Found problems: 85335
2016 Gulf Math Olympiad, 4
4. Suppose that four people A, B, C and D decide to play games of tennis
doubles. They might first play the team A and B against the team C
and D. Next A and C might play B and D. Finally A and D might play
B and C. The advantage of this arrangement is that two conditions are
satisfied.
(a) Each player is on the same team as each other player exactly once.
(b) Each player is on the opposing team to each other player exactly
twice.
Is it possible to arrange a collection of tennis matches satisfying both
condition (a) and condition (b) in the following circumstances?
(i) There are five players.
(ii) There are seven players.
(iii) There are nine players.
2015 Princeton University Math Competition, A2/B3
What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?
2015 Bosnia And Herzegovina - Regional Olympiad, 2
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$
2005 Hungary-Israel Binational, 1
Does there exist a sequence of $2005$ consecutive positive integers that
contains exactly $25$ prime numbers?
1999 National Olympiad First Round, 35
Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k \minus{} 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle?
$\textbf{(A)}\ 14 \qquad\textbf{(B)}\ 53 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 79 \qquad\textbf{(E)}\ 156$
2020 USMCA, 29
Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Given that $AB = 8, AC = 10$, and $\angle BAC = 60^\circ$, find the area of $BCHG$.
[i] Note: this is a modified version of Premier #2 [/i]
2021 CHMMC Winter (2021-22), 7
Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Denote $\Gamma$ the incircle of $ABC$, let $I$ be the center of $\Gamma$ . The circumcircle of $BIC$ intersects $\Gamma$ at $X_1$ and $X_2$. The circumcircle of $CIA$ intersects $\Gamma$ at $Y_1$ and $Y_2$. The circumcircle of $AIB$ intersects $\Gamma$ at $Z_1$ and $Z_2$. The area of the triangle determined by $\overline{X_1X_2}$, $\overline{Y_1Y_2}$, and $\overline{Z_1Z_2}$ equals $\frac{m \sqrt{p}}{n}$ for positive integers $m, n$, and $p$, where $m$ and$ n$ are relatively prime and $p$ is squarefree.
Compute $m+n+ p$.
2010 Contests, 4
Solid camphor is insoluble in water but is soluble in vegetable oil. The best explanation for this behavior is that camphor is a(n)
${ \textbf{(A)}\ \text{Ionic solid} \qquad\textbf{(B)}\ \text{Metallic solid} \qquad\textbf{(C)}\ \text{Molecular solid} \qquad\textbf{(D)}\ \text{Network solid} } $
1999 IMO Shortlist, 8
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
2004 AMC 8, 10
Handy Aaron helped a neighbor $1\frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from $8:20$ to $10:45$ on Wednesday morning, and a half-hour on Friday. He is paid $\$3$ per hour. How much did he earn for the week?
$\textbf{(A)}\ 8\qquad
\textbf{(B)}\ 9\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 15$
1988 IMO Shortlist, 12
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
2015 JBMO Shortlist, 2
The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.
(Moldova)
2012 Saint Petersburg Mathematical Olympiad, 5
$n \geq k$ -two natural numbers. $S$ -such natural, that have not less than $n$ divisors. All divisors of $S$ are written
in descending order. What minimal number of divisors can have number from $k$-th place ?
2016 Czech And Slovak Olympiad III A, 6
We put a figure of a king on some $6 \times 6$ chessboard. It can in one thrust jump either vertically or horizontally. The length of this jump is alternately one and two squares, whereby a jump of one (i.e. to the adjacent square) of the piece begins. Decide whether you can choose the starting position of the pieces so that after a suitable sequence $35$ jumps visited each box of the chessboard just once.
2018 Puerto Rico Team Selection Test, 1
Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.
2020 Bulgaria National Olympiad, P1
On the sides of $\triangle{ABC}$ points $P,Q \in{AB}$ ($P$ is between $A$ and $Q$) and $R\in{BC}$ are chosen. The points $M$ and $N$ are defined as the intersection point of $AR$ with the segments $CP$ and $CQ$, respectively. If $BC=BQ$, $CP=AP$, $CR=CN$ and $\angle{BPC}=\angle{CRA}$, prove that $MP+NQ=BR$.
2021 CCA Math Bonanza, L1.1
Compute
\[
(2+0\cdot 2 \cdot 1)+(2+0-2) \cdot (1) + (2+0)\cdot (2-1) + (2) \cdot \left(0+2^{-1}\right).
\]
[i]2021 CCA Math Bonanza Lightning Round #1.1[/i]
2008 Bosnia and Herzegovina Junior BMO TST, 4
On circle are $ 2008$ blue and $ 1$ red point(s) given. Are there more polygons which have a red point or those which dont have it??
1990 IMO, 3
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
1977 IMO Shortlist, 12
In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.
2012 QEDMO 11th, 1
Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.
LMT Team Rounds 2021+, 10
Aidan and Andrew independently select distinct cells in a $4 $ by $4$ grid, as well as a direction (either up, down, left, or right), both at random. Every second, each of them will travel $1$ cell in their chosen direction. Find the probability that Aidan and Andrew willmeet (be in the same cell at the same time) before either one of them hits an edge of the grid. (If Aidan and Andrew cross paths by switching cells, it doesn’t count as meeting.)
1989 Romania Team Selection Test, 4
Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively.
Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$.
Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.
2024 Saint Petersburg Mathematical Olympiad, 1
The $100 \times 100$ table is filled with numbers from $1$ to $10 \ 000$ as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?
2009 Moldova National Olympiad, 7.4
Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.