Olympiad problems

1977 IMO Shortlist, 8

Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

2024 AMC 10, 20

Tags: set

Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold: - If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$ - If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$. What is the maximum possible number of elements in $S$? $ \textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675 \qquad $

2001 IMO Shortlist, 2

Tags: geometry , circumcenter , triangle , geometric inequality

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

1971 AMC 12/AHSME, 10

Tags:

Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }13$

2023 AMC 10, 12

Tags: polynomial

When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive? $\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

2022 JHMT HS, 5

Tags: geometry

Suppose $\triangle JHU$ satisfies $JH = JU = 44$ and $HU = 32$. There is a unique circle passing through $U$ that is tangent to $\overline{JH}$ at its midpoint; let this circle intersect $\overline{JU}$ and $\overline{HU}$ again at points $X \neq U$ and $Y \neq U$, respectively. Let $Z$ be the unique point on $\overline{JH}$ such that $JZ = XU$. Compute the perimeter of quadrilateral $UXZY$.

2002 District Olympiad, 1

Tags: algebra

Let $x, y, z$ be positive real numbers such that $xyz(x+y+z) = 1$. Show that the following equality holds: $$\sqrt{\left( x^2+\frac{1}{y^2}\right)\left( y^2+\frac{1}{z^2}\right)\left( z^2+\frac{1}{x^2}\right)}=(x+y)(y+z)(z+x)$$ Find some numbers $x ,y ,z$ which satisfy the given property.

1935 Eotvos Mathematical Competition, 3

Tags: geometry , combinatorics , combinatorial geometry , 3d geometry , prism

A real number is assigned to each vertex of a triangular prism so that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. Prove that all six numbers are equal.

2007 IMC, 5

Tags: function , modular arithmetic , algebra , polynomial , induction

Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.

1958 Czech and Slovak Olympiad III A, 4

Tags: locus , geometry , 3d geometry

Consider positive numbers $d,v$ such that $d>v$. Moreover, consider two perpendicular skew lines $p,q$ of distance $v$ (that is direction vectors of both lines are orthogonal and $\min_{X\in p,Y\in q}XY = v$). Finally, consider all line segments $PQ$ such that $P\in p, Q\in q, PQ=d$. a) Find the locus of all points $P$. b) Find the locus of all midpoints of segments $PQ$.

2015 German National Olympiad, 4

Tags: divisibility , number theory

Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions: a) None of the numbers $n_k$ is divisible by $13$. b) Infinitely many of the numbers $n_k$ are divisible by $17$.

2013 Bangladesh Mathematical Olympiad, 4

Tags: algebra

Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.

LMT Team Rounds 2021+, A15 B20

Tags:

Andy and Eddie play a game in which they continuously flip a fair coin. They stop flipping when either they flip tails, heads, and tails consecutively in that order, or they flip three tails in a row. Then, if there has been an odd number of flips, Andy wins, and otherwise Eddie wins. Given that the probability that Andy wins is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Anderw Zhao and Zachary Perry[/i]

2007 Ukraine Team Selection Test, 11

Tags: combinatorics

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

2006 China Team Selection Test, 1

Tags: trapezoid , incenter , geometric transformation , homothety , ratio , geometry

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

2009 Poland - Second Round, 3

Tags: vector , linear algebra , algebra

For every integer $n\ge 3$ find all sequences of real numbers $(x_1,x_2,\ldots ,x_n)$ such that $\sum_{i=1}^{n}x_i=n$ and $\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n$, where $x_0=x_n$ and $x_{n+1}=x_1$.

2005 District Olympiad, 2

Tags: incenter , inradius , symmetry , geometry

Let $ABC$ be a triangle inscribed in a circle of center $O$ and radius $R$. Let $I$ be the incenter of $ABC$, and let $r$ be the inradius of the same triangle, $O\neq I$, and let $G$ be its centroid. Prove that $IG\perp BC$ if and only if $b=c$ or $b+c=3a$.

2021-2022 OMMC, 8

Tags: combinatorics , expected value

Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$. Find $a + b$. [i]Proposed by Isaac Chen[/i]

2019 Germany Team Selection Test, 2

Tags: symmetry , angle chasing , miquel point , geometry

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

1968 All Soviet Union Mathematical Olympiad, 099

Tags: geometry , diagonal

The difference between the maximal and the minimal diagonals of the regular $n$-gon equals to its side ( $n > 5$ ). Find $n$.

1988 Romania Team Selection Test, 16

Tags: combinatorics

The finite sets $A_1$, $A_2$, $\ldots$, $A_n$ are given and we denote by $d(n)$ the number of elements which appear exactly in an odd number of sets chosen from $A_1$, $A_2$, $\ldots$, $A_n$. Prove that for any $k$, $1\leq k\leq n$ the number \[{ d(n) - \sum\limits^n_{i=1} |A_i| + 2\sum\limits_{ i<j} |A_i \cap A_j | - \cdots + (-1)^k2^{k-1} \sum\limits_{i_1 <i_2 <\cdots < i_k} | A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_k}}| \] is divisible by $2^k$. [i]Ioan Tomescu, Dragos Popescu[/i]

2017 India IMO Training Camp, 1

Tags: algebra , polynomial

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

2022-23 IOQM India, 15

Tags: algebra , inequalities

Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\ $\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\. \\ If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$.

1998 National Olympiad First Round, 4

Tags: quadratic , function

$ x,y,z\in \mathbb R$, find the minimal value of $ f\left(x,y,z\right) = 2x^{2} + 5y^{2} + 10z^{2} - 2xy - 4yz - 6zx + 3$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ -3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$

2013 India IMO Training Camp, 2

Tags: modular arithmetic , number theory , equation

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.