Olympiad problems

VII Soros Olympiad 2000 - 01, 8.3

Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

2017 CCA Math Bonanza, L4.2

Tags:

Find $\arctan\left(1\right)+\arctan\left(2\right)+\arctan\left(3\right)$ in radians. [i]2017 CCA Math Bonanza Lightning Round #4.2[/i]

2006 India IMO Training Camp, 3

Tags: number theory unsolved , number theory

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

1999 National Olympiad First Round, 9

Tags: geometry , trigonometry , rectangle

Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$. $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$

2016 Bulgaria JBMO TST, 1

Tags: algebra proposed , algebra

$ a,b,c,d,e,f $ are real numbers. It is true that: $ a+b+c+d+e+f=20 $ $ (a-2)^2+(b-2)^2+...+(f-2)^2=24 $ Find the maximum value of d.

1973 Spain Mathematical Olympiad, 1

Tags: Sequence , algebra

Given the sequence $(a_n)$, in which $a_n =\frac14 n^4 - 10n^2(n - 1)$, with $n = 0, 1, 2,...$ Determine the smallest term of the sequence.

2021 Dutch IMO TST, 1

Tags: combinatorics , game , game strategy , winning strategy , dominos

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

the 14th XMO, P2

Tags: number theory

Let $p$ be a prime. Define $f_n(k)$ to be the number of positive integers $1\leq x\leq p-1$ such that $$\left(\left\{\frac{x}{p}\right\}-\left\{\frac{k}{p}\right\}\right)\left(\left\{\frac{nx}{p}\right\}-\left\{\frac{k}{p}\right\}\right)<0.$$ Let $a_n=f_n\left(\frac 12\right)+f_n\left(\frac 32\right)+\dots+f_n\left(\frac{2p-1}{2}\right)$, find $\min\{a_2, a_3, \dots, a_{p-1}\}$.

2002 China Team Selection Test, 3

Tags: number theory unsolved , number theory

Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let: \[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \} \] Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.

1995 Tournament Of Towns, (448) 4

Tags: number theory

Can the number $a + b + c + d$ be prime if $a, b, c$ and $d$ are positive integers and $ab = cd$?

2013 Cuba MO, 1

Tags: algebra , trinomial

Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.

1986 Spain Mathematical Olympiad, 3

Tags: modular arithmetic

Find all natural numbers $n$ such that $5^n+3$ is a power of $2$

2015 Romania National Olympiad, 4

Tags: number theory , sum of digits

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

2017 Harvard-MIT Mathematics Tournament, 1

Tags: algebra , polynomial

Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$. Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$.

2023 BMT, 20

Tags: number theory

Call a positive integer, $n$, [i]ready [/i] if all positive integer divisors of $n$ have a ones digit of either $1$ or $3$. Let S be the sum of all positive integer divisors of $32!$ that are ready. Compute the remainder when S is divided by $131$.

2021-IMOC, C4

Tags: combinatorics , graph theory , IMOC , 2021

There is a city with many houses, where the houses are connected by some two-way roads. It is known that for any two houses $A,B$, there is exactly one house $C$ such that both $A,B$ are connected to $C$. Show that for any two houses not connected directly by a road, they have the same number of roads adjacent to them. [i]ST[/i]

2002 District Olympiad, 3

Tags: logarithms , calculus , integration , real analysis

[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

2006 Petru Moroșan-Trident, 1

Tags: polynomial , algebra

Prove that the polynom $ X^3-aX-a+1 $ has three integer roots, for an infinite number of integers $ a. $ [i]Liviu Parsan[/i]

2018 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , geometry , tangent circles

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , identity , algebraic identities

Vasya chose a certain number $x$ and calculated the following: $a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$ It turned out that $a_2^2 = a_1a_3$. Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.

1990 All Soviet Union Mathematical Olympiad, 524

Tags: geometry , regular polygon , angles , angle bisector

$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that $XDY = \frac{\pi}{2n}$. Show that $DY$ bisects $\angle XYC$.

1993 Poland - First Round, 5

Tags: algebra , polynomial

Prove that if the polynomial $x^3 + ax^2 + bx + c$ has three distinct real roots, so does the polynomial $x^3 + ax^2 + \frac{1}{4}(a^2 + b)x + \frac{1}{8}(ab-c)$.

2006 BAMO, 3

Tags: Centroid , median , geometry

In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

2011 HMNT, 1

Tags: algebra

Find all ordered pairs of real numbers $(x, y)$ such that $x^2y = 3$ and $x + xy = 4$.

1953 Czech and Slovak Olympiad III A, 4

Tags: 3D geometry , Locus

Consider skew lines $a,b$ and a plane $\rho$ that intersect both of the lines (but does not contain any of them). Choose such points $X\in a,Y\in b$ that $XY\parallel\rho.$ Find the locus of midpoints $M$ of all segments $XY,$ when $X$ moves along line $a$.