Olympiad problems

2009 Croatia Team Selection Test, 2

Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad \equal{} bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.

2014 Iran Team Selection Test, 2

Tags: geometry , incenter , circumcircle , geometric transformation

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

1958 February Putnam, A1

Tags: root , polynomial

If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying $$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$ show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.

2004 Postal Coaching, 3

Tags: number theory , inequalities

Let $a,b,c,d,$ be real and $ad-bc = 1$. Show that $Q = a^2 + b^2 + c^2 + d^2 + ac +bd$ $\not= 0, 1, -1$

1998 Finnish National High School Mathematics Competition, 3

Tags: ratio , geometric sequence , algebra

Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$ Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$

1968 Polish MO Finals, 6

Tags: combinatorial geometry , combinatorics , geometry

Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements: (a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle. (b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.

2019 PUMaC Team Round, 9

Tags: number theory , algebra

Find the integer $\sqrt[5]{55^5 + 3183^5 + 28969^5 + 85282^5}$.

II Soros Olympiad 1995 - 96 (Russia), 11.7

Tags: geometry , 3d geometry , prism , volume

Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.

2001 Junior Balkan MO, 3

Tags: trigonometry , geometry

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? [i]Greece[/i]

2024 Dutch IMO TST, 1

Tags: circumcircle , tangent , geometry

Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.

2021 USMCA, 11

Tags:

Let $f_1 (x) = x^2 - 3$ and $f_n (x) = f_1(f_{n-1} (x))$ for $n \ge 2$. Let $m_n$ be the smallest positive root of $f_n$, and $M_n$ be the largest positive root of $f_n$. If $x$ is the least number such that $M_n \le m_n \cdot x$ for all $n \ge 1$, compute $x^2$.

2013 India IMO Training Camp, 1

Tags: number theory

For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

1962 AMC 12/AHSME, 6

Tags: geometry , perimeter

A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $ \textbf{(A)}\ \frac{9}{2} \qquad \textbf{(B)}\ 2 \sqrt{5} \qquad \textbf{(C)}\ 4 \sqrt{2} \qquad \textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad \textbf{(E)}\ \text{none of these}$

2002 Moldova National Olympiad, 1

Tags: ratio

Volume $ A$ equals one fourth of the sum of the volumes $ B$ and $ C$, while volume $ B$ equals one sixth of the sum of the volumes $ C$ and $ A$. Find the ratio of the volume $ C$ to the sum of the volumes $ A$ and $ B$.

1995 Irish Math Olympiad, 5

Tags: function , vector , algebra

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$: $ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.

2017 AMC 12/AHSME, 19

Tags:

A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$? $\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$

2017 Math Prize for Girls Olympiad, 2

Tags:

Let $n$ be a positive integer. Prove that there exist polynomials $P$ and $Q$ with real coefficients such that for every real number $x$, we have $P(x) \ge 0$, $Q(x) \ge 0$, and \[ 1 - x^n = (1 - x)P(x) + (1 + x)Q(x). \]

1974 IMO Longlists, 19

Tags: geometry

Prove that there exists, for $n \geq 4$, a set $S$ of $3n$ equal circles in space that can be partitioned into three subsets $s_5, s_4$, and $s_3$, each containing $n$ circles, such that each circle in $s_r$ touches exactly $r$ circles in $S.$

1997 Tournament Of Towns, (537) 2

Tags: number theory , divisible , divide

Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

2015 Korea - Final Round, 5

Tags: number theory , recurrence

For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$. They are defined inductively, by the following recurrences. $A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.

1999 Greece Junior Math Olympiad, 2

Tags: number theory , minimum , maximum

Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$. Determine the minimum and the maximum of $x$ in terms of $n$.

2016 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: function , functional equation , algebra

Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$

2021 Saudi Arabia Training Tests, 33

Tags: number theory , inequalities

Call a positive integer $x$ to be [i]remote from squares and cubes [/i] if each integer $k$ satisfies both $|x - k^2| > 10^6$ and $|x - k^3| > 10^6$. Prove that there exist infinitely many positive integer $n$ such that $2^n$ is remote from squares and cubes.

2018 Thailand TST, 4

Tags: collinear , geometry , rectangle , orthocenter , circumcenter

Let $\vartriangle ABC$ be an acute triangle with altitudes $AA_1, BB_1, CC_1$ and orthocenter $H$. Let $K, L$ be the midpoints of $BC_1, CB_1$. Let $\ell_A$ be the external angle bisector of $\angle BAC$. Let $\ell_B, \ell_C$ be the lines through $B, C$ perpendicular to $\ell_A$. Let $\ell_H$ be the line through $H$ parallel to $\ell_A$. Prove that the centers of the circumcircles of $\vartriangle A_1B_1C_1, \vartriangle AKL$ and the rectangle formed by $\ell_A, \ell_B, \ell_C, \ell_H$ lie on the same line.

2021 Junior Balkan Team Selection Tests - Romania, P3

Tags: geometry , quadrilateral

Let $ABCD$ be a convex quadrilateral with angles $\sphericalangle A, \sphericalangle C\geq90^{\circ}$. On sides $AB,BC,CD$ and $DA$, consider the points $K,L,M$ and $N$ respectively. Prove that the perimeter of $KLMN$ is greater than or equal to $2\cdot AC$.