Olympiad problems

2001 Iran MO (2nd round), 2

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

2007 Princeton University Math Competition, 6

Tags: expected value , probability , 3d geometry , geometry

Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

2009 Balkan MO Shortlist, G5

Tags: geometry , equal area , symmetric , convex quadrilateral , area , midpoint

Let $ABCD$ be a convex quadrilateral and $S$ an arbitrary point in its interior. Let also $E$ be the symmetric point of $S$ with respect to the midpoint $K$ of the side $AB$ and let $Z$ be the symmetric point of $S$ with respect to the midpoint $L$ of the side $CD$. Prove that $(AECZ) = (EBZD) = (ABCD)$.

the 9th XMO, 3

Tags: number theory , divisibility

A sequence $\{a_n\} $ satisfies $a_1$ is a positive integer and $a_{n+1}$ is the largest odd integer that divides $2^n-1+a_n$ for all $n\geqslant 1$. Given a positive integer $r$ which is greater than $1$. Is it possible that there exists infinitely many pairs of ordered positive integers $(m,n)$ for which $m>n$ and $a_m = ra_n$? In other words, if you successfully find [b]an[/b] $a_1$ that yields infinitely many pairs of $(m,n)$ which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible $a_1$. @below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.

2018 Math Prize for Girls Problems, 18

Tags:

Evaluate the expression \[ \left| \prod_{k=0}^{15} \left( 1+e^{2\pi i k^2/{31}} \right) \right| \, . \]

2012 Putnam, 5

Tags: function , algebra , vector , linear algebra , matrix , domain

Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.

1996 Irish Math Olympiad, 2

Tags: number theory

Let $ S(n)$ denote the sum of the digits of a natural number $ n$ (in base $ 10$). Prove that for every $ n$, $ S(2n) \le 2S(n) \le 10S(2n)$. Prove also that there is a positive integer $ n$ with $ S(n)\equal{}1996S(3n)$.

2024 Mozambique National Olympiad, P2

Tags: algebra , inequalities , ident , identity

Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$

2017 Baltic Way, 14

Tags: perpendicular lines , angle chase , geometry

Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.

2015 Geolympiad Spring, 6

Tags:

Let $ABC$ be a triangle, $X$ the midpoint of arc $BC$ on the circumcircle. The tangents from $X$ to the incircle meet the circumcircle again at $X_1,X_2$, and $X_1X_2$ intersects the incircle at $P,Q$. Let $M$ be the midpoint of $PQ$, and let $A_1$ be the tangency point of the $A$-mixtillinear incircle with the circumcircle. Show that $A,M,A_1$ are collinear.

2009 Mathcenter Contest, 1

Tags: algebra , inequalities

Let $m,n$ be natural numbers. Prove that $$m^{m^{m^m}}+n^{n^{n^n}}\geq m^{n^{n^n}}+ n^{m^{m^m}}$$ [i](nooonuii)[/i]

2020 Switzerland Team Selection Test, 2

Tags: number theory , coprime

Find all positive integers $n$ such that there exists an infinite set $A$ of positive integers with the following property: For all pairwise distinct numbers $a_1, a_2, \ldots , a_n \in A$, the numbers $$a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n$$ are coprime.

2021 Cono Sur Olympiad, 1

Tags: combinatory , counting

We say that a positive integer is guarani if the sum of the number with its reverse is a number that only has odd digits. For example, $249$ and $30$ are guarani, since $249 + 942 = 1191$ and $30 + 03 = 33$. a) How many $2021$-digit numbers are guarani? b) How many $2023$-digit numbers are guarani?

2010 Korea Junior Math Olympiad, 8

Tags: combinatorics , combinatorial geometry , lattice paths , lattice points

In a rectangle with vertices $(0, 0), (0, 2), (n,0),(n, 2)$, ($n$ is a positive integer) find the number of longest paths starting from $(0, 0)$ and arriving at $(n, 2)$ which satisfy the following: $\bullet$ At each movement, you can move right, up, left, down by $1$. $\bullet$ You cannot visit a point you visited before. $\bullet$ You cannot move outside the rectangle.

2006 Stanford Mathematics Tournament, 7

Tags:

Find all solutions to $aabb=n^4-6n^3$, where $a$ and $b$ are non-zero digits, and $n$ is an integer. ($a$ and $b$ are not necessarily distinct.)

2010 Today's Calculation Of Integral, 555

Tags: function , logarithm , derivative , calculus , integration , calculus computations

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

2012 India National Olympiad, 6

Tags: function , group theory , functional equation , algebra

Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and $(i) f(xy) + f(x)f(y) = f(x) + f(y)$ $(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $ for all $x,y \in \mathbb{Z}$, simultaneously. $(a)$ Find the set of all possible values of the function $f$. $(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.

2006 India IMO Training Camp, 1

Tags: circumcircle , geometry , inradius , inequalities

Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that \[\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.\]

2010 Singapore Junior Math Olympiad, 1

Tags: geometry , inradius , square , tangential

Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.

2019 Bangladesh Mathematical Olympiad, 3

Tags: geometry

Let $\alpha$ and $\omega$ be two circles such that $\omega$ goes through the center of $\alpha$.$\omega$ intersects $\alpha$ at $A$ and $B$.Let $P$ any point on the circumference $\omega$.The lines $PA$ and $PB$ intersects $\alpha$ again at $E$ and $F$ respectively.Prove that $AB=EF$.

2019 Istmo Centroamericano MO, 3

Tags: geometry , tangent circle , equal angles

Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle ABQ = \angle BCA$. Show that the circumcircles of the triangles $ABQ$ and $BHM$ are tangent.

2016 Romanian Master of Mathematics Shortlist, N1

Tags: number theory , quadratic residue , primitive root

Determine all integers $n \ge 3$ whose decimal expansion has less than $20$ digits, such that every quadratic non-residue modulo $n$ is a primitive root modulo $n$. [i]An integer $a$ is a quadratic non-residue modulo $n$, if there is no integer $b$ such that $a - b^2$ is divisible by $n$. An integer $a$ is a primitive root modulo $n$, if for every integer $b$ relatively prime to n there is a positive integer $k$ such that $a^k - b$ is divisible by $n$.[/i]

2022 Princeton University Math Competition, B2

Tags: geometry , 3d geometry

Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and, $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.

2007 Kazakhstan National Olympiad, 3

Tags: number theory

Solve in prime numbers the equation $p(p+1)+q(q+1)=r(r+1)$.

2010 Postal Coaching, 6

Tags: trigonometry , geometry

Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual. $(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$. $(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.