Olympiad problems

1994 North Macedonia National Olympiad, 1

Let $ a_1, a_2, ..., a_ {1994} $ be integers such that $ a_1 + a_2 + ... + a_{1994} = 1994 ^{1994} $ . Determine the remainder of the division of $ a ^ 3_1 + a ^ 3_2 + ... + a ^ 3_{1994} $ with $6$.

1966 IMO Shortlist, 28

Tags: geometry , locus , locus problems , circles

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

1974 Polish MO Finals, 4

Tags: inequalities

Prove that, so have $k$ for $\forall a_1,a_2,...,a_n$ satisfying $$|\sum_{i=1}^k a_i -\sum_{j=k+1}^n a_j |\leq \max_{1\leq m\leq n} |a_m|$$

2022 HMNT, 4

Tags:

Let $ABCD$ be a square of side length $2.$ Let points $X, Y,$ and $Z$ be constructed inside $ABCD$ such that $ABX, BCY,$ and $CDZ$ are equilateral triangles. Let point $W$ be outside $ABCD$ such that triangle $DAW$ is equilateral. Let the area of $WXYZ$ be $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$

2018 Indonesia MO, 4

Tags: algebra

In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial $x^2 + mx + n$ where $m,n \in \mathbb{Z}$, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form $x^2 + ax + b$ into either $x^2 + (a+b)x + b$ or $x^2 + ax + (a+b)$. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of $x$ and the constant of the polynomial. Andi loses if he can't continue to play. Find all $(m,n)$ such that Andi always loses (in finitely many turns).

2018 CMIMC Team, 2-1/2-2

Tags: team

Suppose that $a$ and $b$ are non-negative integers satisfying $a + b + ab + a^b = 42$. Find the sum of all possible values of $a + b$. Let $T = TNYWR$. Suppose that a sequence $\{a_n\}$ is defined via $a_1 = 11, a_2 = T$, and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 3$. Find $a_{19} + a_{20}$.

1988 Tournament Of Towns, (180) 3

Tags: polynomial , algebra , integer polynomial

It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .

2003 Canada National Olympiad, 1

Tags:

Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let $m$ be an integer, with $1 \leq m \leq 720$. At precisely $m$ minutes after 12:00, the angle made by the hour hand and minute hand is exactly $1^\circ$. Determine all possible values of $m$.

Brazil L2 Finals (OBM) - geometry, 2012.3

Tags: geometry , concurrency

Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.

1957 Moscow Mathematical Olympiad, 359

Tags: geometry , locus , perpendicular

Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.

2002 Brazil National Olympiad, 2

Tags: perimeter , cyclic quadrilateral , geometry

$ABCD$ is a cyclic quadrilateral and $M$ a point on the side $CD$ such that $ADM$ and $ABCM$ have the same area and the same perimeter. Show that two sides of $ABCD$ have the same length.

1995 Polish MO Finals, 1

Tags: inequalities

The positive reals $x_1, x_2, ... , x_n$ have harmonic mean $1$. Find the smallest possible value of $x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}$.

2018 China National Olympiad, 1

Tags: number theory , relatively prime , greatest common divisor , inequalities

Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying $$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$ are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$. a) Prove that $A_n$ is finite if and only if $n \not = 2$. b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that $$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$

2014 ASDAN Math Tournament, 1

Tags:

Alex gets $8$ points on an exam, while his friend gets $3$ times as many points as Alex. What is the average of their scores?

2018 Yasinsky Geometry Olympiad, 4

Tags: geometry , angle chasing , isosceles , excircle , circumcircle

Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the intersection point of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles. (Mykola Moroz)

2013 BMT Spring, 4

Tags: semicircle , square , geometry

Let $ABCD$ be a square with side length $2$, and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle?

2013 South East Mathematical Olympiad, 5

Tags: floor function , calculus , algebra

$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called [i]good[/i] if $f(x)=n$ has real root. How many good numbers are in $\{1,3,5,\dotsc,2013\}$?

2013 Online Math Open Problems, 3

Tags:

A [i]palindromic table[/i] is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. \[ \begin{array}[h]{ccc} O & M & O \\ N & M & N \\ O & M & O \end{array} \] How many palindromic tables are there that use only the letters $O$ and $M$? (The table may contain only a single letter.) [i]Proposed by Evan Chen[/i]

2022 Yasinsky Geometry Olympiad, 3

Tags: lemoine point , perpendicular , geometry

Given a triangle $ABC$, in which the medians $BE$ and $CF$ are perpendicular. Let $M$ is the intersection point of the medians of this triangle, and $L$ is its Lemoine point (the intersection point of lines symmetrical to the medians with respect to the bisectors of the corresponding angles). Prove that $ML \perp BC$. (Mykhailo Sydorenko)

1999 Spain Mathematical Olympiad, 1

Tags: parabola , analytic geometry , area of a triangle , tangent , geometry , conic

The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.

2016 Kyiv Mathematical Festival, P1

Tags: equation , algebra

Prove that for every positive integers $a$ and $b$ there exist positive integers $x$ and $y$ such that $\dfrac{x}{y+a}+\dfrac{y}{x+b}=\dfrac{3}{2}.$

2009 Mathcenter Contest, 3

Tags: algebra , inequalities

Let $x,y,z>0$ Prove that $$\frac{x^2+2}{\sqrt{z^2+xy}}+\dfrac{y^2+2}{\sqrt{x ^2+yz}}+\dfrac{z^2+2}{\sqrt{y^2+zx}}\geq 6$$. [i](nooonuii)[/i]

2004 Turkey Team Selection Test, 1

Tags: geometry , geometric transformation , rotation , combinatorics

An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed.

2019 Flanders Math Olympiad, 1

Tags: geometry , 3d geometry , sphere , cylinder

Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$ [img]https://1.bp.blogspot.com/-O4B3P3bghFs/Xy1fDv9zGkI/AAAAAAAAMSQ/ePLVnsXsRi0mz3SWBpIzfGdsizWoLmGVACLcBGAsYHQ/s0/flanders%2B2019%2Bp1.png[/img]

2012 Indonesia TST, 2

Tags: combinatorics

Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.