Olympiad problems

2024 Ukraine National Mathematical Olympiad, Problem 8

Find all polynomials $P(x)$ with integer coefficients, such that for each of them there exists a positive integer $N$, such that for any positive integer $n\geq N$, number $P(n)$ is a positive integer and a divisor of $n!$. [i]Proposed by Mykyta Kharin[/i]

2021 All-Russian Olympiad, 7

Tags: number theory

Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.

2023 Ukraine National Mathematical Olympiad, 11.4

Tags: algebra , functional equation

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following: $$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$ [i]Proposed by Vadym Koval[/i]

2023 LMT Fall, 5

Tags: speed , nt

Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]

2003 Moldova Team Selection Test, 1

Tags: quadratic , diophantine equation , relatively prime , number theory

Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called [i]quadratic [/i] iff at least one of the numbers $ a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1

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In a class, some pupils learn German, the other learn French. The number of girls learning French and the number of boys learning German total to 16. There are 11 pupils learning French, and there are 10 girls in the class. In addition to the girls learning French, there are 16 pupils. How many pupils are there in the class? A. 18 B. 21 C. 23 D. 27 E. 31

2013 Turkey MO (2nd round), 3

Tags: geometry , rectangle

Let $n$ be a positive integer and $P_1, P_2, \ldots, P_n$ be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances $P_iP_1, P_iP_2, \ldots, P_iP_n$ forms the same sequence for all $i=1,2, \ldots, n$ when these numbers are arranged in a non-decreasing order. Find all possible values of $n$.

2012 Turkmenistan National Math Olympiad, 7

Tags: algebra

If $a,b,c$ are positive real numbers and satisfy: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$ then prove that :$ \sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =(\sum_{i=1}^{n} a_{i}b_{i})^2$

2024 Indonesia MO, 5

Tags: number theory , combinatorics , integer

Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties: 1. The sum of a red number and an orange number results in a blue-colored number, 2. The sum of an orange and blue number results in an orange-colored number; 3. The sum of a blue number and a red number results in a red-colored number. (a) Prove that $0$ and $1$ must have distinct colors. (b) Determine all possible colorings of the integers which also satisfy the properties stated above.

2009 Estonia Team Selection Test, 5

Tags: grid , combinatorics , combinatorial geometry

A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip. Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and a) $n = 2008$, b) $n = 2009$.

2012 Today's Calculation Of Integral, 853

Tags: calculus computations , logarithm , integration , calculus , trigonometry , limit

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2014 Harvard-MIT Mathematics Tournament, 10

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[6] Find the number of sets $\mathcal{F}$ of subsets of the set $\{1,\ldots,2014\}$ such that: a) For any subsets $S_1,S_2 \in \mathcal{F}, S_1 \cap S_2 \in \mathcal{F}$. b) If $S \in \mathcal{F}$, $T \subseteq \{1,\ldots,2014\}$, and $S \subseteq T$, then $T \in \mathcal{F}$.

2020 USEMO, 4

Tags: algebra , functional equation

A function $f$ from the set of positive real numbers to itself satisfies $$f(x + f(y) + xy) = xf(y) + f(x + y)$$ for all positive real numbers $x$ and $y$. Prove that $f(x) = x$ for all positive real numbers $x$.

2022 Portugal MO, 4

Tags: median , angle , geometry

Let $[AD]$ be a median of the triangle $[ABC]$. Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$, prove that $\angle BAD = 30^o$.

2016 Postal Coaching, 1

Tags: rational number , number theory

Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$

2019 Pan-African Shortlist, G4

Tags: circumcenter , equilateral triangle , circumcircle , geometry

Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

2011 NIMO Summer Contest, 9

Tags: vieta , algebra , polynomial

The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

2009 AMC 8, 24

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The letters $ A$, $ B$, $ C$ and $ D$ represent digits. If $ \begin{tabular}{ccc} &A&B \\ \plus{}&C&A \\ \hline &D&A \end{tabular}$ and $ \begin{tabular}{ccc} &A&B \\ \minus{}&C&A \\ \hline &&A \end{tabular}$, what digit does $ D$ represent? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2013 Tuymaada Olympiad, 4

Tags: graph theory , induction , combinatorics , algorithm

The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours). Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected. [i]V. Dolnikov[/i] [b]EDIT.[/b] It is confirmed by the official solution that the graph is tacitly assumed to be [b]finite[/b].

2009 Princeton University Math Competition, 8

Tags: reflection , rotation , geometry , geometric transformation

Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?

1991 Baltic Way, 10

Tags: trigonometry

Express the value of $\sin 3^\circ$ in radicals.

2022 Grand Duchy of Lithuania, 3

Tags: concyclic , isosceles , geometry

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

2018 Taiwan TST Round 1, 2

Tags: inequalities

Assume $ a,b,c $ are arbitrary reals such that $ a+b+c = 0 $. Show that $$ \frac{33a^2-a}{33a^2+1}+\frac{33b^2-b}{33b^2+1}+\frac{33c^2-c}{33c^2+1} \ge 0 $$

2001 Federal Math Competition of S&M, Problem 1

Tags: geometry

Let $ABCD$ and $A_1B_1C_1D_1$ be convex quadrangles in a plane, such that $AB=A_1B_1$, $BC=B_1C_1$, $CD=C_1D_1$ and $DA=D_1A_1$. Given that diagonals $AC$ and $BD$ are perpendicular to each other, prove that the same holds for diagonals $A_1C_1$ and $B_1D_1$.

2000 May Olympiad, 5

Tags: combinatorics

In a row there are $12$ cards that can be of three kinds: with both white faces, with both black faces or with one white face and the other black. Initially there are $9$ cards with the black side facing up. The first six cards from the left are turned over, leaving $9$ cards with the black face up. The six cards on the left are then turned over, leaving $8$ cards with the black face up. Finally, six cards are turned over: the first three on the left and the last three on the right, leaving $3$ cards with the black face up. Decide if with this information it is possible to know with certainty how many cards of each kind are in the row