Some points are marked on a sheet of paper so that no three points lie on a single straight line. Some points are connected by segments in such a way that exactly \(4\) segments emanate from each point. It is known that no triangle whose vertices are the given points has been drawn. What is the smallest number of points that can be marked on the sheet of paper?

Uz papīra lapas atzīmēti daži punkti tā, ka nekādi trīs punkti neatrodas uz vienas taisnes. Daži punkti ir savienoti ar nogriežņiem tā, ka no katra punkta iziet tieši \(4\) nogriežņi. Zināms, ka nav uzzīmēts neviens tāds trijstūris, kuram visas virsotnes ir dotajos punktos. Kāds ir mazākais skaits punktu, kas var būt atzīmēti uz papīra lapas?

Show how, by cutting along the grid lines, the figure given in Fig. 1 can be cut into \(4\) identical shapes! The shapes are identical if they can be placed one on top of the other so that they coincide (the shapes may be rotated and flipped).

Parādi, kā, griežot pa rūtiņu līnijām, 1.att. doto figūru var sagriezt \(4\) vienādās figūrās! Figūras ir vienādas, ja tās var uzlikt vienu uz otras tā, ka abas figūras pilnīgi sakrīt (figūras var pagriezt un apmest otrādi).

The dwarfs in the forest have built ten houses and created several paths between them. Each path connects two houses, and the paths may cross. Is it possible that the numbers of paths leaving the houses are respectively: (A) \(2, 2, 2, 2, 2, 3, 3, 3, 3, 7\) paths;
(B) \(2, 2, 2, 2, 4, 4, 4, 4, 4, 4\) paths?

Rūķīši mežā ir uzbūvējuši desmit mājiņas un starp tām izveidojuši vairākas taciņas. Katra taciņa savieno divas mājiņas, taciņas var krustoties. Vai iespējams, ka no mājiņām iziet attiecīgi:
(A) \(2, 2, 2, 2, 2, 3, 3, 3, 3, 7\) taciņas; (B) \(2, 2, 2, 2, 4, 4, 4, 4, 4, 4\) taciņas?

In Novadia there live \(73\) dwarfs, and some of them are friends with each other (if dwarf \(A\) is friends with dwarf \(B\), then \(B\) is also friends with \(A\), that is, friendship is mutual). Can it be that each dwarf has exactly \(9\) friends?

Novadijā dzīvo \(73\) rūḳi un daži no tiem savā starpā draudzējas (ja rūḳis \(A\) draudzējas ar rūḳi \(B\), tad arī \(B\) draudzējas ar \(A\), tas ir, draudzība ir abpusēja). Vai var būt tā, ka katram rūķim ir tieši \(9\) draugi?

In a volleyball tournament, each team played every other team exactly once; there were no draws. It is known that whichever team we choose (denote it by \(K\)), it has won exactly as many matches as the total number of wins by all the teams that \(K\) defeated. What can be the number of teams that participated in this tournament? (In any tournament there are at least \(2\) teams.)

Volejbola turnīrā katra komanda spēlēja ar katru tieši vienu reizi; neizšķirtu nav. Ir zināms: lai kuru komandu mēs izvēlētos (apzīmēsim to ar \(K\)), tā ir izcīnījusi tieši tikpat uzvaru, cik kopā izcīnījušas visas tās komandas, pret kurām \(K\) uzvarēja. Kāds var būt komandu skaits, kas piedalījās šajā turnīrā? (Nevienā turnīrā nav mazāk kā \(2\) komandas.)

If a red coin is inserted into the machine, it gives out \(5\) blue coins, but if a blue coin is inserted into the machine, it gives out \(3\) red coins. Is it possible, by repeatedly using the machine, to obtain equal numbers of red and blue coins if initially one red coin is given?

Ja automātā ievieto sarkanu monētu, tad tas izdod \(5\) zilas monētas, bet, ja automātā ievieto zilu monētu, tad tas izdod \(3\) sarkanas monētas. Vai, atkārtoti izmantojot automātu, ir iespējams iegūt vienāda skaita sarkanās un zilās monētas, ja sākumā ir dota viena sarkana monēta?

Point \(X\) is the midpoint of the diagonal \(AC\) of a convex quadrilateral \(ABCD\). It is known that \(CD \| BX\). Calculate the length of \(AD\) if \(BX=3\), \(BC=7\), and \(CD=6\).

Punkts \(X\) ir izliekta četrstūra \(ABCD\) diagonāles \(AC\) viduspunkts. Zināms, ka \(CD \| BX\). Aprēķināt \(AD\) garumu, ja \(BX=3, BC=7\) un \(CD=6\).

Show how the figure in Fig.2 (a \(6 \times 8\) rectangle, from which a \(3 \times 4\) rectangle has been removed), by cutting along the grid lines, can be subdivided into three identical shapes! The shapes are considered identical if they can be placed one on top of the other so that they coincide (the shapes may be rotated and flipped).

Parādi, kā 2.att. figūru (\(6 \times 8\) rūtiņu taisnstūris, no kura izgriezts \(3 \times 4\) rūtiņu taisnstūris), griežot pa rūtiņu līnijām, var sagriezt trīs vienādās figūrās! Figūras ir vienādas, ja tās var uzlikt vienu uz otras tā, ka abas figūras sakrīt (figūras var pagriezt un apmest otrādi).

The altitudes of an acute triangle \(ABC\) intersect at point \(H\). Calculate the area of quadrilateral \(ABHC\) if \(AH = BC = 8\).

Šaurleņķu trijstūra \(ABC\) augstumi krustojas punktā \(H\). Aprēḳināt četrstūra \(ABHC\) laukumu, ja \(AH = BC = 8\).

Can the number: (A) \(72\), (B) \(73\) be expressed as the sum of three distinct natural numbers such that the sum of any two of these numbers is divisible by the remaining number?

Vai skaitli: (A) \(72\), (B) \(73\) var izteikt kā trīs dažādu naturālu skaitļu summu tā, lai katru divu šo skaitḷu summa dalītos ar atlikušo skaitli?

Prove that the product of no two consecutive natural numbers can be expressed in the form \(27n+11\), where \(n\) is a natural number.

Pierādīt, ka nekādu divu secīgu naturālu skaitļu reizinājums nav izsakāms formā \(27n+11\), kur \(n\) ir naturāls skaitlis.

How many natural numbers \(n\) are there such that the number \(n^{2}\) has as many digits as the number \(n^{3}\)?

Cik ir tādu naturālu skaitļu \(n\), kuriem skaitlim \(n^{2}\) ir tikpat ciparu, cik skaitlim \(n^{3}\)?

Write a number in each empty cell (see Figure 1) so that the sum of the numbers in any three consecutive cells is the same, and the total sum of all the numbers written in the cells (including the two given numbers) is \(223\). It is sufficient to show one example of how to achieve this.

Ieraksti katrā tukšajā rūtiņā (skat. 1.att.) vienu skaitli (skaitļi var būt arī vienādi) tā, lai katrās trīs blakus rūtiņās skaitļu summa būtu viena un tā pati un visu rūtiņās ierakstīto skaitļu (ieskaitot abus dotos skaitļus) summa būtu \(223\). Pietiek parādīt vienu veidu, kā to var izdarīt.

Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\sphericalangle BAC<60^{\circ}\). The circle with center at point \(B\) and radius \(BC\) intersects the sides \(AC\) and \(AB\) of the triangle at points \(D\) and \(E\), respectively. Compute \(\frac{AD}{DC}\) if \(\frac{AE}{EB}=\frac{2}{5}\).

Dots vienādsānu trijstūris \(ABC\), kuram \(AB = AC\) un \(\sphericalangle BAC<60^{\circ}\). Riņka līnija, kuras centrs ir punktā \(B\) un rādiuss \(BC\), krusto trijstūra malas \(AC\) un \(AB\) attiecīgi punktos \(D\) un \(E\). Aprēķināt \(\frac{AD}{DC}\), ja \(\frac{AE}{EB}=\frac{2}{5}\).

Prove that \(x^{2}+y^{2}+\frac{1}{2} \geq x+y\) for all real \(x\) and \(y\).

Pierādīt, ka \(x^{2}+y^{2}+\frac{1}{2} \geq x+y\) visiem reāliem \(x\) un \(y\).

Find the smallest real number \(a\) such that for all real numbers \(x, y, z\) the inequality holds:

\[x^{2}+y^{2}+z^{2}+a \geq x + 2y + 3z\]

Atrast mazāko reālo skaitli \(a\), ar kuru visiem reāliem skaitliem \(x, y, z\) ir spēkā nevienādība:

\[x^{2}+y^{2}+z^{2}+a \geq x + 2y + 3z\]

We start with the number \(17\). We can repeatedly apply any of the following operations:

1. Multiply the current number by \(2\).
2. Delete any of the digits \(0\), \(3\), \(6\), or \(9\) from the number.

Is it possible to obtain (A) the number 1? (B) the number 15?

Naturālu skaitli atļauts reizināt ar \(2\), kā arī izsvītrot no tā pieraksta ciparus \(0,3,6,9\) (varbūt tikai kādu no tiem). Vai, vairākkārt izpildot šādus gājienus, no skaitļa \(17\) var iegūt: (A) skaitli \(1\); (B) skaitli \(15\)?

A sequence with \(N\) elements will be called a permutation of the \(N\) smallest natural numbers if it contains all natural numbers from \(1\) to \(N\).

It is known that the sequence \(\left\{ a_{i} \right\}\) is a permutation of the \(n\) smallest natural numbers (\(n>3\)).

The elements of the sequence \(\left\{ b_{i} \right\}\) (\(1 \leq i \leq n-1\)) are computed by the formula \(b_{i}=\left|a_{i+1}-a_{i}\right|\).

The elements of the sequence \(\left\{ c_{i} \right\}\) (\(1 \leq i \leq n-2\)) are computed by the formula \(c_{i}=\left|b_{i+1}-b_{i}\right|\).

Prove that \(\left\{ b_{i} \right\}\) and \(\left\{ c_{i} \right\}\) cannot both simultaneously be permutations of the \(n-1\) and \(n-2\) smallest natural numbers, respectively!

Skaitļu virkni, kurā ir \(N\) elementi, sauksim par \(N\) mazāko naturālo skaitļu permutāciju, ja tajā atrodami visi naturālie skaitļi no \(1\) līdz \(N\).

Zināms, ka virkne \(\left\{ a_{i} \right\}\) ir \(n\) (\(n>3\)) mazāko naturālo skaitļu permutācija.

Virknes \(\left\{ b_{i} \right\}\) (\(1 \leq i \leq n-1\)) elementus aprēķina pēc formulas \(b_{i}=\left|a_{i+1}-a_{i}\right|\).

Virknes \(\left\{ c_{i} \right\}\) (\(1 \leq i \leq n-2\)) elementus aprēķina pēc formulas \(c_{i}=\left|b_{i+1}-b_{i}\right|\).

Pierādīt, ka \(\left\{ b_{i} \right\}\) un \(\left\{ c_{i} \right\}\) vienlaikus abas nevar būt attiecīgi \(n-1\) un \(n-2\) mazāko naturālo skaitļu permutācijas!

Is it possible to write one natural number in each empty square (see Fig. 1) so that all natural numbers from \(1\) to \(25\) are written and any two numbers whose difference is \(1\) are placed in squares sharing a common side?

Vai tukšajās rūtiņās (skat. 1.att.) var ierakstīt pa vienam naturālam skaitlim tā, lai rezultātā būtu ierakstīti visi naturālie skaitḷi no \(1\) līdz \(25\) un katri divi skaitļi, kuru starpība ir \(1\), būtu ierakstīti rūtiņās ar kopīgu malu?

In a chess tournament, any two players have played at most one game against each other. Must there always be a group of three players where either every pair in the group has played a game, or no pair has played a game? Consider this for a tournament with: (A) \(5\) players, (B) \(6\) players.

Šaha turnīrā katri divi šahisti ir vai nu izspēlējuši tieši vienu šaha partiju, vai arī nav izspēlējuši nevienu partiju. Vai noteikti var atrast tādus trīs šahistus, kas savā starpā ir izspēlējuši vai nu visas \(3\) partijas, vai nevienu partiju, ja turnīrā piedalās (A) \(5\), (B) \(6\) šahisti?

In a nature reserve, the age of each tree in years is expressed as a natural number. The average age of the trees before yesterday’s thunderstorm was exactly \(72\) years. During the storm, due to a lightning strike, one \(2023\)-year-old tree perished, and now the average age of the trees in the reserve is exactly \(71\) years. What is the largest number of \(2023\)-year-old trees that could have been in the reserve before yesterday’s storm?

Note. Over these two days no tree has become older.

Dabas rezervātā katra koka vecums gados izsakāms kā naturāls skaitlis. Koku vidējais vecums pirms vakardienas negaisa bija tieši \(72\) gadi. Negaisa laikā Zibens spēriena dēļ gāja bojā viens \(2023\) gadus vecs koks un tagad rezervāta koku vidējais vecums ir tieši \(71\) gads. Kāds lielākais skaits \(2023\) gadus vecu koku varēja atrasties rezervātā pirms vakardienas negaisa?

Piezīme. Pa šīm divām dienām neviens koks nav kļuvis vecāks.

At an event, \(m\) people met. Each of them is friends with exactly \(3\) other people (if \(A\) is friends with \(B\), then \(B\) is friends with \(A\)). It is known that among any three people one can find two who are not friends with each other. Can it happen that (A) \(m=11\), (B) \(m=10\)?

Pasākumā satikās \(m\) cilvēki. Katrs no tiem draudzējas ar tieši \(3\) citiem cilvēkiem (ja \(A\) draudzējas ar \(B\), tad \(B\) draudzējas ar \(A\)). Zināms, ka no katriem trim cilvēkiem var atrast divus, kuri savā starpā nedraudzējas. Vai var gadīties, ka (A) \(m=11\), (B) \(m=10\)?

Show how the number \(174\) can be written as the sum of \(3\) distinct natural numbers such that the sum of any two of these numbers is divisible by the third number!

Parādi, kā skaitli \(174\) var uzrakstīt kā \(3\) dažādu naturālu skaitļu summu tā, lai katru divu šo skaitļu summa dalītos ar trešo skaitli!

Given a quadrilateral \(ABCD\) with all angles less than \(180^{\circ}\), \(\sphericalangle A=\sphericalangle B\), \(BC=1\) and \(AD=3\). Prove that \(CD>2\).

Dots četrstūris \(ABCD\), kuram visi leņķi ir mazāki nekā \(180^{\circ}\), \(\sphericalangle A=\sphericalangle B\), \(BC=1\) un \(AD=3\). Pierādīt, ka \(CD>2\).

Around a circular track, in one direction Kārlis is running on foot and Sandris is riding a scooter, while in the opposite direction Vilnis is riding a bicycle and Mārtiņš a motorbike (each moves at their own constant speed). It is known that Kārlis meets Vilnis every \(12\) minutes, Sandris overtakes Kārlis every \(20\) minutes, and Mārtiņš overtakes Vilnis every \(5\) minutes. How often does Mārtiņš meet Sandris?

Pa apļveida trasi vienā virzienā Kārlis skrien ar kājām un Sandris brauc ar skrejriteni, bet pretējā virzienā Vilnis brauc ar velosipēdu un Mārtiņš ar mopēdu (katrs brauc ar savu, nemainīgu ātrumu). Zināms, ka Kārlis satiek Vilni ik pēc \(12\) minūtēm, Sandris apdzen Kārli ik pēc \(20\) minūtēm, bet Mārtiņš apdzen Vilni ik pēc \(5\) minūtēm. Cik bieži Mārtiņš satiek Sandri?

Fill each empty square in Figure 1 with a prime number. Numbers may be repeated. The completed grid must satisfy two conditions:

1. The sum of the numbers in any four adjacent squares must be the same.
2. The total sum of all numbers in the grid (including the two given numbers) is \(127\).

Provide one arrangement of numbers that meets these conditions.

Ieraksti katrā tukšajā rūtiņā (skat. 1.att.) vienu pirmskaitli (skaitḷi var būt arī vienādi) tā, lai katrās četrās blakus rūtiņās skaitļu summa būtu viena un tā pati un visu rūtiņās ierakstīto skaitļu (ieskaitot abus dotos skaitļus) summa būtu \(127\). Pietiek parādīt vienu veidu, kā to var izdarīt.

Find all pairs of real numbers \((x;y)\) such that

\[\left(x^{4}+1\right)\left(y^{4}+1\right) = 4 x^{2} y^{2}.\]

Atrast visus tādus reālu skaitļu pārus \((x;y)\), kuriem

\[\left(x^{4}+1\right)\left(y^{4}+1\right)=4 x^{2} y^{2}\]

Can (A) \(90\) light bulbs, (B) \(73\) light bulbs be connected with wires so that each of them is connected by a wire to exactly \(5\) other light bulbs?

Vai (A) \(90\) lampiņas, (B) \(73\) lampiņas ar vadiem var savienot tā, lai katra no tām būtu savienota ar vadu ar tieši \(5\) citām lampiņām?

Determine whether the number \(\sqrt{4+2 \sqrt{3}}-\sqrt{4-2 \sqrt{3}}\) is rational or irrational!

Noskaidrot, vai skaitlis \(\sqrt{4+2 \sqrt{3}}-\sqrt{4-2 \sqrt{3}}\) ir racionāls vai iracionāls!

In a certain land live dwarves; any two of them either are friends or hate each other. It is known that there are no three dwarves who all hate each other. Must there necessarily exist three dwarves who are all pairwise friends if in this land there are (A) \(5\) dwarves, (B) \(6\) dwarves?

Kādā zemē dzīvo rūķi, katri divi no tiem vai nu draudzējas, vai viens otru ienīst. Zināms, ka nav tādu trīs rūḳu, kas visi viens otru ienīst. Vai noteikti var atrast tādus trīs rūḳus, kas visi savā starpā draudzējas, ja šajā zemē ir (A) \(5\) rūķi, (B) \(6\) rūķi?

Find one set of natural numbers to substitute for \(x, y\), and \(z\) so that the equality

\[2+\frac{1}{x+\frac{1}{y+\frac{1}{z}}}=\frac{37}{13}\]

holds!

Atrodi vienu veidu, kādi naturāli skaitḷi jāievieto \(x, y\) un \(z\) vietā, lai vienādība

\[2+\frac{1}{x+\frac{1}{y+\frac{1}{z}}}=\frac{37}{13}\]

būtu patiesa!

Given that \(x\) is a natural number. What is the largest number of the numbers \(x\); \(x+2\); \(x+4\); \(x+6\); \(x+8\) that can simultaneously be prime?

Dots, ka \(x\) ir naturāls skaitlis. Kāds lielākais skaits skaitļu \(x\); \(x+2\); \(x+4\); \(x+6\); \(x+8\) vienlaicīgi var būt pirmskaitḷi?

Show how, by cutting along the grid lines, the figure given in Fig. 2 can be cut into \(4\) identical figures! Two figures are identical if they can be placed one on top of the other so that they coincide (the figures may be rotated and flipped over).

Parādi, kā, griežot pa rūtiņu līnijām, 2.att. doto figūru var sagriezt \(4\) vienādās figūrās! Figūras ir vienādas, ja tās var uzlikt vienu uz otras tā, ka abas figūras sakrīt (figūras var pagriezt un apmest otrādi).

In the forest, gnomes have built eight houses and created several paths between them. Each path connects two houses, and paths can intersect. Is it possible that the houses have the following number of paths coming out of them:
(A) \(2, 2, 2, 4, 4, 4, 4, 4\) paths respectively;
(B) \(1, 2, 2, 2, 2, 3, 4, 5\) paths respectively?

Rūķīši mežā ir uzbūvējuši astoņas mājiņas un starp tām izveidojuši vairākas taciņas. Katra taciņa savieno divas mājiņas, taciņas var krustoties. Vai iespējams, ka no mājiņām iziet attiecīgi: (A) \(2, 2, 2, 4, 4, 4, 4, 4\) taciņas;
(B) \(1,2,2,2,2,3,4,5\) taciņas?

Given an isosceles triangle \(ABC\) with \(AB=AC\) and \(\sphericalangle BAC < 60^{\circ}\). The circle with center at point \(B\) and radius \(BC\) intersects the sides \(AC\) and \(AB\) at the points \(D\) (which does not coincide with \(C\)) and \(E\), respectively. Prove that \(AD < 2AE\).

Dots vienādsānu trijstūris \(ABC\), kuram \(AB=AC\) un \(\sphericalangle BAC < 60^{\circ}\). Rinķa līnija, kuras centrs ir punktā \(B\) un rādiuss \(BC\), krusto trijstūra malas \(AC\) un \(AB\) attiecīgi punktos \(D\) (kas nesakrīt ar \(C\)) un \(E\). Pierādīt, ka \(AD < 2AE\).

There are \(2023\) boxes, initially containing \(1, 2, 3, \ldots, 2023\) candies respectively. In a single move, you may choose a natural number \(n\) and eat \(n\) candies from some boxes (possibly only from one). What is the smallest number of moves needed to make all boxes empty?

Dotas \(2023\) kastes, sākumā tajās ir attiecīgi \(1, 2, 3, \ldots, 2023\) konfektes. Vienā gājienā var izvēlēties naturālu skaitli \(n\) un no dažām kastēm (varbūt tikai no vienas) apēst \(n\) konfektes. Kāds ir mazākais gājienu skaits, ar kuru var panākt, ka visas kastes ir tukšas?

In a box there are white, red, and green balls. In one move, you may take out two balls of different colors from the box and put into the box one ball of the third color (there are always enough balls of any color to put into the box). Is it possible to achieve that only one ball remains in the box, if initially the box contains:
(A) \(10\) white, \(12\) red, and \(16\) green balls;
(B) \(10\) white, \(12\) red, and \(15\) green balls?

Kastē atrodas baltas, sarkanas un zaḷas lodītes. Ar vienu gājienu no kastes var izņemt divas dažādu krāsu lodītes un ielikt kastē vienu trešās krāsas lodīti (vienmēr pietiek jebkuras krāsas lodīšu, ko ielikt kastē). Vai var panākt, ka kastē paliek tikai viena lodīte, ja sākumā kastē atrodas:
(A) \(10\) baltas, \(12\) sarkanas un \(16\) zalas lodītes;
(B) \(10\) baltas, \(12\) sarkanas un \(15\) zaḷas lodītes?

Given an isosceles triangle \(ABC\) with \(AB = BC\). On side \(AB\) a point \(M\) is chosen and on side \(BC\) a point \(K\) is chosen such that \(AM = AK = AC\). It is known that \(AK \perp MC\). Calculate the angles of triangle \(ABC\)!

Dots vienādsānu trijstūris \(ABC\), kuram \(AB = BC\). Uz malas \(AB\) izvēlēts punkts \(M\) un uz malas \(BC\) izvēlēts punkts \(K\) tā, ka \(AM = AK = AC\). Zināms, ka \(AK \perp MC\). Aprēḳināt trijstūra \(ABC\) leņkus!

Prove that the product of any two consecutive natural numbers cannot be expressed in the form \(36n+8\), where \(n\) is a natural number!

Pierādīt, ka nekādu divu secīgu naturālu skaitļu reizinājums nav izsakāms formā \(36n+8\), kur \(n\) ir naturāls skaitlis!

Find the largest natural number \(N\) with the property that for every prime \(p<N\) the number \(N+2p\) is also prime!

Atrast lielāko naturālo skaitli \(N\) ar īpašību - katram pirmskaitlim \(p<N\) skaitlis \(N+2p\) arī ir pirmskaitlis!