Olympiad problems

2022 European Mathematical Cup, 1

Determine all positive integers $n$ for which there exist positive divisors $a$, $b$, $c$ of $n$ such that $a>b>c$ and $a^2 - b^2$, $b^2 - c^2$, $a^2 - c^2$ are also divisors of $n$.

2020-2021 OMMC, 8

Tags: geometry

Let triangle $MAD$ be inscribed in circle $O$ with diameter $85$ such that $MA = 68$ and $DA = 40$. The altitudes from $M, D$ to sides $AD$ and $MA$, respectively, intersect the tangent to circle $O$ at $A$ at $X$ and $Y$ respectively. $XA \times YA$ can be expressed as $\frac{a}{b}$, where $a$ and $ b$ are relatively prime positive integers. Find $a + b$.

2014 Dutch IMO TST, 3

Tags: circumcircle , geometric transformation , reflection , trapezoid , geometry

Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.

2023 JBMO TST - Turkey, 2

Tags: chessboard , combinatorics

A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

2014 Olympic Revenge, 1

Tags: circumcircle , projective geometry , geometry

Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

2013 National Olympiad First Round, 18

Tags: modular arithmetic

What is remainder when the sum \[\binom{2013}{1}+2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\] is divided by $41$? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None} $

2021 Purple Comet Problems, 2

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A furniture store set the sticker price of a table $40$ percent higher than the wholesale price that the store paid for the table. During a special sale, the table sold for $35$ percent less than this sticker price. Find the percent the final sale price was of the original wholesale price of the table.

2008 Balkan MO Shortlist, G3

Tags: geometry , perimeter , area of a triangle , bisector , orthocenter , angle

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

Russian TST 2017, P3

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

2014 Canadian Mathematical Olympiad Qualification, 5

Tags: algebra , polynomial , irreducibility

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

2000 Harvard-MIT Mathematics Tournament, 9

Tags: algebra , polynomial

A cubic polynomial $f$ satisfies $f(0)=0, f(1)=1, f(2)=2, f(3)=4$. What is $f(5)$?

LMT Team Rounds 2010-20, 2020.S30

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Let $ABCD$ be a cyclic quadrilateral such that the ratio of its diagonals is $AC:BD=7:5.$ Let $E$ and $F$ be the intersections of lines $AB$ and $CD$ and lines $BC$ and $AD$, respectively. Let $L$ and $M$ be the midpoints of diagonals $AC$ and $BD$, respectively. Given that $EF=2020,$ the length of $LM$ can be written as $\frac{p}{q}$ where $p,q$ are relatively prime positive integers. Compute $p+q.$

2010 Contests, 1

Tags: number theory

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.

1995 Poland - First Round, 10

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Prove that the equation $x^x = y^3+z^3$ has infinitely many solutions in positive integers $x,y,z$.

1984 IMO Longlists, 30

Tags: geometric sequence , number theory

Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.

2023 Singapore Senior Math Olympiad, 1

Tags: geometry

Let $ABCD$ be a square, $E$ be a point on the side $DC$, $F$ and $G$ be the feet of the altitudes from $B$ to $AE$ and from $A$ to $BE$, respectively. Suppose $DF$ and $CG$ intersect at $H$. Prove that $\angle AHB=90^\circ$.

2022 LMT Spring, 7

Tags: geometry , combinatorics

Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img] Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way. [img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]

2019 Peru Cono Sur TST, P1

Tags: fraction , number theory , inequalities

Find all a positive integers $a$ and $b$, such that $$\frac{a^b+b^a}{a^a-b^b}$$ is an integer

1981 Miklós Schweitzer, 5

Tags: advanced fields , geometry , 3d geometry , vector

Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$. [i]J. Szucs[/i]

2023 Harvard-MIT Mathematics Tournament, 16

Tags: guts

The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.

1984 Tournament Of Towns, (059) A4

Tags: geometry , cut , right isosceles , similar triangles

Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size. (AV Savkin)

1997 Baltic Way, 6

Tags: modular arithmetic , number theory

Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and \[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]

2002 Irish Math Olympiad, 1

Tags: geometry

In a triangle $ ABC$ with $ AB\equal{}20, AC\equal{}21$ and $ BC\equal{}29$, points $ D$ and $ E$ are taken on the segment $ BC$ such that $ BD\equal{}8$ and $ EC\equal{}9$. Calculate the angle $ \angle DAE$.

2018 Bulgaria National Olympiad, 6.

Tags: combinatorics , graph theory

On a planet there are $M$ countries and $N$ cities. There are two-way roads between some of the cities. It is given that: (1) In each county there are at least three cities; (2) For each country and each city in the country is connected by roads with at least half of the other cities in the countries; (3) Each city is connceted with exactly one other city ,that is not in its country; (4) There are at most two roads between cities from cities in two different countries; (5) If two countries contain less than $2M$ cities in total then there is a road between them. Prove that there is cycle of lenght at least $M+\frac{N}{2}$.

2014 PUMaC Individual Finals B, 1

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Let $A, B$ be two points on circle $\gamma$. At point $A$ and $B$ we construct tangents to $\gamma$, $AC$ and $BD$ respectively such that the tangents are both in the clockwise direction. Let the intersection between $AB$ and $CD$ be $P$ . If $AC = BD$, prove that $P$ bisects the line $CD$.