Olympiad problems

2011 Pre - Vietnam Mathematical Olympiad, 4

Tags: combinatorics

For a table $n \times 9$ ($n$ rows and $9$ columns), determine the maximum of $n$ that we can write one number in the set $\left\{ {1,2,...,9} \right\}$ in each cell such that these conditions are satisfied: 1. Each row contains enough $9$ numbers of the set $\left\{ {1,2,...,9} \right\}$. 2. Any two rows are distinct. 3. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.

1988 IMO Longlists, 4

Tags: circumcircle , trigonometry , inradius , geometric inequality , geometry

The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$

2025 Harvard-MIT Mathematics Tournament, 29

Tags: guts

Points $A$ and $B$ lie on circle $\omega$ with center $O.$ Let $X$ be a point inside $\omega.$ Suppose that $XO=2\sqrt{2}, XA=1, XB=3,$ and $\angle{AXB}=90^\circ.$ Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle{AXB}$ and $\triangle{YXZ}$ are similar with the same orientation. Compute $XY.$

2007 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry

Let $ABC$ a triangle and $M,N,P$ points on $AB,BC$, respective $CA$, such that the quadrilateral $CPMN$ is a paralelogram. Denote $R \in AN \cap MP$, $S \in BP \cap MN$, and $Q \in AN \cap BP$. Prove that $[MRQS]=[NQP]$.

1965 IMO Shortlist, 1

Tags: inequalities , trigonometry

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

2022 Chile Junior Math Olympiad, 6

Tags: combinatorics , combinatorial geometry

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?

2001 China Team Selection Test, 2

Tags: circumcircle , incenter , geometry

In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively. $O_1I_1$ intersects $O_2I_2$ at $P$. Find the locus of point $P$ as $D$ moves along $BC$.

2006 Pre-Preparation Course Examination, 1

Tags: limit , vector , probability , invariant , topology , real analysis , advanced fields

Suppose that $X$ is a compact metric space and $T: X\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing sequence $n_i$ such that $\lim_{k\rightarrow \infty} T^{n_k}(x_0)=x_0$ for some $x_0$.

1970 Putnam, A5

Tags: ellipsoid , circles

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$

2014 IMO Shortlist, N1

Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , induction

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

MathLinks Contest 3rd, 3

Tags: combinatorics , geometry

An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$. Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.

2003 Kazakhstan National Olympiad, 6

Tags: geometry , circumcircle , circles

Let the point $ B $ lie on the circle $ S_1 $ and let the point $ A $, other than the point $ B $, lie on the tangent to the circle $ S_1 $ passing through the point $ B $. Let a point $ C $ be chosen outside the circle $ S_1 $, so that the segment $ AC $ intersects $ S_1 $ at two different points. Let the circle $ S_2 $ touch the line $ AC $ at the point $ C $ and the circle $ S_1 $ at the point $ D $, on the opposite side from the point $ B $ with respect to the line $ AC $. Prove that the center of the circumcircle of triangle $ BCD $ lies on the circumcircle of triangle $ ABC $.

2005 Sharygin Geometry Olympiad, 10.3

Tags: circles , parallel , distance , chord , geometry

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$. Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.

2006 Oral Moscow Geometry Olympiad, 1

Tags: equal circles , construction , geometry , cyclic

An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles. (L. Blinkov)

2019 Junior Balkan MO, 3

Tags: perpendicular bisector , circumcircle , geometry

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , number theory , difference , divisor

Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most $n$.

2015 NIMO Summer Contest, 5

Tags: geometry

Let $\triangle ABC$ have $AB=3$, $AC=5$, and $\angle A=90^\circ$. Point $D$ is the foot of the altitude from $A$ to $\overline{BC}$, and $X$ and $Y$ are the feet of the altitudes from $D$ to $\overline{AB}$ and $\overline{AC}$ respectively. If $XY^2$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]

2022 BMT, Tie 3

Tags: algebra

Tej writes $2, 3, ..., 101$ on a chalkboard. Every minute he erases two numbers from the board, $x$ and $y$, and writes $xy/(x+y-1)$. If Tej does this for $99$ minutes until only one number remains, what is its maximum possible value?

1998 USAMTS Problems, 1

Tags: number theory , diophantine equation

Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.

2019 Middle European Mathematical Olympiad, 2

Tags: combinatorial geometry , combinatorics

Let $n\geq 3$ be an integer. We say that a vertex $A_i (1\leq i\leq n)$ of a convex polygon $A_1A_2 \dots A_n$ is [i]Bohemian[/i] if its reflection with respect to the midpoint of $A_{i-1}A_{i+1}$ (with $A_0=A_n$ and $A_1=A_{n+1}$) lies inside or on the boundary of the polygon $A_1A_2\dots A_n$. Determine the smallest possible number of Bohemian vertices a convex $n$-gon can have (depending on $n$). [i]Proposed by Dominik Burek, Poland [/i]

2024 Azerbaijan JBMO TST, 1

Tags: number theory

Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

1959 Putnam, A1

Tags: algebra , polynomial

Let $n$ be a positive integer. Prove that $x^n -\frac{1}{x^{n}}$ is expressible as a polynomial in $x-\frac{1}{x}$ with real coefficients if and only if $n$ is odd.

2003 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine , consecutive

Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

2018 Peru Cono Sur TST, 7

Tags: locus , geometry

Let $ABCD$ be a fixed square and $K$ a variable point on segment $AD$. The square $KLMN$ is constructed such that $B$ is on segment $LM$ and $C$ is on segment $MN$. Let $T$ be the intersection point of lines $LA$ and $ND$. Find the locus of $T$ as $K$ varies along segment $AD$.

2011 Postal Coaching, 5

Tags: logarithm , inequalities

Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$. Prove that \[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\] for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.