Found problems: 160
2017 Iran Team Selection Test, 3
There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes.
What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]?
[i]Proposed by Amin Bahjati[/i]
2017 Iran Team Selection Test, 5
$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s.
[i]Proposed by Aryan Tajmir[/i]
2018 Iran Team Selection Test, 5
$2n-1$ distinct positive real numbers with sum $S $ are given. Prove that there are at least $\binom {2n-2}{n-1}$ different ways to choose $n $ numbers among them such that their sum is at least $\frac {S}{2}$.
[i]Proposed by Amirhossein Gorzi[/i]
2017 Iran Team Selection Test, 5
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively.
Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other.
[i]Proposed by Iman Maghsoudi[/i]
2018 Iran Team Selection Test, 6
Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$
[i]Proposed by Ali Zamani [/i]
2017 Iran Team Selection Test, 1
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
2015 Iran MO (3rd round), 5
Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$:
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$
2017 Iran Team Selection Test, 4
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.
Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.
Is there always a number $x$ that satisfies all the equations?
[i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]
2018 Iran Team Selection Test, 1
Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$ at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$.
[i]Proposed by Ali Zamani [/i]
2017 Iranian Geometry Olympiad, 4
In the isosceles triangle $ABC$ ($AB=AC$), let $l$ be a line parallel to $BC$ through $A$. Let $D$ be an arbitrary point on $l$. Let $E,F$ be the feet of perpendiculars through $A$ to $BD,CD$ respectively. Suppose that $P,Q$ are the images of $E,F$ on $l$. Prove that $AP+AQ\le AB$
[i]Proposed by Morteza Saghafian[/i]
2017 Iran Team Selection Test, 3
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$.
Prove that $X,Y,Z$ are collinear.
[i]Proposed by Hooman Fattahi[/i]
2017 Iranian Geometry Olympiad, 5
Let $X,Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ($X$ is between $B,Y$). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$. Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$, and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$. Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$.
[i]Proposed by Iman Maghsoudi[/i]
2009 Iran MO (3rd Round), 1
Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear.
[b](a)[/b] Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then \[|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|\]
Where $|XY|$ means the length of line segment between $X$ and $Y$.
[b](b)[/b] If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out.
Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then \[|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|\]
Time allowed for this problem was 2 hours.
2016 Iran MO (3rd Round), 1
Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that :
$$p(p(i)) \geq i$$
2017 Iran Team Selection Test, 5
Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as
$$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$
Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that
$$P_{2n}(x)=P_n(x^2+c).$$
[i]Proposed by Navid Safaei[/i]
2021 Iran MO (3rd Round), 2
Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.
2008 Iran Team Selection Test, 5
Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove that:
\[ \sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}. \]
2017 Iran MO (3rd round), 2
Let $P(z)=a_d z^d+\dots+ a_1z+a_0$ be a polynomial with complex coefficients. The $reverse$ of $P$ is defined by
$$P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}$$
(a) Prove that
$$P^*(z)=z^d \overline{ P\left( \frac{1}{\overline{z}} \right) } $$
(b) Let $m$ be a positive integer and let $q(z)$ be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of $q(z)$ lie inside or on the unit circle. Prove that all roots of the polynomial
$$Q(z)=z^m q(z)+ q^*(z)$$
lie on the unit circle.
2015 Iran MO (2nd Round), 3
Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.
2018 Iranian Geometry Olympiad, 5
There are some segments on the plane such that no two of them intersect each other (even at the ending points). We say segment $AB$ [b]breaks[/b] segment $CD$ if the extension of $AB$ cuts $CD$ at some point between $C$ and $D$.
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -5.267474904743955, xmax = 11.572179069738377, ymin = -10.642621257034536, ymax = 4.543526642434019; /* image dimensions */
/* draw figures */
draw((-4,-2)--(1.08,-2.03), linewidth(2));
draw(shift((-2.1866176795507295,-2.0107089507113147))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((-0.16981767035094117,3.225314210196242)--(-2.1866176795507295,-2.0107089507113147), linewidth(2) + linetype("4 4"));
draw((-0.16981767035094117,3.225314210196242)--(-0.8194002739586808,1.538865607509914), linewidth(2));
label("$A$",(-1.2684397405642523,3.860690076971137),SE*labelscalefactor,fontsize(16));
label("$B$",(-1.9211395070170559,2.002590777612728),SE*labelscalefactor,fontsize(16));
label("$C$",(-4.971261820527631,-1.6571211388676117),SE*labelscalefactor,fontsize(16));
label("$D$",(1.08925640451367566,-1.6571211388676117),SE*labelscalefactor,fontsize(16));
/* dots and labels */
dot((-4,-2),dotstyle);
dot((1.08,-2.03),dotstyle);
dot((-0.16981767035094117,3.225314210196242),dotstyle);
dot((-0.8194002739586808,1.538865607509914),dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
$a)$ Is it possible that each segment when extended from both ends, breaks exactly one other segment from each way?
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -6.8, xmax = 8.68, ymin = -10.32, ymax = 3.64; /* image dimensions */
/* draw figures */
draw((-2.56,1.24)--(-0.36,1.4), linewidth(2));
draw((-3.32,-2.68)--(-1.24,-3.08), linewidth(2));
draw(shift((-2.551651190956802,-2.8277593863544612))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw(shift((-0.8889576602618603,1.3615303519809556))*scale(0.17638888888888887)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((-2.551651190956802,-2.8277593863544612)--(-0.8889576602618603,1.3615303519809556), linewidth(2) + linetype("4 4"));
draw((-1.4097008194020806,0.049476186483185636)--(-1.8514772275312024,-1.0636149148218605), linewidth(2));
/* dots and labels */
dot((-2.56,1.24),dotstyle);
dot((-0.36,1.4),dotstyle);
dot((-3.32,-2.68),dotstyle);
dot((-1.24,-3.08),dotstyle);
dot((-1.4097008194020806,0.049476186483185636),dotstyle);
dot((-1.8514772275312024,-1.0636149148218605),dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
$b)$ A segment is called [b]surrounded[/b] if from both sides of it, there is exactly one segment that breaks it.\\
([i]e.g.[/i] segment $AB$ in the figure.) Is it possible to have all segments to be surrounded?
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(7cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -10.70976151557872, xmax = 18.64292748469251, ymin = -16.354300717041443, ymax = 9.136192362141452; /* image dimensions */
/* draw figures */
draw((1.0313140845297686,0.748205038977829)--(-1.3,-4), linewidth(2.8));
draw((-5.780195085389632,-2.13088646583346)--(-2.549994860479401,-2.13088646583346), linewidth(2.8));
draw((4.121070821400425,-3.816208322308361)--(1.78,-1.88), linewidth(2.8));
draw(shift((-0.38228674372374466,-2.13088646583346))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((-2.549994860479401,-2.13088646583346)--(-0.38228674372374466,-2.13088646583346), linewidth(2.8) + linetype("4 4"));
draw(shift((0.32979226045261084,-0.6805897691262632))*scale(0.21166666666666667)*(expi(pi/4)--expi(5*pi/4)^^expi(3*pi/4)--expi(7*pi/4))); /* special point */
draw((4.121070821400425,-3.816208322308361)--(0.32979226045261084,-0.6805897691262632), linewidth(2.8) + linetype("4 4"));
draw((-3.6313140845297687,-8.74820503897783)--(3.600422205681574,5.980726991931396), linewidth(2.8) + linetype("2 2"));
label("$A$",(-0.397698406272906,1.754593418658662),SE*labelscalefactor,fontsize(16));
label("$B$",(-2.6377720405041316,-3.266261278756151),SE*labelscalefactor,fontsize(16));
/* dots and labels */
dot((1.0313140845297686,0.748205038977829),linewidth(6pt) + dotstyle);
dot((-1.3,-4),linewidth(6pt) + dotstyle);
dot((-5.780195085389632,-2.13088646583346),linewidth(6pt) + dotstyle);
dot((-2.549994860479401,-2.13088646583346),linewidth(6pt) + dotstyle);
dot((4.121070821400425,-3.816208322308361),linewidth(6pt) + dotstyle);
dot((1.78,-1.88),linewidth(6pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
[i]Proposed by Morteza Saghafian[/i]
2012 Iran MO (3rd Round), 4
The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$.
[i]Proposed By Pedram Safaei[/i]
2017 Iran Team Selection Test, 1
Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality:
$$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$
[i]Proposed by Mohammad Jafari[/i]
2018 Iran MO (3rd Round), 3
Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.
2017 Iranian Geometry Olympiad, 2
Find the angles of triangle $ABC$.
[asy]
import graph; size(9.115122858763474cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273; /* image dimensions */
/* draw figures */
draw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6));
draw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6));
draw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6));
draw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6));
draw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6));
draw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6));
draw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6));
draw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6));
draw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6));
draw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6));
draw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6));
draw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6));
draw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6));
draw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6));
draw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6));
draw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6));
draw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6));
label("$A$",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14));
label("$B$",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14));
label("$C$",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14));
/* dots and labels */
dot((3.0842,-3.6348),linewidth(3.pt) + dotstyle);
dot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle);
dot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle);
dot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle);
dot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle);
dot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle);
dot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
[i]Proposed by Morteza Saghafian[/i]
2019 Iran MO (3rd Round), 2
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ :
$$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$.
[i]Proposed by Amirhossein Zolfaghari [/i]