Olympiad problems

2020 Junior Balkan Team Selection Tests - Moldova, 2

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

2018 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: perpendicular bisector , geometry , circumcircle

Let $P$ be a point on circumcircle of triangle $ABC$ on arc $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$ intersect at point $E$, and lines $AC$ and $BP$ intersect at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that: ${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$

2007 Korea National Olympiad, 1

Tags: inequalities

For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality? $ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .

2005 Belarusian National Olympiad, 5

Tags: algebra , trigonometry

For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$ Find all possible values of $a+b+c+d$

1986 China Team Selection Test, 2

Tags: inequalities , function , perimeter , geometry , 3d geometry , tetrahedron

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

2020 Purple Comet Problems, 9

Tags: algebra

Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1. Find the product $abc$.

2006 ITAMO, 2

Tags: number theory

Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.

1994 Dutch Mathematical Olympiad, 2

Tags: algebra

A sequence of integers $ a_1,a_2,a_3,...$ is such that $ a_1\equal{}2, a_2\equal{}3$, and $ a_{n\plus{}1}\equal{}2a_{n\minus{}1}$ or $ 3a_n\minus{}2a_{n\minus{}1}$ for all $ n \ge 2$. Prove that no number between $ 1600$ and $ 2000$ can be an element of the sequence.

1997 Iran MO (2nd round), 3

Tags: combinatorics

We have a $n\times n$ table and we’ve written numbers $0,+1 \ or \ -1$ in each $1\times1$ square such that in every row or column, there is only one $+1$ and one $-1$. Prove that by swapping the rows with each other and the columns with each other finitely, we can swap $+1$s with $-1$s.

1992 Tournament Of Towns, (342) 4

Tags: geometry , geometric inequality , angle

(a) In triangle $ABC$, angle $A$ is greater than angle $B$. Prove that the length of side $BC$ is greater than half the length of side $AB$. (b) In the convex quadrilateral $ABCD$, the angle at $A$ is greater than the angle at $C$ and the angle at $D$ is greater than the angle at $B$. Prove that the length of side $BC$ is greater than half of the length of side $AD$. (F Nazarov)

2012 Stars of Mathematics, 2

Tags: geometric transformation , geometry

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$. ([i]Dan Schwarz[/i])

III Soros Olympiad 1996 - 97 (Russia), 9.5

Tags: algebra , geometric progression

For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

1991 Romania Team Selection Test, 7

Tags: algebra , inequalities , sum

Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

2016 Balkan MO, 3

Tags: polynomial , number theory

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

2014 PUMaC Algebra A, 5

Tags:

Real numbers $x$, $y$, and $z$ satisfy the following equality: \[4(x+y+z)=x^2+y^2+z^2\] Let $M$ be the maximum of $xy+yz+zx$, and let $m$ be the minimum of $xy+yz+zx$. Find $M+10m$.

Champions Tournament Seniors - geometry, 2011.2

Tags: right angle , geometry , isosceles

Let $ABC$ be an isosceles triangle in which $AB = AC$. On its sides $BC$ and $AC$ respectively are marked points $P$ and $Q$ so that $PQ\parallel AB$. Let $F$ be the center of the circle circumscribed about the triangle $PQC$, and $E$ the midpoint of the segment $BQ$. Prove that $\angle AEF = 90^o $.

1987 National High School Mathematics League, 3

Tags: irrational number

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are rational numbers, we call it rational point. If $a$ is an irrational number, then in all lines that passes $(a,0)$, $\text{(A)}$There are infinitely many lines, on which there are at least two rational points. $\text{(B)}$There are exactly $n(n\geq2)$ lines, on which there are at least two rational points. $\text{(C)}$There are exactly 1 line, on which there are at least two rational points. $\text{(D)}$Every line passes at least one rational point.

2010 Contests, 2

Tags: function , geometry , rhombus , vieta

Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

2024 Taiwan TST Round 1, G

Tags: geometry

For the quadrilateral $ABCD$, let $AC$ and $BD$ intersect at $E$, $AB$ and $CD$ intersect at $F$, and $AD$ and $BC$ intersect at $G$. Additionally, let $W, X, Y$, and $Z$ be the points of symmetry to $E$ with respect to $AB, BC, CD,$ and $DA$ respectively. Prove that one of the intersection points of $\odot(FWY)$ and $\odot(GXZ)$ lies on the line $FG$. [i]Proposed by chengbilly[/i]

2022 District Olympiad, P2

Tags: inequalities , complex number

Let $z_1,z_2$ and $z_3$ be complex numbers of modulus $1,$ such that $|z_i-z_j|\geq\sqrt{2}$ for all $i\neq j\in\{1,2,3\}.$ Prove that \[|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.\][i]Mathematical Gazette[/i]

2012 Danube Mathematical Competition, 2

Tags: circumcircle , geometry , similar triangles , orthocenter

Let $ABC$ be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.

2015 SDMO (High School), 2

Tags:

$N$ cards are arranged in a circle, with exactly one card face up and the rest face-down. In a turn, choose a proper divisor $k$ of $N$. You may begin at any card on the circle and flip every $k$-th card, counting clockwise, if and only if every $k$-th card begins the turn in the same orientation (either all face-up or all face-down). For example, with $15$ cards, you may start at any position and flip the $3$rd, $6$th, $9$th, $12$th, and $15$th cards around the circle if they all begin the turn face up (or all face-down). For what values of $N$ can all of the cards be flipped face-up in a finite number of turns?

2020 Nordic, 2

Tags: combinatorics

Georg has $2n + 1$ cards with one number written on each card. On one card the integer $0$ is written, and among the rest of the cards, the integers $k = 1, ... , n$ appear, each twice. Georg wants to place the cards in a row in such a way that the $0$-card is in the middle, and for each $k = 1, ... , n$, the two cards with the number $k$ have the distance $k$ (meaning that there are exactly $k - 1$ cards between them). For which $1 \le n \le 10$ is this possible?

2023 HMNT, 1

Tags: algebra

Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler’s series.

2008 International Zhautykov Olympiad, 2

Tags: polynomial , algebra , factorization , sum of cubes

A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good. a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good? b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good? Justify your answers.