Olympiad problems

2023 Israel TST, P1

A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

1965 AMC 12/AHSME, 33

Tags: factorial

If the number $ 15!$, that is, $ 15 \cdot 14 \cdot 13 \dots 1$, ends with $ k$ zeros when given to the base $ 12$ and ends with $ h$ zeros when given to the base $ 10$, then $ k \plus{} h$ equals: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

1985 National High School Mathematics League, 6

Tags:

Let $0<a<1$. $x_1=a,x_2=a^{x_1},\cdots,x_n=a^{x_{n-1}}$. Then sequence $(x_n)$ $\text{(A)}$ Is an increasing sequence. $\text{(B)}$ Is an decreasing sequence. $\text{(C)}$ Increases when $n$ is odd, decreases when $n$ is even. $\text{(D)}$ Decreases when $n$ is odd, increases when $n$ is even.

2024 Macedonian TST, Problem 1

Tags: number theory

Let $p,p_2,\dots,p_k$ be distinct primes and let $a_2,a_3,\dots,a_k$ be nonnegative integers. Define \[ m \;=\; \frac12 \Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr) \Bigl(\prod_{i=1}^k(p_i+1)\;+\;\sum_{i=1}^k(p_i-1)\Bigr), \] \[ n \;=\; \frac12 \Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr) \Bigl(\prod_{i=1}^k(p_i+1)\;-\;\sum_{i=1}^k(p_i-1)\Bigr). \] Prove that \[ p^2-1 \;\bigm|\; p\,m \;-\; n. \]

2018 Brazil Team Selection Test, 4

Tags: algebra , combinatorics , subset , inequalities , boundary conditions

Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$ be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.

1978 Putnam, B4

Tags: integer , equation

Prove that for every real number $N$ the equation $$ x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$ has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3 $ and $x_4$ are all larger than $N.$

1999 Vietnam National Olympiad, 2

Tags: circumcircle , geometry

let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $ A_1,A_2$ respectively. Pairs of point $ (B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $ A_1A_2 \equal{} B_1B_2 \equal{} C_1C_2$ if and only if triangle ABC is equilateral.

1946 Moscow Mathematical Olympiad, 111

Tags: plane , maximum , geometry , 3d geometry , angle

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

2012 Centers of Excellency of Suceava, 1

Tags: group theory , abstract algebra

Let be a natural number $ n\ge 2, $ a group $ G $ and two elements of it $ e_1,e_2 $ such that $ e_2e_1x=xe_2e_1, $ for any element $ x $ of $ G. $ Prove that $ \left( e_1xe_2 \right)^n =e_1x^ne_2, $ for any element $ x $ of $ G, $ if and only if $ e_2e_1=\left( e_2e_1\right)^n. $ [i]Ion Bursuc[/i]

2020 AIME Problems, 4

Tags: divisor

Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020$, and when the last four digits are removed, the result is a divisor of $N$. For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020$. Find the sum of all the digits of all the numbers in $S$. For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.

2005 AMC 10, 25

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A subset $ B$ of the set of integers from $ 1$ to $ 100$, inclusive, has the property that no two elements of $ B$ sum to $ 125$. What is the maximum possible number of elements in $ B$? $ \textbf{(A)}\ 50\qquad \textbf{(B)}\ 51\qquad \textbf{(C)}\ 62\qquad \textbf{(D)}\ 65\qquad \textbf{(E)}\ 68$

1990 Polish MO Finals, 1

Tags: function , functional equation , algebra

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy \[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]

1988 All Soviet Union Mathematical Olympiad, 470

Tags: combinatorics , minimum

There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?

2024 Middle European Mathematical Olympiad, 1

Tags: algebra , monotone , construction , functional inequality , function , sequence

Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.

2019 Saudi Arabia JBMO TST, 4

Tags: inequalities

Let $a, b, c$ be positive reals. Prove that $a/b+b/c+c/a=>(c+a)/(c+b) + (a+b)/(a+c) + (b+c)/(b+a)$

2014 Costa Rica - Final Round, 2

Tags: inequalities , algebra

Let $p_1,p_2, p_3$ be positive numbers such that $p_1 + p_2 + p_3 = 1$. If $a_1 <a_2 <a_3$ and $b_1 <b_2 <b_3$ prove that $$(a_1p_1 + a_2p_2 + a_3p_3) (b_1p_1 + b_2p_2 + b_3p_3)\le (a_1b_1p_1 + a_2b_2p_2 + a_3b_3p_3)$$

1971 Dutch Mathematical Olympiad, 1

Tags: geometry , trapezoid , parallel

Given a trapezoid $ABCD$, where sides $AB$ and $CD$ are parallel; the points $P$ on $AD$ and $Q$ on $BC$ lie such that the lines $AQ$ and $CP$ are parallel. Prove that lines $PB$ and $DQ$ are parallel.

2004 Regional Olympiad - Republic of Srpska, 3

Tags: geometry , isosceles , angle

Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$ and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.

1993 China Team Selection Test, 1

Tags: modular arithmetic , number theory

For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$

2014 Singapore Senior Math Olympiad, 29

Tags:

Find the number of ordered triples of real numbers $(x,y,z)$ that satisfy the following systems of equations: $x^2=4y-4,y^2=4z-4,z^2=4x-4$

2001 South africa National Olympiad, 1

Tags: geometry , perimeter , inequalities , triangle inequality

$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)

2010 Today's Calculation Of Integral, 536

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.

1969 IMO Longlists, 38

Tags: algebra , series summation , trigonometry , trigonometric identities

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

2010 Vietnam National Olympiad, 5

Tags: geometry , geometric transformation , rotation , symmetry , combinatorics

Let a positive integer $n$.Consider square table $3*3$.One use $n$ colors to color all cell of table such that each cell is colored by exactly one color. Two colored table is same if we can receive them from other by a rotation through center of $3*3$ table How many way to color this square table satifies above conditions.

2005 Canada National Olympiad, 2

Tags: inequalities , number theory

Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$. $a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$. $b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.