CMN}$ meets $AB$ at $D$ and $E$ with $D$ closer to $A$ than $E$ is, what is the area of $MNDE$?
[i]2019 CCA Math Bonanza Individual Round #9[/i]
Oliforum Contest I 2008, 3
Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.
2014 VJIMC, Problem 1
Find all complex numbers $z$ such that $|z^3+2-2i|+z\overline z|z|=2\sqrt2.$
2017 HMNT, 9
Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.
2006 Princeton University Math Competition, 10
If $a_1, ... ,a_{12}$ are twelve nonzero integers such that $a^6_1+...·+a^6_{12} = 450697$, what is the value of $a^2_1+...+a^2_{12}$?
2005 Alexandru Myller, 3
Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$.
[i]Dorin Andrica, Eugen Paltanea[/i]
1993 APMO, 5
Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane
with the following properties:
(i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$;
(ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$.
Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
1957 Moscow Mathematical Olympiad, 350
The distance between towns $A$ and $B$ is $999$ km.
At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows:
$\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$
How many signs are there, with both distances written with the help of only two distinct digits?
2015 BMT Spring, 1
The boba shop sells four different types of milk tea, and William likes to get tea each weekday. If William refuses to have the same type of tea on successive days, how many different combinations could he get, Monday through Friday?
2014 NIMO Problems, 5
Let a positive integer $n$ be $\textit{nice}$ if there exists a positive integer $m$ such that \[ n^3 < 5mn < n^3 +100. \] Find the number of [i]nice[/i] positive integers.
[i]Proposed by Akshaj[/i]
1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7
If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals
A. -12
B. 24
C. 35
D. -61
E. -63
2015 Romania National Olympiad, 1
Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $$p^2 + pq + q^2 = r^2 .$$
2006 Sharygin Geometry Olympiad, 9.5
A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.
2012 Stanford Mathematics Tournament, 1
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?
2016 IMO Shortlist, G1
Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.