Olympiad problems

2019 Germany Team Selection Test, 2

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.

LMT Speed Rounds, 8

Tags: combinatorics

To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$. [i]Proposed by Christopher Cheng[/i]

2000 Mongolian Mathematical Olympiad, Problem 6

Tags: geometry , angle bisector , ratio , cyclic quadrilateral

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

1997 Vietnam Team Selection Test, 2

Tags: algebra , logarithm

Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.

2024 Pan-African, 2

Tags: geometry

In triangle $ABC$,let $M$ be the midpoint of the side $BC$,and $N$ is the midpoint of the segment $AM$,the circle going through $N$ and tangent the line $AC$ at $A$ intersects the segment $AB$ again in $P$. prove that the circumcircle of triangle $BPM$ is tangent the line $AM$

1988 Putnam, A3

Tags:

Determine, with proof, the set of real numbers $x$ for which \[ \sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x \] converges.

2012 USA TSTST, 7

Tags: circumcircle , vector , symmetry , rhombus , gliding principle , geometry

Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.

2010 Paenza, 3

Tags: limit , trigonometry , real analysis

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

2014 ELMO Shortlist, 6

Tags: inequalities

Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2005 Spain Mathematical Olympiad, 1

Tags: algebra , number theory

Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$ has an integer solution.

1999 Miklós Schweitzer, 3

Tags: graph theory

Prove that for any finite graph G there is a constant c(G)>0 such that for every n-point graph that does not have an induced subgraph isomorphic to G, there are two disjoint sets of vertices, each with at least $n^{c(G)}$ elements, between which either all edges are connected, or none of the edges are.

2011 Stars Of Mathematics, 3

Tags: induction , inequalities

For a given integer $n\geq 3$, determine the range of values for the expression \[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\] over real numbers $x_1,x_2,\ldots,x_n \geq 1$ satisfying $|x_k - x_{k+1}| \leq 1$ for all $1\leq k \leq n-1$. Do also determine when the extremal values are achieved. (Dan Schwarz)

2017 Princeton University Math Competition, A7

Tags: geometry , cyclic quadrilateral , circumcircle , number theory , relatively prime

Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$. Let $Q$ be the unique point on $\omega$ such that $\overline{QC}$ intersects $\overline{AB}$ at a point $Q_1$, $\overline{QD}$ intersects $\overline{AB}$ at a point $Q_2$, $Q_1$ is closer to $B$ than $P_1$ is to $B$, and $P_2Q_2=2$. The length of $P_1Q_1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2012 China Team Selection Test, 1

Tags: circumcircle , trigonometry , geometry

In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.

1979 Romania Team Selection Tests, 2.

Tags: geometry , 3d geometry , pyramid

Let $VA_1A_2A_3A_4$ be a pyramid with the vertex at $V$. Let $M,\, N,\, P$ be the midpoints of the segments $VA_1$, $VA_3$, and $A_2A_4$. Show that the plane $(MNP)$ cuts the pyramid into two parts with the same volume. [i]Radu Gologan[/i]

1981 Dutch Mathematical Olympiad, 2

Tags: equilateral , geometry

Given is the equilateral triangle $ABC$ with center $M$. On $CA$ and $CB$ the respective points $D$ and $E$ lie such that $CD = CE$. $F$ is such that $DMFB$ is a parallelogram. Prove that $\vartriangle MEF$ is equilateral.

2014 Saudi Arabia Pre-TST, 1.4

Tags: combinatorics , coloring , chessboard

Majid wants to color the cells of an $n\times n$ chessboard into white and black so that each $2\times 2$ subsquare contains two white cells and two black cells. In how many ways can Majid color this $n\times n$ chessboard?

1981 Miklós Schweitzer, 1

Tags: advanced fields

We are given an infinite sequence of $ 1$'s and $ 2$'s with the following properties: (1) The first element of the sequence is $ 1$. (2) There are no two consecutive $ 2$'s or three consecutive $ 1$'s. (3) If we replace consecutive $ 1$'s by a single $ 2$, leave the single $ 1$'s alone, and delete the original $ 2$'s, then we recover the original sequence. How many $ 2$'s are there among the first $ n$ elements of the sequence? [i]P. P. Palfy[/i]

2014 Tuymaada Olympiad, 8

Tags: combinatorics

There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong. [i](K. Kokhas)[/i]