Olympiad problems

2023 AMC 10, 3

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How many positive perfect squares less than $2023$ are divisible by $5$? $\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$

2006 All-Russian Olympiad Regional Round, 10.1

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

2013 India National Olympiad, 5

Tags: rectangle , geometry

In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$

1997 Federal Competition For Advanced Students, P2, 4

Tags: algebra

Determine all quadruples $ (a,b,c,d)$ of real numbers satisfying the equation: $ 256a^3 b^3 c^3 d^3\equal{}(a^6\plus{}b^2\plus{}c^2\plus{}d^2)(a^2\plus{}b^6\plus{}c^2\plus{}d^2)(a^2\plus{}b^2\plus{}c^6\plus{}d^2)(a^2\plus{}b^2\plus{}c^2\plus{}d^6).$

2005 Baltic Way, 16

Tags: greatest common divisor , number theory

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.

2006 Sharygin Geometry Olympiad, 8.6

Tags: projection , geometry , circumcircle , circumcenter , interior

A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.

LMT Guts Rounds, 19

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Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$

2024 Singapore Senior Math Olympiad, Q2

Tags: number theory

Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$ Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]

MBMT Team Rounds, 2020.25

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Let $\left \lfloor x \right \rfloor$ denote the greatest integer less than or equal to $x$. Find the sum of all positive integer solutions to $$\left \lfloor \frac{n^3}{27} \right \rfloor - \left \lfloor \frac{n}{3} \right \rfloor ^3=10.$$ [i]Proposed by Jason Hsu[/i]

1971 IMO Longlists, 49

Tags: polyhedron , 3d geometry , combinatorial geometry , volume , geometry

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

2024 Harvard-MIT Mathematics Tournament, 32

Tags: guts

Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$

2025 Israel TST, P2

Tags: combinatorics , graph theory

Let $ G $ be a graph colored using $ k $ colors. We say that a vertex is [b]forced[/b] if it has neighbors in all the other $ k - 1 $ colors. Prove that for any $ 2024 $-regular graph $ G $ that contains no triangles or quadrilaterals, there exists a coloring using $ 2025 $ colors such that at least $ 1013 $ of the colors have a forced vertex of that color. Note: The graph coloring must be valid, this means no $ 2 $ vertices of the same color may be adjacent.

2000 National Olympiad First Round, 36

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$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$, for every positive integer $n$. If $x_1=-1$, what is $x_{2000}$? $ \textbf{(A)}\ 1999998 \qquad\textbf{(B)}\ 2000998 \qquad\textbf{(C)}\ 2009998 \qquad\textbf{(D)}\ 2000008 \qquad\textbf{(E)}\ 1999999 $

2016 Saudi Arabia IMO TST, 3

Tags: geometry

Let $ABC$ be a triangle inscribed in $(O)$. Two tangents of $(O)$ at $B,C$ meets at $P$. The bisector of angle $BAC $ intersects $(P,PB)$ at point $E$ lying inside triangle $ABC$. Let $M,N$ be the midpoints of arcs $BC$ and $BAC$. Circle with diameter $BC$ intersects line segment $EN$ at $F$. Prove that the orthocenter of triangle $EFM$ lies on $BC$.

2016 BMT Spring, 4

Tags: counting

Three $3$-legged (distinguishable) Stanfurdians take off their socks and trade them with each other. How many ways is this possible if everyone ends up with exactly $3$ socks and nobody gets any of their own socks? All socks originating from the Stanfurdians are distinguishable from each other. All Stanfurdian feet are indistinguishable from other feet of the same Stanfurdian.

2015 Sharygin Geometry Olympiad, P1

Tags: geometry , paper , convex polygon

Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?

2000 District Olympiad (Hunedoara), 2

Tags: floor , number theory , algebra , equation , iteration

[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation $$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$ has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $ [b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.

2013 Sharygin Geometry Olympiad, 3

Tags: incenter , 3d geometry , tetrahedron , sphere , geometry

Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent. [hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]

2011 LMT, 4

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What is the sum of the first $2011$ integers closest in value to $0,$ including $0$ itself?

2011 Junior Macedonian Mathematical Olympiad, 5

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We say that a set $M{}$ containing $4$ elements is "evenly connected" if for each element $x\in M$, at least one of the numbers $x-2$ or $x+2$ belongs to the set $M.$ Let $S_n$ be the number of "evenly connected" subsets of $\{1,2,3\ldots,n\}$. Find the smallest $n{}$ such that $S_n \geq 2011.$

2007 AMC 12/AHSME, 5

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Last year Mr. John Q. Public received an inheritance. He paid $ 20\%$ in federal taxes on the inheritance, and paid $ 10\%$ of what he had left in state taxes. He paid a total of $ \$10,500$ for both taxes. How many dollars was the inheritance? $ \textbf{(A)}\ 30,000 \qquad \textbf{(B)}\ 32,500 \qquad \textbf{(C)}\ 35,000 \qquad \textbf{(D)}\ 37,500 \qquad \textbf{(E)}\ 40,000$

2004 China Team Selection Test, 2

Tags: floor function , number theory , logarithm

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

2015 Benelux, 1

Tags: algebra

Determine the smallest positive integer $q$ with the following property: for every integer $m$ with $1\leqslant m\leqslant 1006$, there exists an integer $n$ such that $$\dfrac{m}{1007}q<n<\dfrac{m+1}{1008}q$$.

2010 AMC 10, 19

Tags: geometry , vieta , ratio , trig identities

Equiangular hexagon $ ABCDEF$ has side lengths $ AB \equal{} CD \equal{} EF \equal{} 1$ and $ BC \equal{} DE \equal{} FA \equal{} r$. The area of $ \triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $ r$? $ \textbf{(A)}\ \frac {4\sqrt {3}}{3} \qquad \textbf{(B)}\ \frac {10}{3} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac {17}{4} \qquad \textbf{(E)}\ 6$

2005 May Olympiad, 4

Tags: combinatorics

At a dance there are $12$ men, numbered $1$ to $12$, and $12$ women numbered $1$ to $12$. Each man is assigned a “secret friend” among the $11$ others. They all danced all the pieces. In the first piece each man danced with the woman who has the same number. From then on, each man danced the new piece with the woman who had danced the piece earlier with his secret friend. In the third piece the couples were: [img]https://cdn.artofproblemsolving.com/attachments/c/d/f5ea0931e5751739c1ba556f84ab5736f2d11a.png[/img] Find the number of each man's secret friend.