Olympiad problems

1975 Dutch Mathematical Olympiad, 3

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$. Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$

2023 Assam Mathematics Olympiad, 8

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If $n$ is a positive even number, find the last two digits of $(2^{6n}+26)-(6^{2n}-62)$.

1987 AIME Problems, 11

Tags: calculus , integration , quadratic , inequalities , algorithm , algebra

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

2021 Bangladeshi National Mathematical Olympiad, 3

Tags: number theory , integer part , floor function

Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$?

2006 National Olympiad First Round, 16

Tags: inequalities

How many positive integer tuples $ (x_1,x_2,\dots, x_{13})$ are there satisfying the inequality $x_1+x_2+\dots + x_{13}\leq 2006$? $ \textbf{(A)}\ \frac{2006!}{13!1993!} \qquad\textbf{(B)}\ \frac{2006!}{14!1992!} \qquad\textbf{(C)}\ \frac{1993!}{12!1981!} \qquad\textbf{(D)}\ \frac{1993!}{13!1980!} \qquad\textbf{(E)}\ \text{None of above} $

LMT Theme Rounds, 10

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Let $S=\{1,2,3,4,5,6\}.$ Find the number of bijective functions $f:S\rightarrow S$ for which there exist exactly $6$ bijective functions $g:S\rightarrow S$ such that $f(g(x))=g(f(x))$ for all $x\in S$. [i]Proposed by Nathan Ramesh

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: number theory , divisor

Find the three-digit number that has the greatest number of different divisors.

2024 Belarusian National Olympiad, 10.7

Tags: number theory

Let's call a pair of positive integers $(k,n)$ [i]interesting[/i] if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$ Find the number of interesting pairs $(k,n)$ with $k \leq 100$ [i]M. Karpuk[/i]

2023 India Regional Mathematical Olympiad, 3

Tags: number theory

For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$ (s(n))^2=m \text { and }(s(m))^2=n . $$

2023 Brazil Team Selection Test, 1

Tags: combinatorics , integer sequence

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

2016 USAJMO, 5

Tags: geometry

Let $\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, respectively. Given that $$AH^2=2\cdot AO^2,$$ prove that the points $O,P,$ and $Q$ are collinear.

2016 Online Math Open Problems, 5

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Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$. What is the maximum possible value for $n$? [i]Proposed by James Lin[/i]

2003 AMC 10, 18

Tags: algebra , polynomial , vieta , big big vieta

What is the sum of the reciprocals of the roots of the equation \[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0? \] $ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$

2018 Online Math Open Problems, 13

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Find the smallest positive integer $n$ for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than $1.999$. [i]Proposed by James Lin

2009 CHKMO, 4

Tags: combinatorics

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.

2024 USAJMO, 5

Tags: function , functional equation , pointwise trap

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[ f(x^2-y)+2yf(x)=f(f(x))+f(y) \] for all $x,y\in\mathbb{R}$. [i]Proposed by Carl Schildkraut[/i]

1997 Portugal MO, 5

Tags: line segment , geometry

A square region of side $12$ contains a water source that supplies an irrigation system constituted by several straight channels forming polygonal lines. Considers the source as a point and each channel as a line segment. Knowing that a point is irrigated if it is not more than $1$ distance from any channel and that the system was designed so that the entire region is irrigated, proves that the total length of irrigation channels exceeds $70$.

2025 Harvard-MIT Mathematics Tournament, 1

Tags: algebra , number theory

Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1.$

Brazil L2 Finals (OBM) - geometry, 2018.4

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a) In $XYZ$ triangle, the incircle touches $XY$ and $XZ$ in $T$ and $W$, respectively. Prove that: $$XT=XW=\frac{XY+XZ-YZ}2$$ Let $ABC$ a triangle and $D$ the foot of the perpendicular of $A$ in $BC$. Let $I$, $J$ be the incenters of $ABD$ and $ACD$, respectively. The incircles of $ABD$ and $ACD$ touch $AD$ in $M$ and $N$, respectively. Let $P$ be where the incircle of $ABC$ touches $AB$. The circle with centre $A$ and radius $AP$ intersects $AD$ in $K$. b) Show that $\triangle IMK \cong \triangle KNJ$. c) Show that $IDJK$ is cyclic.

1990 Chile National Olympiad, 7

Tags: combinatorics , coloring

It is about deciphering the code $C_1C_2C_3C_4$ in which each letter is one of the colors: white $(B)$, blue $(A)$, red $(R)$, green $(V)$, black $(N)$ and brown $(C)$ with allowed repetitions. Four were made attempts to decipher it. $NAVB$ and $ACRC$ have two color hits, but in wrong places. $RBAC$ and $VRBA$ have one color match in the correct place, and two other color matches, in places incorrect. Determine all combinations compatible with the information.

Novosibirsk Oral Geo Oly IX, 2020.5

Tags: geometry , angle bisector , angle

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2018 Finnish National High School Mathematics Comp, 3

Tags: geometry , midpoint , chord , circles

The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.

2002 Estonia Team Selection Test, 2

Tags: geometry , angle bisector , isosceles , trigonometry

Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$

2023 Brazil Team Selection Test, 3

Tags: geometry

Let $ABCD$ be a parallelogram. Let $W, X, Y,$ and $Z$ be points on sides $AB, BC, CD,$ and $DA$, respectively, such that the incenters of triangles $AWZ, BXW, CYX,$ and $DZY$ form a parallelogram. Prove that $WXYZ$ is a parallelogram.

2013 Baltic Way, 9

Tags: combinatorial geometry , combinatorics

In a country there are $2014$ airports, no three of them lying on a line. Two airports are connected by a direct flight if and only if the line passing through them divides the country in two parts, each with $1006$ airports in it. Show that there are no two airports such that one can travel from the first to the second, visiting each of the $2014$ airports exactly once.