Olympiad problems

2025 International Zhautykov Olympiad, 2

Tags: combinatorics

Rose and Brunno play the game on a board shaped like a regular 1001-gon. Initially, all vertices of the board are white, and there is a chip at one of them. On each turn, Rose chooses an arbitrary positive integer $ k $, then Brunno chooses a direction: clockwise or counterclockwise, and moves the chip in the chosen direction by $ k $ vertices. If at the end of the turn the chip stands at a white vertex, this vertex is painted red. Find the greatest number of vertices that Rose can make red regardless of Brunno's actions, if the number of turns is not limited.

1998 USAMTS Problems, 2

Tags: USAMTS , modular arithmetic , number theory , prime factorization

For a nonzero integer $i$, the exponent of $2$ in the prime factorization of $i$ is called $ord_2 (i)$. For example, $ord_2(9)=0$ since $9$ is odd, and $ord_2(28)=2$ since $28=2^2\times7$. The numbers $3^n-1$ for $n=1,2,3,\ldots$ are all even so $ord_2(3^n-1)>0$ for $n>0$. a) For which positive integers $n$ is $ord_2(3^n-1) = 1$? b) For which positive integers $n$ is $ord_2(3^n-1) = 2$? c) For which positive integers $n$ is $ord_2(3^n-1) = 3$? Prove your answers.

2022 CCA Math Bonanza, I5

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Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. $A$, $B$, and $C$ are points on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle such that $\Gamma_2$ is tangent to $AB$ and $BC$ at $Q$ and $R$, and $\Gamma_2$ is also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ intersects $AC$ at $X$ and $Y$. $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$. Find $a+b+c$. [i]2022 CCA Math Bonanza Individual Round #5[/i]

2021 DIME, 6

Tags: DIME P6

Let $ABC$ be a right triangle with right angle at $A$ and side lengths $AC=8$ and $BC=16$. The lines tangent to the circumcircle of $\triangle ABC$ at points $A$ and $B$ intersect at $D$. Let $E$ be the point on side $\overline{AB}$ such that $\overline{AD} \parallel \overline{CE}$. Then $DE^2$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Awesome_guy[/i]

2011 India Regional Mathematical Olympiad, 2

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Let $n$ be a positive integer such that $2n+1$ and $3n+1$ are both perfect squares. Show that $5n+3$ is a composite number.

2022 Taiwan TST Round 2, C

Tags: combinatorics , lattice points

There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$. ([i]Note. An integer point is a point with integer coordinates.[/i]) [i]Proposed by CSJL.[/i]

2019 Auckland Mathematical Olympiad, 5

Tags: Coloring , combinatorics , combinatorial geometry

$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?

1989 IMO Shortlist, 25

Tags: number theory , Diophantine equation , quadratics , equation , IMO Shortlist

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

2023 Assam Mathematics Olympiad, 16

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$n$ is a positive integer such that the product of all its positive divisors is $n^3$. Find all such $n$ less than $100$.

1972 IMO, 1

Tags: combinatorics , pigeonhole principle , IMO , IMO 1972

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

2016 China Team Selection Test, 6

Tags: algebra , function

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

1994 AMC 8, 5

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Given that $\text{1 mile} = \text{8 furlongs}$ and $\text{1 furlong} = \text{40 rods}$, the number of rods in one mile is $\text{(A)}\ 5 \qquad \text{(B)}\ 320 \qquad \text{(C)}\ 660 \qquad \text{(D)}\ 1760 \qquad \text{(E)}\ 5280$

2017 NZMOC Camp Selection Problems, 6

Tags: geometry , Concyclic

Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.

2003 Korea Junior Math Olympiad, 4

Tags: combinatorics , multiple , Integer

When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.

2014 HMIC, 3

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Fix positive integers $m$ and $n$. Suppose that $a_1, a_2, \dots, a_m$ are reals, and that pairwise distinct vectors $v_1, \dots, v_m\in \mathbb{R}^n$ satisfy $$\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0$$ for $i=1,2,\dots,m$. Prove that $$\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.$$

2007 Oral Moscow Geometry Olympiad, 1

Tags: geometry , similar triangles , right triangle

The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled? (S. Markelov)

1975 Chisinau City MO, 99

Tags: trigonometry

Prove the equality: $\sin 54^o -\sin 18^o = 0.5$

2013 Romania National Olympiad, 1

Tags: geometry , angle bisector , median , parallel

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

1999 Mongolian Mathematical Olympiad, Problem 4

Tags: geometry

Is it possible to place a triangle with area $1999$ and perimeter $19992$ in the interior of a triangle with area $2000$ and perimeter $20002$?

2013 Singapore MO Open, 5

Tags: calculus , geometry unsolved , geometry , Triangle , perimeter

Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.

2019 May Olympiad, 1

Tags: number theory , remainder

A positive integer is called [i]piola [/i] if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many [i]piolas[/i] are there between $ 1$ and $100000$?

1986 Vietnam National Olympiad, 1

Tags: inequalities , function , inequalities unsolved

Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0$. Prove that \[ \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1\]

2001 Tuymaada Olympiad, 2

Tags: combinatorics proposed , combinatorics

Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every number is exactly $k$ times less than the sum of all the other numbers in the same cross (i.e., $2n-2$ numbers written in the same row or column with this number). Find all possible $k$. [i]Proposed by D. Rostovsky, A. Khrabrov, S. Berlov [/i]

2020 GQMO, 1

Tags: quadratics , algebra , polynomial

Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities \[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\] hold for all real numbers $X$. [i]Morteza Saghafian, Iran[/i]

2006 IMO Shortlist, 6

Tags: inequalities , function , algebra , IMO Shortlist , IMO , IMO 2006 , Hi

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.