Olympiad problems

2023 India IMO Training Camp, 3

Tags: algebra , polynomial

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

2022 VIASM Summer Challenge, Problem 4

Tags: geometry

Given a triangle $ABC$ inscribed in $(O)$. Choose points $M,N,P$ on the sides $AB,BC,CA$ such that $AMNP$ is a parallelogram. The segment $CM$ intersects $NP$ at $E$; the segment $BP$ intersects $NM$ at $F$; and the segment $BE$ intersects $CF$ at $D.$ a) Prove that: $A,D,N$ are collinear. b) Let $I,J$ be the circumcenters of $\triangle MBF, \triangle PCE,$ respectively. Prove that: $OD$ passes through the midpoint of $IJ.$

LMT Accuracy Rounds, 2023 S5

Tags: number theory

Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?

1998 Tournament Of Towns, 4

Tags: 3-digit , sum , product , digit , number theory

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

2016 AMC 12/AHSME, 2

Tags:

For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

2019 Middle European Mathematical Olympiad, 1

Tags: algebra , functional equation , injective function

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds $$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ [i]Proposed by Patrik Bak, Slovakia[/i]

2012 Albania National Olympiad, 2

Tags: analytic geometry , polynomial , quadratic , algebra

The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.

1997 Miklós Schweitzer, 2

Tags: number theory , real analysis

Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.

2008 Postal Coaching, 6

Tags: point , combinatorial geometry , line

Suppose $n$ straight lines are in the plane so that there exist seven points such that any of these line passes through at least three of these points. Find the largest possible value of $n$.

2012 Czech-Polish-Slovak Match, 3

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$. Let $I, J$ and $K$ be the incentres of the triangles $ABC, ACD$ and $ABD$ respectively. Let $E$ be the midpoint of the arc $DB$ of circle $\omega$ containing the point $A$. The line $EK$ intersects again the circle $\omega$ at point $F$ $(F \neq E)$. Prove that the points $C, F, I, J$ lie on a circle.

2014 Contests, 4

Tags: inequalities , calculus , three variable inequality

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]

2008 ISI B.Math Entrance Exam, 1

Tags: calculus , integration , function , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

1996 German National Olympiad, 2

Tags: inequalities , algebra

Let $a$ and $b$ be positive real numbers smaller than $1$. Prove that the following two statements are equivalent: (i) $a+b = 1$, (ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$, the following inequlity holds: $$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$$

1999 Mexico National Olympiad, 6

Tags: calculus , integration , combinatorics

A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.

2012 Philippine MO, 3

Tags: trigonometry , inequalities

If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that \[ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. \]

2014 Balkan MO Shortlist, G4

Tags: analytic geometry , geometry

Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.

2018 BMT Spring, 2

Tags:

For how many values of $x$ does $20^x \cdot 18^x = 2018^x$?

Kyiv City MO Juniors 2003+ geometry, 2009.89.5

Tags: chord , geometry , orthocenter , circles

A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.

2011 Turkey Junior National Olympiad, 1

Tags: algebra , inequalities

Show that \[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\] holds for all positive real numbers $x,y$.

Kvant 2024, M2813

Tags: geometry

The quadrilateral $ABCD$ is described around a circle centered on $I$. Let the diagonals $AC$ and $BD$ intersect at point $E$. The perpendicular bisectors to the segments $AC$ and $BD$ intersect at the point $P$ lying inside the triangle $BEC$. The circumscribed circles of the triangles $APC$ and $BPD$ intersect at points $P$ and $Q$. Prove that $I$ lies on the line $PQ$. [i] Proposed by Tran Quang Hung [/i]

2014 Federal Competition For Advanced Students, P2, 1

Tags: number of divisors , divisor , number theory

For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

2023 Yasinsky Geometry Olympiad, 3

Tags: geometry , incircle

Points $K$ and $N$ are the midpoints of sides $AC$ and $AB$ of triangle $ABC$. The inscribed circle $\omega$ of the triangle $AKN$ is tangent to $BC$. Find $BC$ if $AC + AB = n$. (Oleksii Karliuchenko)

2007 Singapore Junior Math Olympiad, 5

Tags: algebra , function

For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)

2008 Postal Coaching, 1

Tags: incenter , circumcircle , right angle , arc midpoint , angle chasing , angle , geometry

In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.

2018 Costa Rica - Final Round, 4

Tags: functional , functional inequality , algebra

Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer. Note: $N^*$ denotes the set of positive integers.