Olympiad problems

2012 India PRMO, 14

$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$, and $C$. What is the magnitude of $\angle B AC$ in degrees?

2022 All-Russian Olympiad, 6

Tags: algebra , quadratic

What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac {1}{1000}$?

2023 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

Let $f(x), g(x)$ be real polynomials of degrees $2$ and $3$, respectively. Could it happen that $f(g(x))$ has $6$ distinct roots, which are powers of $2$?

2009 Today's Calculation Of Integral, 470

Tags: calculus , integration , geometry , parabola , calculus computations , conic

Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.

2020 Iran RMM TST, 2

Tags: geometry

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

2015 Cuba MO, 9

Tags: algebra , inequalities

Determine the largest possible value of$ M$ for which it holds that: $$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,$$ for all real numbers $x, y, z > 0$ that satisfy the equation $xy + yz + zx = 1$.

2016 Harvard-MIT Mathematics Tournament, 29

Tags:

Katherine has a piece of string that is $2016$ millimeters long. She cuts the string at a location chosen uniformly at random, and takes the left half. She continues this process until the remaining string is less than one millimeter long. What is the expected number of cuts that she makes?

2014 PUMaC Algebra B, 7

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 $M+10m$.

1991 Bulgaria National Olympiad, Problem 6

Tags: combinatorics , game

White and black checkers are put on the squares of an $n\times n$ chessboard $(n\ge2)$ according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker.

1996 Putnam, 1

Tags: geometry , rectangle

Find the least number $A$ such that for any two squares of combined area $1$, a rectangle of area $A$ exists such that the two squares can be packed in the rectangle (without the interiors of the squares overlapping) . You may assume the sides of the squares will be parallel to the sides of the rectangle.

2017 Junior Regional Olympiad - FBH, 1

Tags: algebra , function , functional equation

It is given function $f(x)=3x-2$ $a)$ Find $g(x)$ if $f(2x-g(x))=-3(1+2m)x+34$ $b)$ Solve the equation: $g(x)=4(m-1)x-4(m+1)$, $m \in \mathbb{R}$

1948 Moscow Mathematical Olympiad, 144

Tags: integer , number theory

Prove that if $\frac{2^n- 2}{n} $ is an integer, then so is $\frac{2^{2^n-1}-2}{2^n - 1}$ .

2021 Miklós Schweitzer, 1

Tags: linear algebra , number theory

Let $n, m \in \mathbb{N}$; $a_1,\ldots, a_m \in \mathbb{Z}^n$. Show that nonnegative integer linear combinations of these vectors give exactly the whole $\mathbb{Z}^n$ lattice, if $m \ge n$ and the following two statements are satisfied: [list] [*] The vectors do not fall into the half-space of $\mathbb{R}^n$ containing the origin (i.e. they do not fall on the same side of an $n-1$ dimensional subspace), [*] the largest common divisor (not pairwise, but together) of $n \times n$ minor determinants of the matrix $(a_1,\ldots, a_m)$ (which is of size $m \times n$ and the $i$-th column is $a_i$ as a column vector) is $1$. [/list]

2018 Bosnia and Herzegovina EGMO TST, 2

Tags: positive integer , divide , number theory

Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

2023-24 IOQM India, 8

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Given a $2 \times 2$ tile and seven dominoes ( $2 \times 1$ tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a $2 \times 7$ rectangle using some of these tiles.

2020 Brazil Cono Sur TST, 3

Tags: combinatorics

Between the states of Alinaesquina and Berlinda, each road connects one city of Alinaesquina to one city of Berlinda. All the roads are in two-ways, and between any two cities, it is possible to travel from one to the other, using only these (possibly more than one) roads. Furthermore, it is known that, from any city of anyone of the two states, the same number of $k$ roads are going out. We know that $k\geq 2$. Prove that governments of the states can close anyone of the roads, and there will still be a route (possibly through several roads) between any two cities.

2013 Putnam, 1

Tags: geometry , 3d geometry , icosahedron , probability , expected value

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.

V Soros Olympiad 1998 - 99 (Russia), 11.7

Tags: inequalities , algebra , trigonometry

Prove that for all positive and admissible values of $x$ the following inequality holds: $$\sin x + arc \sin x>2x$$

1982 National High School Mathematics League, 3

Tags:

If $\log_2(\log_{\frac{1}{2}}(\log_2x))=\log_3(\log_{\frac{1}{3}}(\log_3y))=\log_5(\log_{\frac{1}{5}}(\log_5z))=0$, then $\text{(A)}z<x<y\qquad\text{(B)}x<y<z\qquad\text{(C)}y<z<x\qquad\text{(D)}z<y<x$

III Soros Olympiad 1996 - 97 (Russia), 11.5

Tags: geometry , parallelogram

The area of a convex quadrilateral is $S$, and the angle between the diagonals is $a$. On the sides of this quadrilateral, as on the bases, isosceles triangles with vertex angle equal to $\phi$, wherein two opposite triangles are located on the other side of the corresponding side of the quadrilateral than the quadrilateral itself, and the other two are located on the other side. Prove that the vertices of the constructed triangles, different from the vertices of the quadrilateral, serve as the vertices of a parallelogram. Find the area of this parallelogram.

1998 National Olympiad First Round, 9

Tags: geometry

$ C_{1}$ and $ C_{2}$ be two externally tangent circles with diameter $ \left[AB\right]$ and $ \left[BC\right]$, with center $ D$ and $ E$, respectively. Let $ F$ be the intersection point of tangent line from A to $ C_{2}$ and tangent line from $ C$ to $ C_{1}$ (both tangents line on the same side of $ AC$). If $ \left|DB\right|\equal{}\left|BE\right|\equal{}\sqrt{2}$, find the area of triangle $ AFC$. $\textbf{(A)}\ \frac{7\sqrt{3} }{2} \qquad\textbf{(B)}\ \frac{9\sqrt{2} }{2} \qquad\textbf{(C)}\ 4\sqrt{2} \qquad\textbf{(D)}\ 2\sqrt{3} \qquad\textbf{(E)}\ 2\sqrt{2}$

2000 Tournament Of Towns, 1

Tags: sum , combinatorics , table , square table

Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table. (R Zhenodarov)

1999 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , probability

A circle is randomly chosen in a circle of radius $1$ in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?

2018 Singapore MO Open, 2

Tags: geometry , smo

Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.

Kvant 2024, M2780

Tags: graph theory , chromatic number , combinatorics

Consider a natural number $n\geqslant 3$ and a graph $G{}$ with a chromatic number $\chi(G)=n$ which has more than $n{}$ vertices. Prove that there exist two vertex-disjoint subgraphs $G_1{}$ and $G_2{}$ of $G{}$ such that $\chi(G_1)+\chi(G_2)\geqslant n+1.$ [i]Proposed by V. Dolnikov[/i]