Found problems: 85335
1999 Chile National Olympiad, 5
Consider the numbers $x_1, x_2,...,x_n$ that satisfy:
$\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$
$\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$
Prove that $n$ is a multiple of $4$.
2014 Sharygin Geometry Olympiad, 3
Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15^o$ and $\angle BAN = 45^o$. Find the value of angle $\angle ABM$.
(A. Blinkov)
2013-2014 SDML (Middle School), 6
The base $5$ number $32$ is equal to the base $7$ number $23$. There are two $3$-digit numbers in base $5$ which similarly have their digits reversed when expressed in base $7$. What is their sum, in base $5$? (You do not need to include the base $5$ subscript in your answer).
2011 Kosovo National Mathematical Olympiad, 3
If $a,b,c$ are real positive numbers prove that the inequality holds:
\[ \frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}} \]
2021 Nigerian Senior MO Round 2, 4
let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$
1961 All-Soviet Union Olympiad, 1
Consider the figure below, composed of 16 segments. Prove that there is no curve intersecting each segment exactly once. (The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.)
[asy]
draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0));
draw((0,3/2)--(6,3/2));
draw((2,0)--(2,3/2));
draw((4,0)--(4,3/2));
draw((3,3/2)--(3,3));
[/asy]
2014 Bosnia Herzegovina Team Selection Test, 3
Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.
2023 Korea - Final Round, 4
Find all positive integers $n$ satisfying the following.
$$2^n-1 \text{ doesn't have a prime factor larger than } 7$$
2023 HMNT, 5
A complex quartic polynomial $Q$ is [i]quirky [/i] if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x) = x^4-kx^3-x^2-x-45$ is quirky. Compute the product of these four values of $k$.
2006 Stanford Mathematics Tournament, 11
Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$ has roots $r_1,r_2,\ldots,r_{2006}$. The coefficients satisfy $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i,j\le2006$. Given that $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$, determine $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$.
2017 F = ma, 20
20) A particle of mass m moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.
If the collision is completely $inelastic$ under what condition will the fractional momentum transfer between the two objects be a maximum?
A) $\frac{m}{M} \ll 1$
B) $0.5 < \frac{m}{M} < 1$
C) $m = M$
D) $1 < \frac{m}{M} < 2$
E) $\frac{m}{M} \gg 1$
2013 Iran Team Selection Test, 1
In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.
2013 Princeton University Math Competition, 2
Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.
2016 IMO Shortlist, C7
There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if $n$ is odd.
(b) Prove that Geoff can never fulfill his wish if $n$ is even.
1999 Slovenia National Olympiad, Problem 1
Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by
$$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$
2008 Mathcenter Contest, 1
In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then
$$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$
2016 Oral Moscow Geometry Olympiad, 5
From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).
1983 IMO Shortlist, 20
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
2009 China Team Selection Test, 1
Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$
1998 German National Olympiad, 6b
Prove that the following statement holds for all odd integers $n \ge 3$:
If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.
2007 IberoAmerican Olympiad For University Students, 7
The [i]height[/i] of a positive integer is defined as being the fraction $\frac{s(a)}{a}$, where $s(a)$ is the sum of all the positive divisors of $a$. Show that for every pair of positive integers $N,k$ there is a positive integer $b$ such that the [i]height[/i] of each of $b,b+1,\cdots,b+k$ is greater than $N$.
2005 Sharygin Geometry Olympiad, 2
Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.
2003 AMC 12-AHSME, 16
A point $ P$ is chosen at random in the interior of equilateral triangle $ ABC$. What is the probability that $ \triangle ABP$ has a greater area than each of $ \triangle ACP$ and $ \triangle BCP$?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{4} \qquad
\textbf{(C)}\ \frac{1}{3} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{2}{3}$
2017 Baltic Way, 20
Let $S$ be the set of all ordered pairs $(a,b)$ of integers with $0<2a<2b<2017$ such that $a^2+b^2$ is a multiple of $2017$. Prove that \[\sum_{(a,b)\in S}a=\frac{1}{2}\sum_{(a,b)\in S}b.\]
Proposed by Uwe Leck, Germany
2004 Estonia National Olympiad, 3
The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.