Found problems: 85335
2010 National Olympiad First Round, 10
How many integers $n$ with $0\leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 8
$
VMEO III 2006 Shortlist, G1
Given a circle $(O)$ and a point $P$ outside that circle. $M$ is a point running on the circle $(O)$. The circle with center $I$ and diameter $PM$ intersects circle $(O)$ again at $N$. The tangent of $(I)$ at $P$ intersects $MN$ at $Q$. The line through $Q$ perpendicular to $PO$ intersects $PM$ at $ A$. $AN$ intersects $(O)$ further at $ B$. $BM$ intersects $PO$ at $C$. Prove that $AC$ is perpendicular to $OQ$.
2002 National Olympiad First Round, 27
The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys?
$
\textbf{a)}\ 18
\qquad\textbf{b)}\ 20
\qquad\textbf{c)}\ 22
\qquad\textbf{d)}\ 24
\qquad\textbf{e)}\ 25
$
1997 Hungary-Israel Binational, 1
Determine the number of distinct sequences of letters of length 1997 which use each of the letters $A$, $B$, $C$ (and no others) an odd number of times.
2005 All-Russian Olympiad Regional Round, 10.1
The cosines of the angles of one triangle are respectively equal to the sines of the angles of the other triangle. Find the largest of these six angles of triangles.
2019 India IMO Training Camp, P1
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2008 Kyiv Mathematical Festival, 3
Prove that among any 7 integers there exist three numbers $ a,b,c$ such that $ a^2\plus{}b^2\plus{}c^2\minus{}ab\minus{}bc\minus{}ac$ is divisible by 7.
2023 Brazil Team Selection Test, 3
Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$
1999 Brazil National Olympiad, 4
On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$, then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$. In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$.
The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs.
2003 Spain Mathematical Olympiad, Problem 5
How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?
2013 Princeton University Math Competition, 1
Let $a_1=2013$ and $a_{n+1} = 2013^{a_n}$ for all positive integers $n$. Let $b_1=1$ and $b_{n+1}=2013^{2012b_n}$ for all positive integers $n$. Prove that $a_n>b_n$ for all positive integers $n$.
2010 Sharygin Geometry Olympiad, 25
For two different regular icosahedrons it is known that some six of their vertices are vertices of a regular octahedron. Find the ratio of the edges of these icosahedrons.
1982 IMO Shortlist, 19
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
2021 Indonesia TST, G
let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $
let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $
suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $
now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $
2017 Regional Olympiad of Mexico Northeast, 6
Find all triples of real numbers $(a, b, c)$ that satisfy the system of equations $$\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}$$
1996 Turkey Team Selection Test, 3
Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.
2021 China Team Selection Test, 6
Find the smallest positive real constant $a$, such that for any three points $A,B,C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A,B,C$ lie on the interior or boundary of $\triangle PQR$.
1971 IMO Longlists, 19
In a triangle $P_1P_2P_3$ let $P_iQ_i$ be the altitude from $P_i$ for $i = 1, 2,3$ ($Q_i$ being the foot of the altitude). The circle with diameter $P_iQ_i$ meets the two corresponding sides at two points different from $P_i.$ Denote the length of the segment whose endpoints are these two points by $l_i.$ Prove that $l_1 = l_2 = l_3.$
1992 IMO Longlists, 33
Let $a, b, c$ be positive real numbers and $p, q, r$ complex numbers. Let $S$ be the set of all solutions $(x, y, z)$ in $\mathbb C$ of the system of simultaneous equations
\[ax + by + cz = p,\]\[ax2 + by2 + cz2 = q,\]\[ax3 + bx3 + cx3 = r.\]
Prove that $S$ has at most six elements.
2000 IMO Shortlist, 6
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.
2009 QEDMO 6th, 10
Let $n \in N$. The land of Draconis has more than $2^n$ dungeons. Between two different Dungeons have exactly one path, but each path is a one-way street. Total $n$ Dragon cults share the territory; each path is controlled by exactly one cult. It is said that a dragon cult $K$ has established itself in a dungeon $D$ if there is both one a path beginning in $D$ and one a path ending in $D$, both of which are controlled by $K$ . Prove that there is a cult $K$, which has at least one dungeon controlled.
1995 AMC 8, 18
The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square?
[asy]
draw((2,2)--(2,-2)--(-2,-2)--(-2,2)--cycle);
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
draw((0,1)--(0,2));
draw((1,0)--(2,0));
draw((0,-1)--(0,-2));
draw((-1,0)--(-2,0));
[/asy]
$\text{(A)}\ 25 \qquad \text{(B)}\ 44 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 62 \qquad \text{(E)}\ 75$
2023 Indonesia TST, N
Find all triplets natural numbers $(a, b, c)$ satisfied
\[GCD(a, b) + LCM(a,b) = 2021^c\]
with $|a - b|$ and $(a+b)^2 + 4$ are both prime number
2024 AMC 8 -, 4
When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. Which number did Yunji leave out?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2016 Harvard-MIT Mathematics Tournament, 10
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.