Olympiad problems

2006 Sharygin Geometry Olympiad, 24

Tags: projection , geometry , locus , 3d geometry , plane , sphere , circles

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane

2019 USEMO, 1

Tags: geometry , cyclic quadrilateral , angle chasing

Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar. [i]Robin Son[/i]

2014 IMAR Test, 4

Tags: steiner trees , combinatorics

Let $n$ be a positive integer. A Steiner tree associated with a finite set $S$ of points in the Euclidean $n$-space is a finite collection $T$ of straight-line segments in that space such that any two points in $S$ are joined by a unique path in $T$ , and its length is the sum of the segment lengths. Show that there exists a Steiner tree of length $1+(2^{n-1}-1)\sqrt{3}$ associated with the vertex set of a unit $n$-cube.

2012 AMC 12/AHSME, 6

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The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $

1998 Bulgaria National Olympiad, 1

Tags: combinatorics

Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.

2010 Contests, 2

Tags: trigonometry , geometry

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

2006 Estonia Math Open Senior Contests, 5

Tags: combinatorics

Two players A and B play the following game. Initially, there are $ m$ equal positive integers $ n$ written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with $ k\minus{}1$; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?

2021-IMOC, G10

Tags: geometry , circumcenter , incenter , tangent circle

Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.

1992 Tournament Of Towns, (343) 1

Tags: combinatorics

Numbers in an $n$ by $n$ table may be changed by adding $1$ to each number on an arbitrary closed non-selfintersecting “rook path” (a broken line with segments parallel to the borders of the table). Originally $1$’s stand on one of the diagonals, and $0S’s in the other cells of the table. Can one get (after several transformations) a table in which all numbers are equal to each other? (A “rook path” contains all cells through which it passes.) (AA Egorov)

2017 Saudi Arabia JBMO TST, 1

Tags: polynomial , algebra

Given a polynomial $f(x) = x^4+ax^3+bx^2+cx$. It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct). Prove that if the roots of the first equation satisfy the equality $x_1 + x_2 = x_3 + x_4$, then the same equation holds for the roots of the second equation

2007 Estonia National Olympiad, 4

Tags: table , square table , combinatorics

The figure shows a figure of $5$ unit squares, a Greek cross. What is the largest number of Greek crosses that can be placed on a grid of dimensions $8 \times 8$ without any overlaps, with each unit square covering just one square in a grid?

1982 AMC 12/AHSME, 3

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Evaluate $(x^x)^{(x^x)}$ at $x = 2$. $\textbf{(A)} \ 16 \qquad \textbf{(B)} \ 64 \qquad \textbf{(C)} \ 256 \qquad \textbf{(D)} \ 1024 \qquad \textbf{(E)} \ 65,536$

2017 NZMOC Camp Selection Problems, 7

Tags: composition , number theory

Let $a, b, c, d, e$ be distinct positive integers such that $$a^4 + b^4 = c^4 + d^4 = e^5.$$ Show that $ac + bd$ is composite.

Croatia MO (HMO) - geometry, 2018.7

Tags: incenter , circumcircle , arc midpoint , geometry

Given an acute-angled triangle $ABC$ in which $|AB| <|AC|$. Point $D$ is the midpoint of the shorter arc $BC$ of its circumcircle. The point $I$ is the center of its incircle, and the point $J$ is symmetric point of $I$ wrt line $BC$. The line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ belonging to the arc $AB$. Prove that $|AI |= |IE|$.

1992 Spain Mathematical Olympiad, 1

Tags: number theory , divisibility , combinatorics , congruence

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

2015 Belarus Team Selection Test, 3

Tags: geometry , incircle , concyclic

Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quadrilateral. I.Gorodnin

1958 Miklós Schweitzer, 6

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[b]6.[/b] Prove that if $a_n \geq 0$ and $\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k$ $(n=1, 2, \dots)$ , then $\sum_{k=1}^{\infty} a_k $ is convergent and its sum is less than $2ea_1$. [b](S. 9)[/b]

2024 HMNT, 10

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Let $S = \{1, 2, 3, . . . , 64\}.$ Compute the number of ways to partition $S$ into $16$ arithmetic sequences such that each arithmetic sequence has length $4$ and common difference $1, 4,$ or $16.$

1987 Traian Lălescu, 2.3

Tags: group theory , abstract algebra , centralizer , normalizer

Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $ [i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $

1950 AMC 12/AHSME, 34

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When the circumference of a toy balloon is increased from $20$ inches to $25$ inches, the radius is increased by: $\textbf{(A)}\ 5\text{ in} \qquad \textbf{(B)}\ 2\dfrac{1}{2}\text{ in} \qquad \textbf{(C)}\ \dfrac{5}{\pi}\text{ in} \qquad \textbf{(D)}\ \dfrac{5}{2\pi}\text{ in} \qquad \textbf{(E)}\ \dfrac{\pi}{5}\text{ in}$

2004 CHKMO, 1

Tags: inequalities

Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$

2012 USAMTS Problems, 5

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A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.

2019 Sharygin Geometry Olympiad, 15

Tags: geometry

The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. The perpendicular from $E$ to $DF$ meets $BC$ at point $X$, and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$. The segment $AD$ meets $\omega$ for the second time at point $Z$. Prove that the circumcircle of the triangle $XYZ$ touches $\omega$.

2009 Jozsef Wildt International Math Competition, W. 15

Tags: trigonometry , inequalities

Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$

1985 IMO Longlists, 57

Tags: 3d geometry , sphere , tetrahedron , geometry

[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ [i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$