Olympiad problems

2007 Junior Balkan MO, 3

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

2020 CMIMC Team, 3

Tags: team , 2020

Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.

2006 Belarusian National Olympiad, 6

Tags: combinatorics , table , Sum , max

An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum? (V. Lebed) [img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]

2009 Romania Team Selection Test, 3

Tags: function , complex analysis , trigonometry , inequalities , floor function , complex numbers , algebra proposed

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

2015 ELMO Problems, 5

Tags: Elmo , combinatorics

Let $m, n, k > 1$ be positive integers. For a set $S$ of positive integers, define $S(i,j)$ for $i<j$ to be the number of elements in $S$ strictly between $i$ and $j$. We say two sets $(X,Y)$ are a [i]fat[/i] pair if \[ X(i,j)\equiv Y(i,j) \pmod{n} \] for every $i,j \in X \cap Y$. (In particular, if $\left\lvert X \cap Y \right\rvert < 2$ then $(X,Y)$ is fat.) If there are $m$ distinct sets of $k$ positive integers such that no two form a fat pair, show that $m<n^{k-1}$. [i]Proposed by Allen Liu[/i]

2015 Purple Comet Problems, 11

Tags: Purple Comet

Suppose that the vertices of a polygon all lie on a rectangular lattice of points where adjacent points on the lattice are a distance 1 apart. Then the area of the polygon can be found using Pick’s Formula: $I + \frac{B}{2}$ −1, where I is the number of lattice points inside the polygon, and B is the number of lattice points on the boundary of the polygon. Pat applied Pick’s Formula to ﬁnd the area of a polygon but mistakenly interchanged the values of I and B. As a result, Pat’s calculation of the area was too small by 35. Using the correct values for I and B, the ratio n = $\frac{I}{B}$ is an integer. Find the greatest possible value of n.

2023 Harvard-MIT Mathematics Tournament, 9

Tags: geometry

Point $Y$ lies on line segment $XZ$ such that $XY = 5$ and $Y Z = 3$. Point $G$ lies on line $XZ$ such that there exists a triangle $ABC$ with centroid $G$ such that $X$ lies on line $BC$, $Y$ lies on line $AC$, and $Z$ lies on line $AB$. Compute the largest possible value of $XG$.

LMT Speed Rounds, 2016.13

Tags:

Find the area enclosed by the graph of $|x|+|2y|=12$. [i]Proposed by Nathan Ramesh

2010 Federal Competition For Advanced Students, P2, 1

Tags: algebra , inequalities

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

1975 Chisinau City MO, 93

Tags: algebra

Prove that $(a^2 + b^2 + c^2)^ 2 = 2 (a^4 + b^4 + c^4)$ if $a + b + c = 0$.

2016 District Olympiad, 3

Tags: inequalities

Let be nonnegative real numbers $ a,b,c, $ holding the inequality: $ \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. $ Prove that $ \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1. $

1985 AMC 8, 1

Tags:

$ \frac{3 \times 5}{9 \times 11} \times \frac{7 \times 9 \times 11}{3 \times 5 \times 7}\equal{}$ \[ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ \frac{1}{49} \qquad \textbf{(E)}\ 50 \]

2005 All-Russian Olympiad Regional Round, 8.2

Tags: combinatorics , game , game strategy

In the middle cell of the $1 \times 2005$ strip there is a chip. Two players each queues move it: first, the first player moves the piece one cell in any direction, then the second one moves it $2$ cells, the $1$st - by $4$ cells, the 2nd by $8$, etc. (the $k$-th shift occurs by $2^{k-1}$ cells). That, whoever cannot make another move loses. Who can win regardless of the opponent's play?

2015 Junior Regional Olympiad - FBH, 3

Tags: geometry , midpoint

Let $D$ be a midpoint of $BC$ of triangle $ABC$. On side $AB$ is given point $E$, and on side $AC$ is given point $F$ such that $\angle EDF = 90^{\circ}$. Prove that $BE+CF>EF$

1966 IMO Shortlist, 21

Tags: geometry , 3D geometry , area , Volume , geometric inequality , IMO Longlist , IMO Shortlist

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

2013 India National Olympiad, 6

Tags: algebra , polynomial , function , inequalities , logarithms , inequalities proposed

Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$

1927 Eotvos Mathematical Competition, 3

Tags: geometry , inradius , exradius , Geometric Inequalities

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

2024 Princeton University Math Competition, 3

Tags: Team Round

Let $f(x)=x^2-3x+1,$ and let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be the $4$ roots of $f(f(x))=x.$ Evaluate $\lfloor 10\alpha_1\rfloor+ $ $\lfloor 10\alpha_2\rfloor$ $+$ $\lfloor 10\alpha_3\rfloor+\lfloor 10\alpha_4\rfloor.$

2018 Balkan MO Shortlist, A6

Tags: inequalities

Let $ x_1, x_2, \cdots, x_n$ be positive real numbers . Prove that: $$\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}. $$

1992 India National Olympiad, 2

Tags: inequalities

If $x , y, z \in \mathbb{R}$ such that $x+y +z =4$ and $x^2 + y^2 +z^2 = 6$, then show that each of $x, y, z$ lies in the closed interval $\left[ \dfrac{2}{3} , 2 \right]$. Can $x$ attain the extreme value $\dfrac{2}{3}$ or $2$?

1993 AMC 12/AHSME, 13

Tags: geometry , perimeter , AMC

A square of perimeter $20$ is inscribed in a square of perimeter $28$. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? $ \textbf{(A)}\ \sqrt{58} \qquad\textbf{(B)}\ \frac{7\sqrt{5}}{2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ \sqrt{65} \qquad\textbf{(E)}\ 5\sqrt{3} $

1985 All Soviet Union Mathematical Olympiad, 395

Tags: geometry , hexagon , areas

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

Durer Math Competition CD Finals - geometry, 2017.D4

Tags: geometry , angles

The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$; a) Find the angle $\angle DPB$? b) Prove that $P$ lies on the diagonal $AC$.

2005 Serbia Team Selection Test, 3

Tags: polynomial , algebra , Sequence

Find all polynomial with real coefficients such that: P(x^2+1)=P(x)^2+1

2014 Singapore Senior Math Olympiad, 7

Tags: trigonometry , inequalities

Find the largest number among the following numbers: $ \textbf{(A) }\tan47^{\circ}+\cos47^{\circ}\qquad\textbf{(B) }\cot 47^{\circ}+\sqrt{2}\sin 47^{\circ}\qquad\textbf{(C) }\sqrt{2}\cos47^{\circ}+\sin47^{\circ}\qquad\textbf{(D) }\tan47^{\circ}+\cot47^{\circ}\qquad\textbf{(E) }\cos47^{\circ}+\sqrt{2}\sin47^{\circ} $