Olympiad problems

Kyiv City MO 1984-93 - geometry, 1986.7.5

Prove that the sum of the lengths of the diagonals of an arbitrary quadrilateral is less than the sum of the lengths of its sides.

2006 District Olympiad, 3

Tags: parallelogram , geometry

Let $ABCD$ be a convex quadrilateral, $M$ the midpoint of $AB$, $N$ the midpoint of $BC$, $E$ the intersection of the segments $AN$ and $BD$, $F$ the intersection of the segments $DM$ and $AC$. Prove that if $BE = \frac 13 BD$ and $AF = \frac 13 AC$, then $ABCD$ is a parallelogram.

1977 Germany Team Selection Test, 2

Tags: polynomial , functional equation , algebra

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

2022-23 IOQM India, 12

Tags: geometry , trapezoid

Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$, let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$.\\ Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygon $S$.

2002 India IMO Training Camp, 15

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2017 Cono Sur Olympiad, 1

Tags: number theory

A positive integer $n$ is called [i]guayaquilean[/i] if the sum of the digits of $n$ is equal to the sum of the digits of $n^2$. Find all the possible values that the sum of the digits of a guayaquilean number can take.

2022 Iran Team Selection Test, 2

Tags: number theory

For a positive integer $n$, let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$, respectively. let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n\in \mathbb{N}$. Prove that each prime factor of $\tau(a)$ divides $\tau(b)$. Proposed by MohammadAmin Sharifi

2005 Bosnia and Herzegovina Team Selection Test, 5

Tags: permutation , identical , set theory

If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$, $k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$

1983 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , inequalities

Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n\plus{}1}\minus{}1)$ in base $ 10$ are $ 28$.

1949-56 Chisinau City MO, 5

Tags: number theory , perfect square , last digit

Prove that the square of any integer cannot end with two fives.

the 15th XMO, 2

Tags: inequalities

$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of $$ \sum_{1\leq i , j \leq n , i\ne j}|a_{i}-a^2_j|$$

1973 All Soviet Union Mathematical Olympiad, 178

Tags: algebra , trinomial , inequalities

The real numbers $a,b,c$ satisfy the condition: for all $x$, such that for $ -1 \le x \le 1$, the inequality $$| ax^2 + bx + c | \le 1$$ is held. Prove that for the same $x$ , $$| cx^2 + bx + a | \le 2$$

2013 Harvard-MIT Mathematics Tournament, 16

Tags: hmmt , geometric transformation , reflection , pythagorean theorem , geometry

The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball.

1984 Iran MO (2nd round), 6

Tags:

Let $D$ and $D'$ be two lines with the equations \[\frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \quad \text{and} \quad \frac{x+1}{2} = \frac{y+2}{4} = \frac{z-1}{3}.\] Find the length of their common perpendicular.

1992 China Team Selection Test, 2

Tags: geometry , geometric transformation , rotation , induction , rectangle , combinatorics

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

2013 Dutch BxMO/EGMO TST, 4

Tags: functional equation , determine all functions , algebra

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

1999 China Second Round Olympiad, 2

Tags: trigonometry , complex number , algebra

Let $a$,$b$,$c$ be real numbers. Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$ Find $|az_{1}+bz_{2}+cz_{3}|$.

2020 BMT Fall, 1

Tags: probability , combinatorics

Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2011 Indonesia TST, 2

Tags: floor function , graph , graph theory , combinatorics

A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$. Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.

2009 Danube Mathematical Competition, 3

Tags: combinatorics , combinatorial geometry , equilateral triangle , equilateral , minimum

Let $n$ be a natural number. Determine the minimal number of equilateral triangles of side $1$ to cover the surface of an equilateral triangle of side $n +\frac{1}{2n}$.

2005 Tournament of Towns, 3

Tags:

Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings. [i](4 points)[/i]

2023 Sharygin Geometry Olympiad, 2

Tags: rectangle , perpendicular lines , geometry

The diagonals of a rectangle $ABCD$ meet at point $E$. A circle centered at $E$ lies inside the rectangle. Let $CF$, $DG$, $AH$ be the tangents to this circle from $C$, $D$, $A$; let $CF$ meet $DG$ at point $I$, $EI$ meet $AD$ at point $J$, and $AH$ meet $CF$ at point $L$. Prove that $LJ$ is perpendicular to $AD$.

1942 Putnam, A5

Tags: ratio , torus

A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$. For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?

2023 Romania Team Selection Test, P3

Tags: geometry , arc midpoint , cyclic quadrilateral

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

1979 USAMO, 5

Tags: modular arithmetic

A certain organization has $n$ members, and it has $n\plus{}1$ three-member committees, no two of which have identical member-ship. Prove that there are two committees which share exactly one member.