Olympiad problems

1984 Swedish Mathematical Competition, 4

Tags: algebra , polynomial

Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.

1986 IMO Shortlist, 15

Tags: geometry , circumcircle , ratio , trigonometry

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$ [b](a)[/b] Prove that $ABCD$ and $A''B''C''D''$ are similar. [b](b) [/b]The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

May Olympiad L2 - geometry, 2012.4

Tags: combinatorics , combinatorial geometry , triangle inequality

Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.

1967 IMO Longlists, 42

Tags: trigonometry , factorization , trigonometric equation , algebra

Decompose the expression into real factors: \[E = 1 - \sin^5(x) - \cos^5(x).\]

2010 NZMOC Camp Selection Problems, 3

Tags: number theory , prime

Find all positive integers n such that $n^5 + n + 1$ is prime.

2022 Bulgaria EGMO TST, 6

Tags: combinatorics , induction , subset

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.

2009 Greece National Olympiad, 4

Tags: trigonometry , complex number , geometry

Consider pairwise distinct complex numbers $z_1,z_2,z_3,z_4,z_5,z_6$ whose images $A_1,A_2,A_3,A_4,A_5,A_6$ respectively are succesive points on the circle centered at $O(0,0)$ and having radius $r>0.$ If $w$ is a root of the equation $z^2+z+1=0$ and the next equalities hold \[z_1w^2+z_3w+z_5=0 \\ z_2w^2+z_4w+z_6=0\] prove that [b]a)[/b] Triangle $A_1A_3A_5$ is equilateral [b]b)[/b] \[|z_1-z_2|+|z_2-z_3|+|z_3-z_4|+|z_4-z_5|+z_5-z_6|+|z_6-z_1|=3|z_1-z_4|=3|z_2-z_5|=3|z_3-z_6|.\]

2016 Harvard-MIT Mathematics Tournament, 10

Tags: incircle , collinearity

Let $ABC$ be a triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$. Point $P$ lies on $\overline{EF}$ such that $\overline{DP} \perp \overline{EF}$. Ray $BP$ meets $\overline{AC}$ at $Y$ and ray $CP$ meets $\overline{AB}$ at $Z$. Point $Q$ is selected on the circumcircle of $\triangle AYZ$ so that $\overline{AQ} \perp \overline{BC}$. Prove that $P$, $I$, $Q$ are collinear.

2000 Vietnam National Olympiad, 2

Tags: geometric transformation , circumcircle , geometry

Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity. (a) Determine the locus of the midpoint of $ M_1M_2$. (b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.

2025 Junior Balkan Team Selection Tests - Romania, P1

Tags: number theory

A positive integer $n\geqslant 3$ is [i]almost squarefree[/i] if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.

2018 Romania Team Selection Tests, 4

Tags: greatest common divisor , bezout s identity , number theory

Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.

2017 CMIMC Algebra, 3

Tags: algebra , polynomial , quadratic

Suppose $P(x)$ is a quadratic polynomial with integer coefficients satisfying the identity \[P(P(x)) - P(x)^2 = x^2+x+2016\] for all real $x$. What is $P(1)$?

1996 Tournament Of Towns, (506) 3

Tags: combinatorics

(a) Can it happen that in a group of $10$ girls and $9$ boys, ball the girls know a different number of boys while all the boys know the same number of girls? (b) What if there are $11$ girls and $10$ boys? (NB Vassiliev)

2021 Romania Team Selection Test, 2

Tags: set , combinatorics

Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$

1994 IMO Shortlist, 1

Tags: algebra , sequence , recurrence relation

Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$

1963 IMO Shortlist, 2

Tags: geometry , 3d geometry , locus , locus problems , sphere

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

2016 AMC 10, 6

Tags:

Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's? $\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$

2008 Tuymaada Olympiad, 2

Tags: number theory , prime number

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2018 Balkan MO Shortlist, G3

Tags: geometry , inequalities , geometric inequality , semiperimeter , maximum value , minimum

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

1999 Switzerland Team Selection Test, 5

Tags: geometry , rectangle , angle bisector

In a rectangle $ABCD, M$ and $N$ are the midpoints of $AD$ and $BC$ respectively and $P$ is a point on line $CD$. The line $PM$ meets $AC$ at $Q$. Prove that MN bisects the angle $\angle QNP$.

2022 Caucasus Mathematical Olympiad, 2

Tags: geometry , number theory , parallelogram

In parallelogram $ABCD$, points $E$ and $F$ on segments $AD$ and $CD$ are such that $\angle BCE=\angle BAF$. Points $K$ and $L$ on segments $AD$ and $CD$ are such that $AK=ED$ and $CL=FD$. Prove that $\angle BKD=\angle BLD$.

1964 AMC 12/AHSME, 30

Tags: quadratic

If $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$, the larger root minus the smaller root is: $ \textbf{(A)}\ -2+3\sqrt{3}\qquad\textbf{(B)}\ 2-\sqrt{3}\qquad\textbf{(C)}\ 6+3\sqrt{3}\qquad\textbf{(D)}\ 6-3\sqrt{3}\qquad\textbf{(E)}\ 3\sqrt{3}+2 $

2014 National Olympiad First Round, 12

Tags:

If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 12 $

2014 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , inequalities , inequality

Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$

1980 Putnam, B5

Tags: convexity , multiplication

For each $t \geq 0$ let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)).$$ Define necessary and sufficient conditions on $ t$ for $S_t $ to be closed under multiplication.