June 12, 2025 Blue Acquisition Corp.

Exhibit 10.5

June 12, 2025

Blue Acquisition Corp.

▇▇▇▇ ▇▇▇▇▇ ▇▇▇▇

Newport Beach CA, 92660-4803

Re: Initial Public Offering

Ladies and Gentlemen:

This letter (this “Letter Agreement”) is being delivered to you in accordance with the Underwriting Agreement (the “Underwriting Agreement”) entered into by and among Blue Acquisition Corp., a Cayman Islands exempted company (the “Company”) and BTIG, LLC., as representative (the “Representative”) of the underwriters (the “Underwriters”), relating to an underwritten initial public offering (the “Public Offering”), of up to 20,125,000 of the Company’s units (including up to 2,625,000 units which may be purchased to cover over-allotments, if any) (the “Units”), each comprised of one Class A ordinary share, par value $0.0001 per share, of the Company (the “Class A Ordinary Shares”) and one right (each right, a “Share Right”). Each Share Right entitles the holder thereof to receive one-tenth (1/10) of one Class A Ordinary Share upon the consummation of the Company’s initial business combination. The Units shall be sold in the Public Offering pursuant to the registration statement on Form S-1 (File No. 333-287281) and prospectus (the “Prospectus”) filed by the Company with the U.S. Securities and Exchange Commission (the “Commission”) and the Company shall apply to have the Units listed on the Nasdaq Global Market. Certain capitalized terms used herein are defined in paragraph 11 hereof.

In order to induce the Company and the Underwriters to enter into the Underwriting Agreement and to proceed with the Public Offering and for other good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, Blue Holdings Sponsor LLC, a Delaware limited liability company (the “Sponsor”) and each of the undersigned individuals, each of whom is, or will be, a member of the Company’s board of directors and/or management team (each an “Insider” and, collectively, the “Insiders”), hereby agree with the Company as follows:

1. The Sponsor and each Insider agree that if the Company seeks shareholder approval of a proposed Business Combination, then in connection with such proposed Business Combination, it, he or she shall (i) vote all Founder Shares, Private Placement Shares, and any shares acquired by it, him or her in the Public Offering or the secondary public market in favor of such proposed Business Combination, except that it, he or she shall not vote any Class A Ordinary Shares that it, he or she purchased after the Company publicly announces its intention to engage in such proposed Business Combination for or against such proposed Business Combination and (ii) not redeem any Class A Ordinary Shares owned by it, him or her in connection with such shareholder approval. If the Company seeks to consummate a proposed Business Combination by engaging in a tender offer, the Sponsor and each Insider agrees that it, he or she will not sell or tender any Ordinary Shares owned by it, him or her in connection herewith.

2. The Sponsor and each Insider agree that in the event that the Company fails to consummate a Business Combination by the date that is 21 months after the closing of the Public Offering, or such earlier date as Company’s board of directors may approve, or such later date as the Company’s shareholders may approve, in each case in accordance with the Company’s amended and restated memorandum and articles of association, as may be amended from time to time (the “Completion Window” and the “Memorandum and Articles,” respectively), the Sponsor and each Insider shall take all reasonable steps to cause the Company to (i) cease all operations except for the purpose of winding up, (ii) as promptly as reasonably possible but not more than ten (10) business days thereafter, subject to lawfully available funds therefor, redeem 100% of the Class A Ordinary Shares sold as part of the Units in the Public Offering (the “Offering Shares”), at a per-share price, payable in cash, equal to the aggregate amount then on deposit in the Trust Account, including interest earned on the funds held in the Trust Account (which interest shall be net of taxes payable and less up to $100,000 of interest to pay dissolution expenses), divided by the number of Offering Shares then in issue, which redemption will completely extinguish the Public Shareholders’ rights as shareholders (including the right to receive further liquidation distributions, if any), subject to applicable law and (iii) as promptly as reasonably possible following such redemption, subject to the approval of the Company’s remaining shareholders and the Company’s board of directors, dissolve and liquidate, subject in each case to the Company’s obligations under Cayman Islands law to provide for claims of creditors and other requirements of applicable law. The Sponsor and the Insiders agree to not propose any amendment to the Memorandum and Articles not for the purposes of approving, or in conjunction with the consummation of, a Business Combination (A) to modify the substance or timing of the Company’s obligation to allow redemption in connection with a Business Combination or to redeem one hundred per cent (100%) of the Offering Shares if the Company has not consummated a Business Combination within the Completion Window or (B) with respect to any other material provisions relating to the rights of holders of Class A Ordinary Shares or pre-initial Business Combination activity, unless the Company provides its Public Shareholders with the opportunity to redeem their Offering Shares upon effectiveness of any such amendment at a per share price, payable in cash, equal to the aggregate amount then on deposit in the Trust Account including interest earned on the Trust Account and not previously released to the Company to pay its taxes, divided by the number of Offering Shares then in issue, subject to applicable law. The Sponsor and each Insider acknowledges that it, he or she will not be entitled to rights to liquidating distributions from the Trust Account with respect to any Founder Shares or Private Placement Shares held by it, him or her if the Company fails to complete a Business Combination within the Completion Window; although it, he or she will be entitled to liquidating distributions from the Trust Account with respect to any Offering Shares it, he or she holds if the Company fails to complete a Business Combination within the prescribed time frame. The Sponsor and each Insider hereby further acknowledge that it, he or she will not be entitled to (a) redemption rights with respect to any Founder Shares, Private Placement Shares, and Offering Shares held by it, him or her, in connection with the consummation of a Business Combination, or (b) redemption rights with respect to Founder Shares, Private Placement Shares, and Offering Shares held by it, him or her in connection with a shareholder vote to amend the Memorandum and Articles in the manner described above.

3. To the fullest extent permitted by applicable law and the Memorandum and Articles, the Company hereby agrees to defend, indemnify, hold harmless and exonerate (including the advancement of expenses to the fullest extent permitted by applicable law) the Sponsor and its members (present and former), managers and affiliates and their respective present and former officers and directors (each, a “Sponsor Indemnitee”) from any and all costs, fees, expenses, judgments, liabilities, fines, penalties, reasonable attorneys’ fees and amounts paid in settlement (including all interest, assessments and other charges paid or payable in connection with or in respect of such costs, fees, expenses, judgments, liabilities, fines, penalties and amounts paid in settlement) actually, and reasonably, incurred by a Sponsor Indemnitee or on a Sponsor Indemnitee’s behalf in connection with any threatened, pending or completed action, suit, arbitration, mediation, alternate dispute resolution mechanism, investigation, inquiry, hearing or any other actual, threatened or completed proceeding instituted by the Company or any third party, whether civil, criminal, administrative or investigative in nature, in respect of any investment opportunities sourced by a Sponsor Indemnitee for the Company or any liability arising with respect to a Sponsor Indemnitee’s activities in connection with the affairs of the Company (in each case to the extent that such indemnification, hold harmless and exoneration obligations with respect to such matters are not expressly covered by a separate written agreement between the Company and the applicable Sponsor Indemnitee); provided, that in no event shall a Sponsor Indemnitee be entitled to be indemnified or held harmless hereunder in respect of any costs, fees, expenses, judgments, liabilities, fines, penalties and amounts paid in settlement (if any) that a Sponsor Indemnitee may incur by reason of such person’s own actual fraud or intentional misconduct; provided, further, that, for the avoidance of doubt, under no circumstance shall a Sponsor Indemnitee have a claim to any monies or assets held in the Trust Account, and the Company shall not be permitted to procure monies or assets held in the Trust Account for the satisfaction of its obligations to any Sponsor Indemnitee in respect of the indemnification provided hereunder. The Sponsor Indemnitees shall be third party beneficiaries of this paragraph.

4. During the period commencing on the effective date of the Underwriting Agreement and ending 180 days after such date, the undersigned shall not, without the prior written consent of the Representative, (i) sell, offer to sell, contract or agree to sell, hypothecate, pledge, grant any option to purchase or otherwise dispose of or agree to dispose of, directly or indirectly, or establish or increase a put equivalent position or liquidate or decrease a call equivalent position within the meaning of Section 16 of the Securities Exchange Act of 1934, as amended, and the rules and regulations of the Commission promulgated thereunder, any Units, Class A Ordinary Shares, the Company’s Class B ordinary shares, par value $0.0001 per share (the “Class B Ordinary Shares” and, together with the Class A Ordinary Shares, the “Ordinary Shares”), Share Rights or any securities convertible into, or exercisable, or exchangeable for, Class A Ordinary Shares owned by him, her or it; provided, however, that the foregoing shall not apply to transfers to the Sponsor by the Insiders, (ii) enter into any swap or other arrangement that transfers to another, in whole or in part, any of the economic consequences of ownership of any Units, Class A Ordinary Shares, Founder Shares, Share Rights or any securities convertible into, or exercisable, or exchangeable for, Class A Ordinary Shares owned by him, her or it, whether any such transaction is to be settled by delivery of such securities, in cash or otherwise or (iii) publicly announce any intention to effect any transaction specified in clause (i) or (ii). If the undersigned is an officer or director of the Company, the undersigned further agrees that the forgoing restrictions shall be equally applicable to any issuer-directed Units that the undersigned may purchase in the Public Offering. In addition, the 300,000 Ordinary Shares issuable to ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇ are subject to a lock-up for a period of 180 days immediately following the effective date of the registration statement pursuant to Rule 5110(e)(1) of the FINRA. Pursuant to FINRA Rule 5110(e)(1), these securities will not be sold during the offering, or sold, transferred, assigned, pledged, or hypothecated, or be the subject of any hedging, short sale, derivative, put or call transaction that would result in the economic disposition of the securities by any person for a period of 180 days immediately following the effective date of the registration statement, except to any underwriter and selected dealer participating in the Public Offering and their bona fide officers or partners, provided that all securities so transferred remain subject to the lockup restriction above for the remainder of the time period.

2

5. In the event of the liquidation of the Trust Account, the Sponsor (which for purposes of clarification shall not extend to any officer, member or manager of the Sponsor) agrees to indemnify and hold harmless the Company against any and all loss, liability, claim, damage and expense whatsoever (including, but not limited to, any and all legal or other expenses reasonably incurred in investigating, preparing or defending against any litigation, whether pending or threatened, or any claim whatsoever) to which the Company may become subject as a result of any claim by (i) any third party (other than the Company’s independent public accountants) for services rendered or products sold to the Company or (ii) a prospective target business with which the Company has entered into a letter of intent, confidentiality or other similar agreement or business combination agreement (a “Target”); provided, however, that such indemnification of the Company by the Sponsor shall apply only to the extent necessary to ensure that such claims by a third party for services rendered (other than the Company’s independent public accountants) or products sold to the Company or a Target do not reduce the amount of funds in the Trust Account to below (A) $10.00 per Offering Share or (B) such lesser amount per Offering Share held in the Trust Account as of the date of the liquidation of the Trust Account due to reductions in the value of the trust assets, in each case including interest earned on the funds held in the Trust Account and net of taxes payable, except as to any claims by a third party or Target that executed an agreement waiving claims against and all rights to seek access to the Trust Account whether or not such agreement is enforceable. In the event that any such executed waiver is deemed to be unenforceable against such third party, the Sponsor shall not be responsible for any liability as a result of any such third-party claims. Notwithstanding any of the foregoing, such indemnification of the Company by the Sponsor shall not apply as to any claims under the Company’s obligation to indemnify the Underwriters against certain liabilities, including liabilities under the Securities Act of 1933, as amended (the “Securities Act”). The Sponsor shall have the right to defend against any such claim with counsel of its choice reasonably satisfactory to the Company if, within fifteen (15) days following written receipt of notice of the claim to the Sponsor, the Sponsor notifies the Company in writing that it shall undertake such defense.

6. To the extent that the Underwriters do not exercise their over-allotment option to purchase an additional 2,625,000 Units (as described in the Prospectus), the Sponsor agrees, upon the expiration or waiver of such option, to forfeit and surrender for no consideration for cancellation, a number of Founder Shares equal to the product of 922,613 multiplied by a fraction, (i) the numerator of which is 2,625,000 minus the number of Units purchased by the Underwriters upon the exercise of their over-allotment option, and (ii) the denominator of which is 2,625,000. The forfeiture and surrender will be adjusted to the extent that the over-allotment option is not exercised in full by the Underwriters so that the Founder Shares will represent approximately 26% of the Company’s issued and outstanding Ordinary Shares after the Public Offering (not including the Private Placement Shares). The Sponsor further agrees that to the extent that the size of the Public Offering is increased or decreased and the Sponsor has either purchased or sold Ordinary Shares or an adjustment to the number of Founder Shares has been effected by way of a share dividend or share capitalization, or a surrender for no consideration or share contribution back to capital, or otherwise, in each case in connection with such increase or decrease in the size of the Public Offering, then (A) the references to 2,625,000 in the numerator and denominator of the formula in the first sentence of this paragraph 6 shall be changed to a number equal to 15% of the number of Class A Ordinary Shares included in the Units issued in the Public Offering and (B) the reference to 922,163 in the formula set forth in the first sentence of this paragraph 6 shall be adjusted to such number of Founder Shares that the Sponsor would have to collectively return to the Company in order for all holders of Founder Shares to hold an aggregate of approximately 26% of the Company’s issued and outstanding Ordinary Shares after the Public Offering (not including the Private Placement Shares).

3

7. The Sponsor and each Insider hereby agrees and acknowledges that: (i) each of the Underwriters and the Company would be irreparably injured in the event of a breach by the Sponsor of its obligations (as applicable) under paragraphs 1, 2, 4, 5, 6, 8(a) and 8(b) or by each Insider of its obligations under paragraphs 1, 2, 4, 8(a) and 8(b), (ii) monetary damages may not be an adequate remedy for such breach and (iii) the non-breaching party shall be entitled to injunctive relief, in addition to any other remedy that such party may have in law or in equity, in the event of such breach.

8. Transfer Restrictions.

(a) Subject to the exceptions set forth herein, the Sponsor and each Insider agree not to Transfer, directly or indirectly, any Founder Shares or the Class A Ordinary Shares issuable upon conversion of the Founder Shares held by it, him or her until the earlier of (i) six months after the completion of a Business Combination or earlier if, subsequent to a Business Combination, the closing price of the Class A Ordinary Shares equals or exceeds $12.00 per share (as adjusted for share sub-divisions, share consolidations, share capitalizations, reorganizations, recapitalizations and the like) for any 20 trading days within any 30-trading day period after the Business Combination and (ii) subsequent to a Business Combination, the date on which the Company consummates a subsequent liquidation, merger, share exchange or other similar transaction which results in all of the Company’s shareholders having the right to exchange their Class A Ordinary Shares for cash, securities or other property (the “Lock-up”).

(b) Subject to the exceptions set forth herein, the Sponsor and each Insider agree not to Transfer, directly or indirectly, any Private Placement Units (including the underlying Private Placement Shares and Private Placement Rights) held by it, he or she until thirty (30) days after the completion of a Business Combination.

(c) Notwithstanding the provisions set forth in paragraphs 8(a) and 8(b), transfers of the Founder Shares (including the Class A Ordinary Shares issued or issuable upon the conversion of the Founder Shares) and Private Placement Units (including the underlying Private Placement Shares and Private Placement Rights) that are held by the Sponsor, any Insider or any of their permitted transferees, as applicable (that have complied with any applicable requirements of this paragraph 8(c)), are permitted (i) to the Company’s or the Underwriters’ respective officers, directors, advisors or consultants, any affiliate or family member of any of the Company’s or the Underwriters’ respective officers, directors, advisors or consultants, any members or partners of the Sponsor or their affiliates and funds and accounts advised by such members or partners, any affiliates of the Sponsor, or any employees of such affiliates, (ii) in the case of an individual, as a gift to such person’s immediate family or to a trust, the beneficiary of which is a member of such person’s immediate family, an affiliate of such person or to a charitable organization; (iii) in the case of an individual, by virtue of laws of descent and distribution upon death of such person; (iv) in the case of an individual, pursuant to a qualified domestic relations order; (v) by private sales or transfers made in connection with any forward purchase agreement or similar arrangement, in connection with an extension of the Completion Window or in connection with the consummation of a Business Combination at prices no greater than the price at which the shares or units were originally purchased; (vi) pro rata distributions from the Sponsor or the Representative to its respective members, partners or shareholders pursuant to the Sponsor’s or the Underwriters’ respective limited liability company agreement or other charter documents; (vii) by virtue of the laws of the Cayman Islands or the Sponsor’s limited liability company agreement upon dissolution of the Sponsor or upon dissolution of the Underwriter, (viii) in the event of the Company’s liquidation prior to consummation of a Business Combination; (ix) in the event that, subsequent to the consummation of a Business Combination, the Company completes a liquidation, merger, share exchange or other similar transaction which results in all of its shareholders having the right to exchange their Class A ordinary shares for cash, securities or other property or (x) to a nominee or custodian of a person or entity to whom a transfer would be permissible under clauses (i) through (vii); provided, however, that, in the case of clauses (i) through (vii), these permitted transferees must enter into a written agreement agreeing to be bound by these transfer restrictions herein and the other restrictions contained in this Agreement (including provisions relating to voting, the Trust Account and liquidating distributions).

9. Each Insider’s biographical information furnished to the Company and the Representative that is included in the Prospectus is true and accurate in all respects and does not omit any material information with respect to such Insider’s background and contains all of the information required to be disclosed pursuant to Item 401 of Regulation S-K, promulgated under the Securities Act. Each Insider’s questionnaire furnished to the Company and the Representative including any such information that is included in the Prospectus is true and accurate in all respects. Each Insider represents and warrants that: (i) such Insider is not subject to or a respondent in any legal action for, any injunction, cease-and-desist order or order or stipulation to desist or refrain from any act or practice relating to the offering of securities in any jurisdiction; (ii) such Insider has never been convicted of, or pleaded guilty to, any crime (A) involving fraud, (B) relating to any financial transaction or handling of funds of another person or (C) pertaining to any dealings in any securities and such Insider is not currently a defendant in any such criminal proceeding; and (iii) none of the Sponsor or any such Insider has ever been suspended or expelled from membership in any securities or commodities exchange or association or had a securities or commodities license or registration denied, suspended or revoked.

4

10. The Sponsor and each Insider has full right and power, without violating any agreement to which it, he or she is bound (including, without limitation, any non-competition or non-solicitation agreement with any employer or former employer), to enter into this Letter Agreement and, as applicable, to serve as an officer of the Company or as a director on the board of directors of the Company and each Insider hereby consents to being named in the Prospectus as an officer and/or director of the Company, as applicable.  

11. As used herein, (i) “Business Combination” shall mean a merger, amalgamation, share exchange, asset acquisition, share purchase, reorganization or similar business combination, involving the Company and one or more businesses or entities; (ii) “Founder Shares” shall mean the Class B Ordinary Shares held by the Sponsor prior to the consummation of the Public Offering; (iii) “Private Placement Units” shall mean the aggregate of 539,750 private placement units (or up to 592,250 private placement units if the underwriters’ over-allotment option is exercised in full) that BTIG, LLC and ▇▇▇▇▇▇▇ & ▇▇▇▇, Inc. and Sponsor have agreed to purchase for an aggregate purchase price of $5,397,500 (or up to $5,922,500 if the underwriters’ over-allotment option is exercised in full), or $10.00 per unit, in a private placement that shall occur simultaneously with the consummation of the Public Offering. Each Private Placement Unit consists of one Class A Ordinary Share (the “Private Placement Share”) and one Share Right (the “Private Placement Rights”) to receive one-tenth (1/10) of one Class A Ordinary Share upon the consummation of the Company’s Business Combination; (iv) “Public Shareholders” shall mean the holders of Offering Shares other than the Sponsor and the Insiders; (v) “Trust Account” shall mean the trust fund into which a portion of the net proceeds of the Public Offering and the sale of the Private Placement Units shall be deposited; and (vi) “Transfer” shall mean the (a) sale of, offer to sell, contract or agreement to sell, hypothecate, pledge, grant of any option to purchase or otherwise dispose of or agreement to dispose of, directly or indirectly, or establishment or increase of a put equivalent position or liquidation with respect to or decrease of a call equivalent position within the meaning of Section 16 of the Exchange Act, and the rules and regulations of the Commission promulgated thereunder with any respect to, any security, (b) entry into any swap or other arrangement that transfers to another, in whole or in part, any of the economic consequences of ownership of any security, whether any such transaction is to be settled by delivery of such securities, in cash or otherwise, or (c) public announcement of any intention to effect any transaction specified in clause (a) or (b).

12. This Letter Agreement constitutes the entire agreement and understanding of the parties hereto in respect of the subject matter hereof and supersedes all prior understandings, agreements or representations by or among the parties hereto, written or oral, to the extent they relate in any way to the subject matter hereof or the transactions contemplated hereby. This Letter Agreement may not be changed, amended, modified or waived (other than to correct a typographical error) as to any particular provision, except by a written instrument executed by all parties hereto. Each of the parties hereto hereby acknowledges and agrees that each Representative is a third-party beneficiary of this Letter Agreement.

13. No party hereto may assign either this Letter Agreement or any of its rights, interests or obligations hereunder without the prior written consent of the other parties. Any purported assignment in violation of this paragraph 13 shall be void and ineffectual and shall not operate to transfer or assign any interest or title to the purported assignee. This Letter Agreement shall be binding on the Sponsor, each Insider and each of their respective successors, heirs and assigns and permitted transferees.

14. This Letter Agreement shall be governed by and construed and enforced in accordance with the laws of the State of New York, without giving effect to conflicts of law principles that would result in the application of the substantive laws of another jurisdiction. The parties hereto (i) all agree that any action, proceeding, claim or dispute arising out of, or relating in any way to, this Letter Agreement shall be brought and enforced in the courts of the State of New York located in the City and County of New York, Borough of Manhattan, and irrevocably submit to such jurisdiction and venue, which jurisdiction and venue shall be exclusive and (ii) waive any objection to such exclusive jurisdiction and venue or that such courts represent an inconvenient forum.

15. Any notice, consent or request to be given in connection with any of the terms or provisions of this Letter Agreement shall be in writing and shall be sent by express mail or similar private courier service, by certified mail (return receipt requested), by hand delivery or facsimile transmission.

5

16. This Letter Agreement shall terminate on the earlier of (i) the expiration of the Lock-up or (ii) the liquidation of the Company; provided, however, that this Letter Agreement shall earlier terminate in the event that the Public Offering is not consummated and closed by August 31, 2025; provided, further, that paragraph 5 of this Letter Agreement shall survive such liquidation.

Sincerely,
BLUE ACQUISITION SPONSOR LLC By: Blue Holdings Management LLC
By:	/s/ ▇▇▇▇▇ ▇▇▇▇
Name:	▇▇▇▇▇ ▇▇▇▇
Title:	Managing Member

/s/ ▇▇▇▇▇ ▇▇▇▇

▇▇▇▇▇ ▇▇▇▇

/s/ ▇▇▇▇▇ ▇▇▇▇▇

▇▇▇▇▇ ▇▇▇▇▇

/s/ ▇▇▇▇▇▇ ▇▇▇▇▇

▇▇▇▇▇▇ ▇▇▇▇▇

/s/ ▇▇▇▇▇ ▇▇▇▇ ▇▇▇▇▇▇▇

▇▇▇▇▇ ▇▇▇▇ ▇▇▇▇▇▇▇

/s/ ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇

▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇

/s/ ▇▇▇▇▇ ▇▇▇▇▇▇▇

▇▇▇▇▇ ▇▇▇▇▇▇▇

/s/ ▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇

▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇

[SIGNATURE PAGE TO LETTER AGREEMENT]

6

Acknowledged and Agreed:

BLUE ACQUISITION CORP.

By:	/s/ ▇▇▇▇▇ ▇▇▇▇
Name:	▇▇▇▇▇ ▇▇▇▇
Title:	Chief Executive Officer

[SIGNATURE PAGE TO LETTER AGREEMENT]

7