Olympiad problems

1977 IMO Longlists, 11

Let $n$ and $z$ be integers greater than $1$ and $(n,z)=1$. Prove: (a) At least one of the numbers $z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1,$ is divisible by $n$. (b) If $(z-1,n)=1$, then at least one of the numbers $z_i$ is divisible by $n$.

1986 AMC 12/AHSME, 17

Tags: pigeonhole principle

A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.) $ \textbf{(A)}\ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 50$

1955 AMC 12/AHSME, 23

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In checking the petty cash a clerk counts $ q$ quarters, $ d$ dimes, $ n$ nickels, and $ c$ cents. Later he discovers that $ x$ of the nickels were counted as quarters and $ x$ of the dimes were counted as cents. To correct the total obtained the clerk must: $ \textbf{(A)}\ \text{make no correction} \qquad \textbf{(B)}\ \text{subtract 11 cents} \qquad \textbf{(C)}\ \text{subtract 11}x\text{ cents} \\ \textbf{(D)}\ \text{add 11}x\text{ cents} \qquad \textbf{(E)}\ \text{add }x\text{ cents}$

2014 AMC 12/AHSME, 12

Tags: ratio , geometry , trigonometry , trig identities , law of cosines

Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle? $\textbf{(A) }2\qquad \textbf{(B) }1+\sqrt3\qquad \textbf{(C) }3\qquad \textbf{(D) }2+\sqrt3\qquad \textbf{(E) }4\qquad$

2021-2022 OMMC, 21

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For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \tfrac pq$ for coprime positive integers $p$, $q$, find $p + q$. [i]Proposed by Justin Lee[/i]

2007 Korea National Olympiad, 1

Tags: inequalities

For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality? $ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .

1952 Moscow Mathematical Olympiad, 227

Tags: combinatorial geometry , plane , lines in plane

$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.

1979 Dutch Mathematical Olympiad, 3

Tags: recurrence relation , sequence , number theory , last digit

Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.

2013 NIMO Problems, 8

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Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$. [i]Based on a proposal by Ivan Koswara[/i]

2008 Sharygin Geometry Olympiad, 6

Tags: geometric transformation , reflection , circumcircle , geometry

(B.Frenkin) Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.

2010 Contests, 2

Tags: ratio , geometry

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.

2023/2024 Tournament of Towns, 7

Tags: combinatorics

7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , max , area , midpoint , sidelengths

Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$ lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.

2010 Contests, 3

Tags: geometry , parallelogram , circumcircle , trigonometry , trapezoid , trig identities , law of sines

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.

1999 Spain Mathematical Olympiad, 6

Tags: combinatorial geometry , parallel lines , minimum , plane , combinatorics

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

1983 Bulgaria National Olympiad, Problem 2

Tags: inequalities

Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$, $$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$

2004 Tournament Of Towns, 3

Tags: 3d geometry , geometry , projection , pyramid

The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.

2017 Indonesia MO, 5

Tags: algebra , polynomial

A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.

2000 Canada National Olympiad, 1

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At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length $300$ meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne's speed is different from the other two speeds, then at some later time Anne will be at least $100$ meters from each of the other runners. (Here, distance is measured along the shorter of the two arcs separating two runners.)

2021 Brazil National Olympiad, 4

Tags: algebra , square , minimum value , combinatorics

A set $A$ of real numbers is framed when it is bounded and, for all $a, b \in A$, not necessarily distinct, $(a-b)^{2} \in A$. What is the smallest real number that belongs to some framed set?

2008 AMC 8, 17

Tags: geometry , rectangle , perimeter

Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? $\textbf{(A)}\ 76\qquad \textbf{(B)}\ 120\qquad \textbf{(C)}\ 128\qquad \textbf{(D)}\ 132\qquad \textbf{(E)}\ 136$

2015 Azerbaijan JBMO TST, 3

Tags: geometry , circumcircle

Acute-angled $\triangle{ABC}$ triangle with condition $AB<AC<BC$ has cimcumcircle $C^,$ with center $O$ and radius $R$.And $BD$ and $CE$ diametrs drawn.Circle with center $O$ and radius $R$ intersects $AC$ at $K$.And circle with center $A$ and radius $AD$ intersects $BA$ at $L$.Prove that $EK$ and $DL$ lines intersects at circle $C^,$.

2023 Indonesia TST, 3

Tags: algebra , geometric sequence

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

2011 AMC 12/AHSME, 17

Tags: geometry , analytic geometry , trigonometry , similar triangles , area of a triangle

Circles with radii $1, 2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency? $ \textbf{(A)}\ \frac{3}{5} \qquad \textbf{(B)}\ \frac{4}{5} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{6}{5} \qquad \textbf{(E)}\ \frac{4}{3} $

2020 Iranian Our MO, 6

Tags: functional equation , algebra , polynomial

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$ [i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]