Olympiad problems

2000 Romania National Olympiad, 4

Tags: polynomial , algebra

Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.

2016 NIMO Problems, 3

Tags:

Find the sum of all positive integers $n$ such that exactly $2\%$ of the numbers in the set $\{1, 2, \ldots, n\}$ are perfect squares. [i]Proposed by Michael Tang[/i]

1997 Moldova Team Selection Test, 9

Tags: function , algebra

Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that: $f(1997)=1998$ $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$

2013 Math Prize For Girls Problems, 1

Tags: geometry

The figure below shows two equilateral triangles each with area 1. [asy] unitsize(40); draw(polygon(3)); draw(rotate(60) * polygon(3)); [/asy] The intersection of the two triangles is a regular hexagon. What is the area of the union of the two triangles?

1973 AMC 12/AHSME, 26

Tags: arithmetic sequence

The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 8$

2006 Thailand Mathematical Olympiad, 11

Tags: number theory , remainder , product , prime

Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

2020 HK IMO Preliminary Selection Contest, 9

Tags: geometry

In $\Delta ABC$, $\angle B=46.6^\circ$. $D$ is a point on $BC$ such that $\angle BAD=20.1^\circ$. If $AB=CD$ and $\angle CAD=x^\circ$, find $x$.

2015 Azerbaijan JBMO TST, 3

Tags: geometry , circumcircle , symmetry

There is a triangle $ABC$ that $AB$ is not equal to $AC$.$BD$ is interior bisector of $\angle{ABC}$($D\in AC$) $M$ is midpoint of $CBA$ arc.Circumcircle of $\triangle{BDM}$ cuts $AB$ at $K$ and $J,$ is symmetry of $A$ according $K$.If $DJ\cap AM=(O)$, Prove that $J,B,M,O$ are cyclic.

Cono Sur Shortlist - geometry, 2018.G6

Tags: geometry , orthocenter , circumcircle , concurrency , tangent circle

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.

2008 Tournament Of Towns, 7

Tags: angle , geometry

A convex quadrilateral $ABCD$ has no parallel sides. The angles between the diagonal $AC$ and the four sides are $55^o, 55^o, 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.

2011 Junior Balkan MO, 3

Tags: geometry , rhombus , symmetry , ratio , combinatorics

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

1989 Spain Mathematical Olympiad, 5

Tags: complex number , number theory

Consider the set $D$ of all complex numbers of the form $a+b\sqrt{-13}$ with $a,b \in Z$. The number $14 = 14+0\sqrt{-13}$ can be written as a product of two elements of $D$: $14 = 2 \cdot 7$. Find all possible ways to express $14$ as a product of two elements of $D$.

2023 Stanford Mathematics Tournament, 10

Tags: geometry

Let $\vartriangle ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, and $CA = 15$. The angle bisector of $\angle BAC$, the angle bisector of $\angle ABC$, and the angle bisector of $\angle ACB$ intersect the circumcircle of $\vartriangle ABC$ again at points $D$, $E$ and $F$, respectively. Compute the area of hexagon $AF BDCE$.

2014 Tuymaada Olympiad, 3

Tags: geometric transformation , homothety , reflection , geometry

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

2017 Korea USCM, 8

Tags: differential equation , ordinary differential equation , trigonometry

$u(t)$ is solution of the following initial value problem. $$\begin{cases} u''(t) + u'(t) = \sin u(t) &\;\;(t>0),\\ u(0)=1,\;\; u'(0)=0 & \end{cases}$$ (1) Show that $u(t)$ and $u'(t)$ are bounded on $t>0$. (2) Find $\lim\limits_{t\to\infty} u(t)$ with proof.

2004 Iran MO (3rd Round), 12

Tags: algebra , polynomial , algorithm , geometry , geometric transformation , induction , number theory

$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$) Also we easily have $+,*$ in $\mathbb{N}_{10}$. first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

2014 Online Math Open Problems, 3

Tags: geometry , floor function

Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$). [i]Proposed by Yang Liu[/i]

2005 Moldova Team Selection Test, 3

Tags: linear algebra , matrix , combinatorics , extremal combinatorics

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

1980 Bulgaria National Olympiad, Problem 4

Tags: inequality , inequalities

Let $a $, $b $, and $c $ be non-negative reals. Prove that $a^3+b^3+c^3+6abc\ge \frac{(a+b+c)^3}{4} $.

2013 Singapore Junior Math Olympiad, 2

Tags: parallel , geometry

In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.

1985 IMO Longlists, 70

Tags: function , algebra

Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in \mathbb N$. ($[x]$ is the integer part of $x$.) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$, show that the function $\max(f(x) - g(x), 0)$ is in $C$, for all $f; g \in C$.

Durer Math Competition CD Finals - geometry, 2018.D4

Tags: geometry , collinear , similar

Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB \parallel A'B' $, $BC \parallel B'C'$ and $CA \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.

2024 Polish MO Finals, 1

Tags: rectangle , geometry , angle

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

2004 Cuba MO, 2

Tags: number theory , consecutive

When we write the number $n > 2$ as the sum of some integers consecutive positives (at least two addends), we say that we have an [i]elegant decomposition[/i] of $n$. Two [i]elegant decompositions[/i] will be different if any of them contains some term that does not contains the other. How many different elegant decompositions does the number $3^{2004}$ have?

2005 Bulgaria National Olympiad, 6

Tags: algebra , polynomial , symmetry , modular arithmetic , number theory

Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.