Olympiad problems

1984 IMO Shortlist, 10

Tags: number theory , equation , Product , Perfect Square , IMO Shortlist

Prove that the product of five consecutive positive integers cannot be the square of an integer.

2001 Poland - Second Round, 3

Tags: combinatorics proposed , combinatorics

For a positive integer $n$, let $A_n$ and $B_n$ be the families of $n$-element subsets of $S_n=\{1,2,\ldots ,2n\}$ with respectively even and odd sums of elements. Compute $|A_n|-|B_n|$.

2021 Harvard-MIT Mathematics Tournament., 5

Tags: 3D geometry , combinatorics

A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\circ}, 72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.

2015 Brazil Team Selection Test, 2

Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2006 Putnam, B4

Tags: Putnam , analytic geometry , vector , induction , linear algebra , college contests

Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$.) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$

2006 Indonesia MO, 4

Tags: induction , combinatorics proposed , combinatorics

A black pawn and a white pawn are placed on the first square and the last square of a $ 1\times n$ chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.

1997 Canada National Olympiad, 1

Tags: number theory , least common multiple , number theory proposed

Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.

2006 USA Team Selection Test, 2

Tags: geometry , circumcircle , trigonometry , incenter

In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$

2009 AMC 12/AHSME, 24

Tags: trigonometry , function , summer program , Mathcamp , analytic geometry , AMC

For how many values of $ x$ in $ [0,\pi]$ is $ \sin^{\minus{}1}(\sin 6x)\equal{}\cos^{\minus{}1}(\cos x)$? Note: The functions $ \sin^{\minus{}1}\equal{}\arcsin$ and $ \cos^{\minus{}1}\equal{}\arccos$ denote inverse trigonometric functions. $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

2018-2019 Winter SDPC, 5

Tags: algebra , polynomial

Prove that there exists a positive integer $N$ such that for every polynomial $P(x)$ of degree $2019$, there exist $N$ linear polynomials $p_1,p_2, \ldots p_N$ such that $P(x)=p_1(x)^{2019}+p_2(x)^{2019}+ \ldots + p_N(x)^{2019}$. (Assume all polynomials in this problem have real coefficients, and leading coefficients cannot be zero.)

2018 Balkan MO Shortlist, N2

Tags: factorial , number theory , BMO Shortlist

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P5

Tags: algebra , Binary , Perfect Square , memorable

We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares? [i]Proposed by Nikola Velov[/i]

2017 Online Math Open Problems, 8

Tags: Online Math Open

A five-digit positive integer is called [i]$k$-phobic[/i] if no matter how one chooses to alter at most four of the digits, the resulting number (after disregarding any leading zeroes) will not be a multiple of $k$. Find the smallest positive integer value of $k$ such that there exists a $k$-phobic number. [i]Proposed by Yannick Yao[/i]

2018 Macedonia JBMO TST, 1

Tags: JMMO , Junior , 2018 , Macedonia , number theory

Determine all positive integers $n>2$, such that $n = a^3 + b^3$, where $a$ is the smallest positive divisor of $n$ greater than $1$ and $b$ is an arbitrary positive divisor of $n$.

2015 Thailand TSTST, 1

Tags: inequalities

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}}+\frac{b}{b+\sqrt{(b+c)(b+a)}}+\frac{c}{c+\sqrt{(c+a)(c+b)}}\leq\frac{2a^2+ab}{(b+\sqrt{ca}+c)^2}+\frac{2b^2+bc}{(c+\sqrt{ab}+a)^2}+\frac{2c^2+ca}{(a+\sqrt{bc}+b)^2}.$$

1998 AMC 12/AHSME, 28

Tags: trigonometry , ratio , number theory , relatively prime , geometry , angle bisector , Pythagorean Theorem

In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

1998 May Olympiad, 4

Tags: combinatorics

A regular octagon is drawn on the patio floor. Emiliano writes in the vertices the numbers from $1$ to $8$ in any order. Put a stone at point $1$. He walks towards point $2$, having traveled $1/2$ of the way he stops and leaves the second stone. From there he walks to point $3$, having traveled $1/3$ of the way, he stops and leaves the third stone. From there he walks to point $4$, having traveled $1/4$ of the way, he stops and leaves the fourth stone. This goes on until, after leaving the seventh stone, he walks towards point 8 and having traveled $1/8$ of the way, he leaves the eighth stone. The number of stones left in the center of the octagon depends on the order in which you wrote the numbers on the vertices. What is the greatest number of stones that can remain in that center?

1964 Spain Mathematical Olympiad, 4

Tags: geometry

We are given an equilateral triangle $ABC$, of side $a$, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?

2024 Argentina Cono Sur TST, 2

Tags: number theory

There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.

2005 Estonia National Olympiad, 4

Tags: geometry , parallelogram , Circumcenter , convex

In a fixed plane, consider a convex quadrilateral $ABCD$. Choose a point $O$ in the plane and let $K, L, M$, and $N$ be the circumcentres of triangles $AOB, BOC, COD$, and $DOA$, respectively. Prove that there exists exactly one point $O$ in the plane such that $KLMN$ is a parallelogram.

2022 CCA Math Bonanza, I14

Tags:

Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$. Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #14[/i]

Kvant 2021, M2645

Tags: number theory , Kvant

Vitya wrote down $n{}$ different natural numbers in his notebook. For each pair of numbers from the notebook, he wrote out their smallest common multiple on the board. Could it happen for some $n>100$ that $n(n-1)/2$ numbers on the board are (in some order) consecutive terms of a non-constant arithmetic progression? [i]Proposed by S. Berlov[/i]

2010 Postal Coaching, 5

Tags: inequalities , rearrangement inequality , inequalities unsolved

For any positive real numbers $a, b, c$, prove that \[\sum_{cyclic} \frac{(b + c)(a^4 - b^2 c^2 )}{ab + 2bc + ca} \ge 0\]

2010 China Team Selection Test, 1

Tags: geometry , circumcircle , geometric transformation , reflection , geometry unsolved

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

2020-21 KVS IOQM India, 27

Tags: geometry , reflection , orthocenter , angle

Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$, and $AB$, respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments$ PA, PB$. and $PC$ is $x^o$, determine the value of $x$.